issn 0022-1120 · 2016. 12. 6. · issn 0022-1120 25 december 2016 volume 809 volume 809 25 dec....

ISSN 0022-1120 25 December 2016

VOLUME 809

VOLUME

809

25 Dec2016

80925 December 2016

1 Axial interaction of a vortex ring with acylinderD Das A Manghnani M Bansal ampP Sohoni

31 Laminarisation of flow at low Reynoldsnumber due to streamwise body forceS He K He amp M Seddighi

72 The effect of Stokes number on particlevelocity and concentration distributionsin a well-characterised turbulentco-flowing two-phase jetT C W Lau amp G J Nathan

111 The onset of vortex-induced vibrations of aflexible cylinder at large inclination angleR Bourguet amp M S Triantafyllou

135 Three-dimensional vorticity momentum andheat transport in a turbulent cylinder wakeJ G Chen Y Zhou T M Zhou ampR A Antonia

S 168 Segregation-induced finger formation ingranular free-surface flowsJ L Baker C G Johnson ampJ M N T Gray

213 Compressible integral representation ofrotational and axisymmetric rocket flowM Akiki amp J Majdalani

240 Physical interpretation of probability densityfunctions of bubble-induced agitationF Risso

264 A multiscale method to calculate filterblockageM P Dalwadi M Bruna amp I M Griffiths

290 A statistical state dynamics-based study ofthe structure and mechanism of large-scalemotions in plane Poiseuille flowB F Farrell P J Ioannou J JimeacutenezN C Constantinou A Lozano-Duraacuten ampM-A Nikolaidis

316 On the generation of large-scale eddy-drivenpatterns the average eddy modelT Radko

345 Lift force on spherical nanoparticles in shearflows of rarefied binary gas mixturesS Luo J Wang G Xia amp Z Li

360 A poroelastic fluidndashstructure interaction modelof syringomyeliaM Heil amp C D Bertram

390 Turbulent thermal convection driven by heatedinertial particlesR Zamansky F Coletti M Massot ampA Mani

438 Nonlinear coupling of interfacial instabilitieswith resonant wave interactions in horizontaltwo-fluid plane CouettendashPoiseuille flowsnumerical and physical observationsB K Campbell K Hendrickson amp Y Liu

480 Deformation and orientation statistics ofneutrally buoyant sub-Kolmogorov ellipsoidaldroplets in turbulent TaylorndashCouette flowV Spandan D Lohse amp R Verzicco

502 Variational treatment of inertiandashgravity wavesinteracting with a quasi-geostrophic mean flowR Salmon

530 Stability and dynamics of two-dimensionalfully nonlinear gravityndashcapillary solitarywaves in deep waterZ Wang

Contents continued on inside back cover

J Fluid Mech (2016) vol 809 pp 168ndash212 ccopy Cambridge University Press 2016This is an Open Access article distributed under the terms of the Creative Commons Attributionlicence (httpcreativecommonsorglicensesby40) which permits unrestricted re-use distribution andreproduction in any medium provided the original work is properly citeddoi101017jfm2016673

168

Segregation-induced finger formation ingranular free-surface flows

J L Baker1dagger C G Johnson1 and J M N T Gray1dagger1School of Mathematics and Manchester Centre for Nonlinear Dynamics University of Manchester

Oxford Road Manchester M13 9PL UK

(Received 19 April 2016 revised 30 September 2016 accepted 12 October 2016)

Geophysical granular flows such as landslides pyroclastic flows and snow avalanchesconsist of particles with varying surface roughnesses or shapes that have a tendency tosegregate during flow due to size differences Such segregation leads to the formationof regions with different frictional properties which in turn can feed back on the bulkflow This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects The full segregation equation for dense granular flows isintegrated through the avalanche thickness by assuming inversely graded layers withlarge particles above fines and a Bagnold shear profile The resulting large particletransport equation is then coupled to depth-averaged equations for conservation ofmass and momentum with the feedback arising through a basal friction law that iscomposition dependent implying greater friction where there are more large particlesThe new system of equations includes viscous terms in the momentum balance whichare derived from the micro(I)-rheology for dense granular flows and represent a singularperturbation to previous models Linear stability calculations of the steady uniformbase state demonstrate the significance of these higher-order terms which ensurethat unlike the inviscid equations the growth rates remain bounded everywhere Thenew system is therefore mathematically well posed Two-dimensional simulations ofbidisperse material propagating down an inclined plane show the development of anunstable large-rich flow front which subsequently breaks into a series of finger-likestructures each bounded by coarse-grained lateral levees The key properties of thefingers are independent of the grid resolution and are controlled by the physicalviscosity This process of segregation-induced finger formation is observed inlaboratory experiments and numerical computations are in qualitative agreement

Key words fingering instability geophysical and geological flows granular media

1 IntroductionThe process of particle size segregation whereby mixtures of different sized

particles separate into distinct grain-size classes during flow can be very pronounced

dagger Email addresses for correspondence jamesbakeralumnimanchesteracuknicograymanchesteracuk

httpwwwcambridgeorgcoreterms httpdxdoiorg101017jfm2016673Downloaded from httpwwwcambridgeorgcore IP address 10915911065 on 12 Nov 2016 at 145245 subject to the Cambridge Core terms of use available at

Segregation-induced finger formation in granular free-surface flows 169

with experiments showing rapid vertical segregation into regions of nearly pure smalland large particles (Savage amp Lun 1988 Vallance amp Savage 2000 Golick amp Daniels2009) When this is combined with periodic deposition it can lead to the formationof striking alternating stratified layers (Gray amp Hutter 1997 Makse et al 1997 Grayamp Ancey 2009) in heaps as well as petal-like patterns in rotating drums (Hill et al1999 Gray amp Chugunov 2006 Zuriguel et al 2006) For dense granular flows thedominant physical mechanisms driving segregation are thought to be kinetic sievingand squeeze expulsion (Middleton 1970 Savage amp Lun 1988 van der Vaart et al2015) As a polydisperse material is sheared smaller particles are more likely to beable to percolate down through cavities that open up which in turn exerts an upwardforce on the larger particles Several models have been proposed to capture this effect(eg Bridgwater Foo amp Stephens 1985 Savage amp Lun 1988 Dolgunin amp Ukolov1995 Gray amp Thornton 2005 Gray amp Chugunov 2006 Gray amp Ancey 2011 MarksRognon amp Einav 2012 Gray amp Ancey 2015) which all have a similar structure anddescribe the evolving particle size distribution for a given bulk flow A recent reviewcan be found in Gray Gajjar amp Kokelaar (2015)

Field studies (eg Pierson 1986 Iverson 2003 Lube et al 2007) have providedstrong evidence for the occurrence of particle size segregation in geophysical flowsIn particular debris flow deposits show self-organisation into leveed channels withlarge particles being vertically segregated to the free surface sheared to the flowfront and then shouldered aside into coarse-grained static regions (Feacutelix amp Thomas2004 Johnson et al 2012) The finer material forms a lining on the inside wallof these lateral levees (Kokelaar et al 2014) which reduces the friction in thechannel and enhances the mobility of the mixed interior Experiments at the UnitedStates Geological Survey (USGS) debris flow flume in Oregon USA (Johnsonet al 2012) as well as smaller-scale laboratory investigations (Deboeuf et al 2006Goujon Dalloz-Dubrujeaud amp Thomas 2007 Kokelaar et al 2014) have been able toreproduce these feedback effects with runout distances for a bidisperse material beinggreater than for either type of particle in pure phase A related phenomenon is theformation of segregation-induced fingering instabilities in granular free-surface flows(Pouliquen Delour amp Savage 1997 Pouliquen amp Vallance 1999 Aranson Malloggiamp Clement 2006 Malloggi et al 2006 Woodhouse et al 2012) These studies canbe motivated by field observations of geophysical flows advancing as a series oflobate structures for example the pyroclastic currents following the Mount St Helenseruption in July 1980 (figure 1)

Experiments are carried out using a bidisperse mixture of spherical ballotini (white75ndash150 microm diameter) and angular carborundum (brown 305ndash355 microm) flowingdown a plane inclined at 27 which is roughened by attaching a single layer ofturquoise ballotini (750ndash1000 microm) to the base with double-sided tape (figure 2 andsupplementary movie 1 available at httpsdoiorg101017jfm2016673) Initiallywell-mixed material is released from rest using a double gate system with an inflowheight of 2 mm As it flows down the slope the large particles are segregated tothe surface and preferentially sheared to the front This front becomes unstable dueto greater frictional forces and splits into a number of different channels or fingerswith the internal structure of each finger resembling that of a single leveed channel

The time scales associated with this instability are relatively short with the earlytraces of fingers beginning to appear after approximately 1 s However fingers onlydevelop after the segregation of large particles to the free surface and subsequentaccumulation at the front and hence the fingering time scale must necessarily beslower than that of particle size segregation There have been several attempts to


170 J L Baker C G Johnson and J M N T Gray

Fine more mobile interior

Coarse-rich flow head

~10 m

Lateral levees channelise the flowand enhance run-out distance

FIGURE 1 Pyroclastic flow deposits from the eruption of Mount St Helens on July 22nd1980 showing evidence of particle size segregation and finger formation during runout(Photo courtesy Dan Miller and USGS)

calculate this segregation rate for example in large-scale experimental debris flowswhere Johnson et al (2012) found large particles rising at approximately 35 cm sminus1or 1 of the typical bulk downslope velocity In laboratory experiments of dryglass beads Wiederseiner et al (2011) measured percolation rates of 15 mm sminus1compared to average bulk velocities of 30 mm sminus1 This ratio is consistent withthe discrete element model (DEM) simulations used by Staron amp Phillips (2014) tocalculate segregation time scales Such segregation rates suggest that these thin flows(less than 2 mm or approximately 10 particle diameters) rapidly segregate before theonset of the fingering instability

There is an important distinction to be made between two different finger formationregimes that occur for different inflow conditions For the experiments shown infigure 2 and movie 1 large quantities of granular material are loaded into the hoppermeaning grains are supplied at a constant flux for the entire observed duration Theresulting fingers are bounded by coarse-rich levees and also have regions of purecarborundum at the rear of each channel wall which are eroded by oncoming materialfrom the inflow This erosion process is particularly apparent in movie 1 where it canbe seen that the lsquolarge particle islandsrsquo creep downslope in a series of discrete surgesThese islands move more slowly than the flow front leading to finger elongationalthough levees of adjacent fingers typically remain in contact Figure 3(a) showsa close up of the experimental frontal zone and a schematic of this behaviour isgiven by figure 4(a) The continuous inflow regime is representative of early fingeringinstability experiments (Pouliquen et al 1997 Pouliquen amp Vallance 1999) whereaslater work (Woodhouse et al 2012 Gray et al 2015) used only a finite amount ofmaterial in the hopper In this case the initial onset of finger formation is identicalbut as the inflow stops and the supply wanes erosion of the large particle islandsceases These stationary regions then act as a barrier between adjacent channelspreventing contact and allowing distinct separated fingers to form as the remainder ofthe material lengthens the pre-existing fingers (figures 3b and 4b) This final phasecan lead to unexpectedly long run-out distances such as in figure 1 before eventually



~5 cm

FIGURE 2 Experiments on a plane inclined at 27 using 80 ballotini (white75ndash150 microm) 20 carborundum (brown 305ndash355 microm) released from rest through adouble gate system of inflow thickness 2 mm The chute is roughened with turquoiseballotini (750ndash1000 microm) Images show snapshots at approximate times t= 09 s t= 26 st= 41 s t= 60 s and t= 79 s Supplementary movie 1 available online

coming to rest and revealing the lubricating fine-grained levee lining (Kokelaar et al2014)

Continuously supplied experiments are also conducted using a monodisperse flowof small ballotini (figure 5 and supplementary movie 2) There are some smallirregularities as the front advances most likely due to imperfections of the inflowlayer and on the channel bed as well as the formation of roll waves but the samefinger structures do not form and propagation is approximately uniform across theslope This is consistent with the work of Pouliquen (1999b) who showed that amonodisperse granular front flows with a constant velocity and well-defined shapeon a rough inclined plane The process of finger formation is therefore driven byparticle size segregation Note that experiments using pure large particles do not flow



~5 cm

~5 cm

(a)

(b)

FIGURE 3 Close ups of the experimental flow fronts for (a) a continuous supply ofparticles from the inflow gate and (b) a finite release of granular material where thesupply has already been cutoff In both cases a bidisperse mixture of 80 white ballotini(75ndash150 microm) 20 brown carborundum (305ndash355 microm) is used and the inflow thicknessis 2 mm

at this slope inclination of 27 because the angular carborundum in pure phase istoo resistive This highlights another key component of the instability mechanismwhich requires the larger particles to have a higher effective friction coefficient thanthe smaller ones In natural flows the interstitial pore pressure is dissipated morerapidly through large particles meaning that large-particle-rich regions experiencegreater frictional forces even if the particles themselves are not more angular likein the experiments shown here (Iverson 1997 Johnson et al 2012) The equivalentexperiments have also been carried out using a bidisperse mixture of different sizedspheres and a frontal instability does still form although the resulting fingers have



Continuous inflow Finite inflow

Erodingmaterial

Contact betweenadjacent levees

Staticmaterial

Grain-free regions

(a) (b)

FIGURE 4 Schematic illustrating the difference between the initial onset of fingerformation and fully developed fingers (a) A continuous supply of material from the inflowgate causes the large particles at the back of the levees to be slowly eroded and movedownstream The front of the fingers propagates faster meaning they lengthen over timeand adjacent fingers remain in contact with each other (b) When the inflow is cutoff theregions at the rear of the levees come to rest and all remaining material flows down thepre-established channels This leads to elongated distinct fingers with grain-free zones inbetween which will eventually arrest as the flow wanes In both diagrams shaded regionscorrespond to coarse-rich areas and dotted lines denote extent of the fingers at an earliertime

weaker less stable levee walls In this case the geometrical properties of the twospherical species are the same but the large particles are slightly more resistive dueto their interaction with the bed roughness (Goujon Thomas amp Dalloz-Dubrujeaud2003) On the other hand the fingering instability does not form in experimentsusing rough small particles and smooth large grains where it is found that the largerparticles shear off the top of the fines which are deposited on the chute without theformation of fingers

The above observations suggest that any theoretical model should account for boththe bulk flow and the effect of particle size segregation in particular the relativefrictional differences Pouliquen amp Vallance (1999) proposed a model for thesesegregation-mobility feedback effects in bidisperse granular flows based on theirexperimental work Depth-averaged mass and momentum balance equations werecoupled to the depth-averaged concentration (representing the distribution of largeand small particles) through a basal friction law that was weighted according to theevolving composition However this work did not explicitly model the size-segregationprocess instead prescribing an initial concentration distribution and allowing itto be advected with the bulk flow The work of Gray amp Kokelaar (2010ab) indepth integrating previous three-dimensional segregation equations (eg Gray ampThornton 2005) allowed the development of fully coupled avalanche-segregationmodels This was exploited by Woodhouse et al (2012) where the coupling was



~5 cm

FIGURE 5 Experiments on a plane inclined at 27 using monodisperse granular materialconsisting of 100 ballotini (75ndash150 microm) released from rest through a double gatesystem of inflow thickness 2 mm Images show snapshots at approximate times t =04 s t = 17 s t = 30 s t = 43 s and t = 57 s Note the time scales are shorterthan the equivalent bidisperse experiments (figure 2) as pure small particles travel fasterSupplementary movie 2 available online

achieved through a concentration-dependent version of Pouliquenrsquos (1999a) frictionlaw This model was able to capture the qualitative features of spontaneous leveedfinger formation but the authors showed that at a critical concentration the equationswere mathematically ill posed in the sense of Joseph amp Saut (1990) ie a linearstability analysis produced unbounded growth rates in the high wavenumber limitThe critical Froude number at which this occurred corresponded to where one ofthe characteristics of the shallow water equations coincided with that from the largeparticle transport equation (Gray amp Kokelaar 2010ab) and the system loses stricthyperbolicity Consequently at a specific concentration any numerical grid-scale noisegrows unboundedly as the grid size tends to zero and the ill posedness manifests



itself in the form of grid-dependent simulations with the number of fingers beinggoverned by the numerical viscosity

The Woodhouse et al (2012) model suggests that additional physics is required toregularise the depth-averaged governing equations Gray amp Edwards (2014) recentlydevised a strategy to achieve this using the micro(I)-rheology for dense granular flows(GDR MiDi 2004 da Cruz et al 2005 Jop Forterre amp Pouliquen 2005 2006)To leading order they showed that this three-dimensional constitutive law onlycontributed via an effective basal friction equivalent to the dynamic friction law forrough beds (Pouliquen 1999a Pouliquen amp Forterre 2002) and the depth-averagedequations reduce to a standard hyperbolic avalanche model (eg Gray Tai amp Noelle2003) Using the steady uniform Bagnold velocity and lithostatic pressure profiles(GDR MiDi 2004) they were able to include the gradient of the depth-averagedin-plane deviatoric stress into the downstream momentum balance These higher-orderviscous terms represent a singular perturbation to the system and in many situationsthey can be neglected However strong evidence for their inclusion is provided byroll waves where the standard shallow water avalanche equations are unable topredict the cutoff frequency observed in experiments (Forterre amp Pouliquen 2003)With viscous terms the depth-averaged micro(I)-rheology is able to predict this cutofffor a wide range of Froude numbers and slope angles without any fitting parameters(Gray amp Edwards 2014)

In addition Edwards amp Gray (2015) showed that the extra terms play a crucial rolein the formation of steadily propagating erosionndashdeposition waves on erodible bedsBaker Barker amp Gray (2016) recently proposed a two-dimensional extension of theequations to account for lateral variation and applied the model to steady uniformchannel flows The generalised viscous terms give rise to downslope velocities withcross-slope profiles another physical feature not captured by classical shallow-watermodels These very promising results for monodisperse flows suggest that Grayamp Edwardsrsquo (2014) depth-averaged micro(I)-rheology could provide the dissipativemechanism to regularise the depth-averaged segregation-mobility feedback equationsThis paper therefore describes how to generalise their work into a bidisperse set-upand shows that the resulting model is mathematically well posed A two-dimensional(downslope and lateral) extension of the system of equations based on the work ofBaker et al (2016) admits numerical solutions showing the formation of fingeringinstabilities on an inclined plane with the key finger characteristics being independentof the grid resolution and controlled by the newly introduced physical viscosity

2 A depth-averaged model for particle size-segregation

Consider a Cartesian coordinate system Oxz with the x-axis pointing downslopeat an angle ζ to the horizontal and the z-axis being the upward pointing normal(figure 6) A bidisperse mass of granular material is assumed to lie between a freesurface at z= s(x t) and rigid base at z= b(x) so that the flow thickness is h(x t)=s minus b Denoting the volume fraction of small particles as φ isin [0 1] (so that theproportion of large particles is 1minus φ) the evolving concentration distribution can bemodelled by a general segregation-diffusive-remixing equation (eg Bridgwater 1976Savage amp Lun 1988 Dolgunin amp Ukolov 1995 Gray amp Chugunov 2006 Gray ampAncey 2011 Gajjar amp Gray 2014)

partφ

partt+ part

partx(φu)+ part

partz(φw)minus part

partz(Q(φ))= part

partz

(Dpartφ

partz

) (21)



z

x

0

FIGURE 6 A schematic diagram of the coordinate axes Oxz inclined at an angle ζ tothe horizontal so that the x-axis points downslope and the z-axis is the upward pointingnormal The granular material lies between the base z= b(x) and free surface z= s(x t)giving a flow thickness h(x t)= sminus b At z= l(x t) there is an interface separating a layerof pure small particles (φ= 1) of thickness η(x t)= lminus b at the bottom of the flow froma layer of pure large particles (φ = 0) lying on top

where the bulk velocity u has components (u w) in the downslope and normaldirections respectively The first three terms on the left-hand side represent theadvection of the concentration with the bulk flow whereas the fourth term accountsfor vertical segregation The flux function Q(φ)gt 0 satisfies Q(0)=Q(1)= 0 to ensurethe segregation mechanism shuts off in the monodisperse limits Different functionalforms for Q have been proposed including a simple quadratic Q(φ) = qφ(1 minus φ)(Gray amp Thornton 2005) or skewed cubic Q(φ) = qφ(1 minus φ)(1 minus γφ) (Gajjar ampGray 2014 van der Vaart et al 2015) the latter being motivated by experimentalobservations of asymmetric segregation which has also been found from discreteparticle method simulations (Tunuguntla Bokhove amp Thornton 2014) The exactdependence will not be important in this paper The right-hand side of (21) representsdiffusive remixing where the diffusivity D may in general depend on the flowvariables

The segregation equation (21) is subject to kinematic boundary conditions

ubpartbpartxminuswb = 0 at z= b(x) (22)

partspartt+ us

partspartxminusws = 0 at z= s(x t) (23)

where subscripts b and s denote evaluation of the velocity field at the base and freesurface respectively In addition there is no flux of either large or small particlesacross the boundaries ie

Q(φ)+Dpartφ

partz= 0 at z= b(x) and z= s(x t) (24)



Following Gray amp Kokelaar (2010ab) the segregation-diffusive-remixing equation(21) may be integrated through the avalanche thickness using Leibnizrsquo rule(Abramowitz amp Stegun 1970) to interchange the order of differentiation andintegration giving

part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)

are the depth-averaged small particle concentration and small particle flux respectivelyThe kinematic and no-flux boundary conditions (22)ndash(24) ensure that the square-bracketed terms disappear and the depth-integrated segregation equation (25) reducesto

part

partt(hφ)+ part

partx(hφu)= 0 (27)

The model is closed by deriving expressions relating the depth-averaged concentrationflux to the depth-averaged downslope velocity the latter being defined analogously to(26) as

u(x t)= 1h

int s

bu(x z t) dz (28)

Since bidisperse flows have been observed to rapidly segregate into inversely gradedlayers (Gray amp Hutter 1997 Gray amp Ancey 2009) Gray amp Kokelaar (2010ab)suggested using a concentration profile

φ =

0 llt zlt s1 blt zlt l

(29)

representing a layer of pure small particles lying on top of a layer of pure largeparticles where z= l(x t) denotes the height of the separating interface In additionthe bulk velocity is assumed to take the form

u(x z t)= u(x t)f (z) (210)

where z = (z minus b)h is the rescaled vertical coordinate and f is the vertical shearprofile which should be an increasing function to ensure surface velocities aregreater than those at the base and should also satisfyint 1

0f (z) dz= 1 (211)

to be consistent with the definition (28) Gray amp Kokelaar (2010ab) used familiesof linear shear profiles given by

f (z)= fL(z)equiv α + 2(1minus α)z (212)



to derive their depth-averaged segregation equation where the parameter α isin [0 1]controls the relative amount of shear and basal slip These were also employed byJohnson et al (2012) to reconstruct the full velocity field at the USGS flume Whilstsimple linear profiles capture the basic features of the flow a more physically accuratechoice is the Bagnold velocity profile

f (z)= fB(z)equiv 53(1minus (1minus z)32) (213)

which can be derived as the steady uniform solution to the three-dimensionalmicro(I)-rheology for granular flows (eg GDR MiDi 2004 Gray amp Edwards 2014)Substituting the inversely graded concentration (29) and velocity profile (210) intothe flux integral in (26) gives

φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)

which may then be inserted into the depth-integrated segregation equation (27) togive

part

partt(hφ)+ part

partx(hφu)minus part

partx(huG(φ))= 0 (215)

where

G(φ)= φ minusint φ

0f (z) dz (216)

The first two terms in (215) represent advection of the depth-averaged concentrationwith the bulk flow and the third term captures the preferential shearing of the largeparticles to the flow front (the minus sign implies that fines are transported to therear) For this reason it is referred to as the lsquolarge particle transport equationrsquo and is amore general version of that derived by Gray amp Kokelaar (2010ab) and Woodhouseet al (2012) The form of the lsquotransport functionrsquo G depends on the choice of shearprofile with the linear shear profile (212) leading to the quadratic

G(φ)=GL(φ)equiv (1minus α)φ(1minus φ) (217)

as in Gray amp Kokelaar (2010a) and the Bagnold shear profile (213) giving

GB(φ)equiv 23(1minus φ)(1minus (1minus φ)32) (218)

The functions (217) and (218) have similar forms with both satisfying G(0)=G(1)=0 meaning the concentration is simply advected at the same speed as the bulk flowin both of the monodisperse limits The Bagnold transport function (218) is skewedslightly towards smaller concentrations of small particles However the difference isrelatively small (lt7 of the maximum amplitude) and (218) may be approximatedusing a quadratic of the form (217) (figure 7) A value α = 17 is chosen toensure that the total area under the two curves and hence the mean transport rateacross all different concentrations is the same and such a fitted quadratic for Gshall be assumed throughout this paper This makes subsequent computations morestraightforward since the (1 minus φ)32 term in (218) results in complex values ifround-off errors cause φ to be slightly greater than unity Though the linear profile



05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)

FIGURE 7 (a) Plots of the linear (212) and Bagnold (213) shear profiles f (z) (b) Thecorresponding transport functions G(φ) given by (217) and (218) respectively The valueα = 17 is chosen for the linear profiles so that the area under the curves in (b) is thesame

(212) with α = 17 is qualitatively different to the Bagnold shear (213) due to thenon-zero basal slip velocity the remainder of this work does not distinguish betweenthe velocity at different vertical positions meaning this simplification is appropriatewhen dealing with depth-averaged quantities

Note the similar structure of the original segregation equation (21) and the largeparticle transport equation (215) with the vertical segregation in the former beingreplaced by lateral segregation in the latter Also note that it is possible to reformulate(215) in terms of the small particle layer thickness η(x t) = l minus b using the factthat η= hφ or the thickness of the large particle layer κ(x t)= hminus η as describedin Gray amp Kokelaar (2010ab) Here it shall be left in terms of the depth-averagedconcentration of small particles φ because this is more representative of what wouldactually be seen in overhead views of bidisperse experiments

3 Segregation-mobility couplingThe large particle transport equation (215) may be solved for the depth-averaged

concentration φ for a prescribed flow thickness h and bulk velocity u (eg Grayamp Kokelaar 2010ab) In some cases h and u can be inferred from experimentalmeasurements (Johnson et al 2012) but typically they are unknown and need tobe solved for as part of the problem Furthermore it is expected that the evolvingconcentration distribution will feed back on the bulk motion and this couplingshould be built into the model The equations representing conservation of mass andmomentum for the bulk flow are (Gray amp Edwards 2014)

parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12

gh2 cos ζ)= ghS+ part

partx

(νh32 part u

partx

) (32)

where g is the constant of gravitational acceleration The shape factor χ = u2u2 in(32) depends on the form of the velocity profile with depth The Bagnold profile(213) gives a value χ = 54 but it shall be assumed that χ = 1 for simplicity



since non-unity values change the characteristic structure of the inviscid equationsand cause problems near zero-thickness regions (Hogg amp Pritchard 2004) This iscommon across the granular flow literature (Grigorian Eglit amp Iakimov 1967 Savageamp Hutter 1989 Gray Wieland amp Hutter 1999 Pouliquen amp Forterre 2002) eventhough it is formally inconsistent with the sheared velocity profile The source termsS are due to a combination of gravity effective basal friction and changes in basaltopography (eg Gray et al 2003)

S= cos ζ(

tan ζ minusmicrobsgn(u)minus partbpartx

) (33)

where sgn is the sign function and ensures friction always opposes the direction ofmotion The effective basal friction coefficient microb provides a mechanism to incorporatesegregation-mobility feedback effects into the governing equations As noted in sect 1the different species of particle have different frictional properties and for fingers todevelop it is required that the larger particles experience greater resistance to motionThis is accounted for by taking a concentration-weighted sum (eg Pouliquen ampVallance 1999 Woodhouse et al 2012)

microb(h Fr φ)= φmicroSb(h Fr)+ (1minus φ)microL

b(h Fr) (34)

where

microSb(h Fr) lt microL

b(h Fr) (35)

are the basal friction coefficients for smooth small and frictional large particlesrespectively and are written as functions of thickness and Froude number

Fr= |u|radicgh cos ζ

(36)

It is assumed that the friction laws for the individual constituents are given by thedynamic friction law of Pouliquen amp Forterre (2002)

microNb (h Fr)=microN

1 +microN

2 minusmicroN1

(βN h)(LN Fr)+ 1 FrgtβN (37)

where N = SL denotes small or large particles respectively The values microN1 = tan ζN1

and microN2 = tan ζN2 are constants where angles ζN1 and ζN2 correspond to the minimum

and maximum slope angles for which steady uniform flows are observed for amonodisperse material of constituent N The length scales LN and dimensionlessconstants βN are found empirically and may depend on both the granular materialand bed composition These constants are estimated for the laboratory set-up offigures 2ndash5 and are given in table 1 along with the other parameters that are keptconstant in this paper

Strictly speaking the individual basal friction laws (37) only hold providing FrgtβN For slower flows the extended law of Pouliquen amp Forterre (2002) should beimplemented which accounts for arresting and static regions (see eg Johnson amp Gray2011 Edwards amp Gray 2015) For simplicity it shall be assumed that (37) is valideverywhere for both types of particle The implications of this assumption will bediscussed in sectsect 6 and 7



ζ = 270 ζ S1 = 200 ζ S

2 = 300 βS = 0150 LS = 20times 10minus4 mχ = 1 ζ L

1 = 290 ζ L2 = 400 βL = 072 LL = 50times 10minus4 m

TABLE 1 Material parameters that will remain constant throughout this paper

The form of the final viscous term in the momentum equation (32) is motivatedby the work done by Gray amp Edwards (2014) for monodisperse flows who usedthe micro(I)-rheology (GDR MiDi 2004 da Cruz et al 2005 Jop et al 2005 2006)to incorporate more of the specific material properties into the depth-averagedgoverning equations They showed that to leading order the micro(I)-rheology onlycontributes via the basal friction coefficient which is equivalent to (37) The resultingshallow-water-like equations are similar to those that have been successfully used inmany granular flow configurations (Grigorian et al 1967 Pouliquen 1999b Grayet al 2003) Higher-order viscous terms were introduced using the steady-stateBagnold velocity profile and lithostatic pressure distribution to derive an expressionfor the depth-averaged in-plane deviatoric stress which Gray amp Edwards (2014) thenwrote in the same form as in (32) using the relationship between the depth-averagedBagnold velocity and flow thickness In this formulation νh122 may be interpretedas the kinematic viscosity which acts in the depth-integrated momentum balanceequation on the gradient term hpart upartx Gray amp Edwards (2014) were able to writethe controlling coefficient ν = νN explicitly in terms of the friction parameters of themonodisperse material as

νN = 2LNradicg9βN

sin ζradiccos ζ

(microN

2 minus tan ζtan ζ minusmicroN

1

) ζN1 lt ζ lt ζN2 (38)

For the bidisperse flows being considered here it might be sensible to choose

ν = ν(φ)= φνS + (1minus φ)νL (39)

in an analogous manner to (34) where νS and νL are the coefficients for small andlarge particles and are given by (38) However the coefficients νS and νL are onlyvalid for slope angles ζN1 ltζ ltζN2 where steady uniform flows are possible Outsideof this range the coefficient of viscosity is negative and therefore the monodispersedepth-averaged theory is ill posed and must be regularised This reflects the underlyingill posedness of the micro(I)-rheology (Barker et al 2015) In order to get levee andfinger formation the slope angle must be such that large particles in pure phase arebrought to rest whilst small particles and mixtures may still flow ie

ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)

In this range the coefficient of viscosity for large particles is undefined and it is notcurrently clear how to extend (38) to all slope angles Instead of using (38) and (39)a constant bulk value ν gt 0 is imposed in this paper which may now be consideredas a free parameter The effect of changing this constant will be investigated anddiscussed

The large particle transport equation (215) together with the mass and momentumbalances (31) (32) define a fully coupled system for the flow thickness anddepth-averaged velocity and concentration Segregation-mobility feedback effects



are achieved through the effective basal friction in the momentum equation (32)with higher concentrations of large particles resulting in greater friction From themonodisperse expressions it is known that the viscous terms are typically small inmagnitude compared to the standard shallow-water contributions The importanceof these terms should not be underestimated however as they represent a singularperturbation to the inviscid equations (Woodhouse et al 2012) which are ill posedat a critical Froude number It will be shown here that the inclusion of viscosity issufficient to regularise the equations

4 Steady uniform flowsA simple solution to the system of equations (215) (31) and (32) is given by

h= h0 u= u0 φ = φ0 (41aminusc)

for constants h0 gt 0 u0 gt 0 φ0 isin [0 1] This represents a steady fully developedflowing layer Upon substitution into the governing equations conservation of mass(31) and the large particle transport equation (215) are automatically satisfiedAssuming there are no topography gradients the momentum equation (32) reducesto a force balance between gravity and basal friction

tan ζ =microb(h0 F φ0) (42)

where

F= Fr0 = u0radicgh0 cos ζ

(43)

is the steady uniform Froude number Treating h0 and φ0 as known control parametersequation (42) can be solved for F as a function of thickness and concentrationSubstituting the friction law (34) and (37) into the force balance (42) leads to thequadratic equation

AF2 + Bh0F+Ch20 = 0 (44)

where the coefficients are given by

A(φ0)= φ0microS2 + (1minus φ0)micro

L2 minus tan ζ (45)

B(φ0)= φ0(MSmicroS1 +MLmicroS

2)+ (1minus φ0)(MSmicroL2 +MLmicroL

1)minus (MS +ML) tan ζ (46)

C(φ0)= (φ0microS1 + (1minus φ0)micro

L1 minus tan ζ )MSML (47)

with MN = βN LN For a slope angle in the range given by (310) it can be seenthat A(φ0) gt 0 for all φ0 isin [0 1] whereas C(φ0) gt 0 for φ0 lt φ

lowast0 and C(φ0) lt 0 for

φ0 gt φlowast0 where

φlowast0 =microL

1 minus tan ζmicroL

1 minusmicroS1 (48)

Consequently the steady-state Froude number found by taking the positive root of(44)

F= h0

(minusB+radicB2 minus 4AC

2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10

FIGURE 8 Contour plots of the steady uniform Froude number F(h0 φ0) given by (49)The shaded regions represent where φ0 lt φ

lowast0 (given by (48)) meaning there are too many

frictional large particles for steady uniform flow

is only positive providing that φ0 gt φlowast0 meaning steady uniform flow is not possible

if there are too many frictional large particles Figure 8 shows a contour plot of thetwo-parameter family of steady states F(h0 φ0) along with the regions where φ0lt φ

lowast0

In the pure small limit (φ0 = 1) the expression (49) reduces to that given in Gray ampEdwards (2014)

F= F(h0)= MSh0(tan ζ minusmicroS1)

microS2 minus tan ζ

(410)

which can also be derived from the more straightforward force balance tan ζ =microS

b(h0 F) The corresponding steady uniform velocities u0(h0 φ0) may be recoveredfrom the Froude number (49) using the relation (43) As a final point the inclusionof higher-order terms into the momentum balance (32) does not change thesteady-state values derived above allowing direct comparisons to be made withthe inviscid equations in subsequent sections

5 Linear stability analysis51 Non-dimensionalisation

Assume the values h0 and φ0 are chosen such that a steady state h= h0 φ = φ0 u=u0(h0 φ0) gt 0 exists with corresponding Froude number F gt 0 as described in theprevious section It is then convenient to introduce the scalings

h= h0h u= u0 ˆu x= h0x t= h0

u0t (51aminusd)



where the hats denote dimensionless quantities Note that the depth-averagedconcentration φ is already non-dimensional Using these scalings the governingequations may be written as

parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h

(tan ζ minusmicrob(h u φ)

)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part


where the hats have been dropped for brevity and conservation of mass (52) has beenused to simplify the momentum and large particle transport equations (53) and (54)respectively In (53) it has also implicitly been assumed that ugt 0 The basal frictioncoefficient is now written in terms of the non-dimensional variables as

microb(h u φ)= φmicroSb(h u)+ (1minus φ)microL

b(h u) (55)

where the individual friction coefficients for each type of particle are given by

microNb (h u)=microN

1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)

for constants γN = (βN h0)(LN F) The granular Reynolds number in (53) is definedas

R= u0radic

h0

ν (57)

and will typically take large values since the viscous terms are small in magnitudecompared to the standard shallow-water contributions as shown by Gray amp Edwards(2014) The Reynolds number is a function of both steady uniform flow thickness h0and concentration φ0

52 Linearised equations and the characteristic polynomialBy construction there is a one-parameter family of steady-state solutions to (52)ndash(54)given by h= 1 u= 1 φ= φ0 and so the variables are perturbed about this base state

h(x t)= 1+ h1(x t) |h1| 1 (58)u(x t)= 1+ u1(x t) |u1| 1 (59)φ(x t)= φ0 + φ1(x t) |φ1| 1 (510)

The linearised governing equations then become

parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1

partx=minus(h1microh + u1microu + φ1microφ)+ F2

Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)

where the simplified notation

G0 equivG(φ0)= 67 φ0(1minus φ0) Gprime0 equivGprime(φ0)= 6

7(1minus 2φ0) (514ab)

has been introduced to denote the steady-state values of the transport function and itsderivative The basal friction law contributes via terms

microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)

= φ0ΓS + (1minus φ0)Γ

L gt 0 (516)

microφ = partmicrob

partφ

∣∣∣∣(11φ0)

=microSb(1 1)minusmicroL

b(1 1) lt 0 (517)

where the positive constants Γ S Γ L are given by

Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)

for N = L S Now seek normal mode solutions of the form

(h1 u1 φ1)= (HU Φ)eσ teikx (519)

where for temporal stability analysis the wavenumber k is real and σ(k)= σR(k)+iσI(k) is complex Substituting this ansatz into the linearised equations (511)ndash(513)allows the system to be written as an eigenvalue problem

AW = σW (520)

where W = (HU Φ)T and the matrix A is given by

A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2

ikG0 ikG0 ik(Gprime0 minus 1)

(521)

Equation (520) has non-zero solutions for W if and only if |Aminus σ I| = 0 (where I isthe 3times 3 identity matrix) leading to a cubic characteristic polynomial

f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)

where the coefficients f0 f1 and f2 are functions of the wavenumber k and base stateand are given in appendix A Equation (522) can be solved to give three roots σ (1)σ (2) σ (3) with corresponding real parts σ (1)R σ (2)R σ (3)R The growth rate σM is thengiven by the maximum of these values

σM =max (σ (1)R σ(2)R σ

(3)R ) (523)

If σM(k) lt 0 for all values of k then all perturbations decay exponentially in time andthe base state is linearly stable On the other hand if there exists a wavenumber such



3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)

FIGURE 9 Plots of the growth rates σM(k) for both the inviscid and viscous equations(a) When F lt Fc both growth rates are positive for all wavenumbers k meaning thatperturbations grow in time and the base state is unstable In the large wavenumber limitsthe inviscid curves tend to a positive constant (530) (dash-dot lines) whereas the viscousvalues decay to zero according to the asymptotics (534) giving an internal maximum σMaxat finite wavenumber kM (b) For FgtFc the inviscid values tend to the constant (529) askminusrarrinfin In the viscous case there is an internal maximum as well as a cutoff wavenumberkc above which all perturbations are stable The different cases are established by fixingthe depth-averaged concentration φ0= 08 giving a critical Froude number Fc= 194 andvarying the flow thickness h0 The values h0 = 2 mm and h0 = 3 mm give correspondingFroude numbers F= 157 and F= 236 for (a) and (b) respectively The coefficient in theeffective viscosity is set to be ν = 0001 m32 sminus1 but qualitatively similar behaviour isfound for all positive values

that σM(k) gt 0 then this perturbation grows in time and the steady flow is linearlyunstable Example plots of σM(k) are shown in figures 9 and 10 for both the newviscous equations and the inviscid equivalent (ν = 0) This inviscid case correspondsto a one-dimensional version of Woodhouse et alrsquos (2012) model although they usedthe original exponential form of the basal friction law (Pouliquen 1999a) as opposedto (37) In the viscous regime the coefficient ν is set to be 0001 m32 sminus1 for allstability results shown on figures 9 and 10 but qualitatively similar behaviour is foundfor different values of ν gt 0



3 6 90

01

02

InviscidViscous

FIGURE 10 Plots of the growth rates σM(k) at the critical Froude number F = Fc Theinviscid growth rates grow unboundedly σM prop k12 as k minusrarrinfin according to the scaling(531) The viscous values are also unstable for all wavenumbers but σM remains boundedand decays to zero like 1k4 as k minusrarr infin (535) giving an internal maximum σMax =max (σM) at k= kM The parameters used are ν = 0001 m32 sminus1 and φ0 = 08 giving acritical Froude number Fc = 194 at flow thickness h0 = 247 mm

All of the cases considered in figures 9 and 10 show regions of positive growth ratefor small k meaning these base states are unstable to small wavenumber perturbationsHowever there are major differences at larger values of k depending on the Froudenumber and whether the viscous or inviscid equations are being considered Inthe inviscid regime the growth rates remain positive for all wavenumbers and areincreasing functions of k whereas the viscous growth rates take their maximumat a finite value of k and in some cases have a cutoff wavenumber above whichperturbations are stable (figure 9b)

An important result in Woodhouse et al (2012) was the discovery of unboundedgrowth rates σM minusrarrinfin as kminusrarrinfin (figure 10) at a critical Froude number meaningthe inviscid system of equations was ill posed at this specific point in parameterspace (Joseph amp Saut 1990) Although the model was well posed almost everywherenumerical simulations of the fingering instability showed that the width of the fingerswas grid dependent This was because the computations always encompassed a line ofpoints where the depth-averaged concentration was equal to the critical concentrationfor ill posedness and refining the computational domain introduced increasinglyunstable small wavelength perturbations in these regions The critical regime can berelated to the characteristics of the inviscid equations which are given in (x t) spaceby the lines

dxdt= λ (524)

where λ is the characteristic wavespeed For the non-dimensional mass and momentumbalance equations (52) (53) there are two wavespeeds given by

λ(1) = u+ 1F

radich λ(2) = uminus 1

F

radich (525ab)



(see for example Courant amp Hilbert 1962) whereas for the large particle transportequation (54) the wavespeed is

λ(3) = u(1minusGprime(φ)) (526)

Note that it is only possible to separate the two sets of characteristics in thisway because the segregation-mobility coupling arises through the source terms ofthe momentum equation which do not contribute to the characteristic structureEvaluating at the steady state (h u φ)= (1 1 φ0) and noting that Gprime0 lt 1 it can beseen that λ(1) gt 0 and λ(3) gt 0 The other wavespeed λ(2) is positive for F gt 1 andnegative for Flt 1 Most importantly at the critical Froude number

F= Fc equiv 1|Gprime0|

(527)

the large particle transport equationrsquos characteristic wavespeed (526) coincides withone of those from the shallow water equations (525) depending on the sign of Gprime0and the system loses strict hyperbolicity This is the point at which unbounded growthrates are found in the linear stability analysis for the inviscid model It is thereforevital to check that the growth rate for this new viscous model remains bounded for allwavenumbers by conducting an asymptotic analysis of the characteristic polynomial(522) for k 1 The reader is referred to appendix A for the full details but asummary of the key results is given in the following sections

53 Inviscid high wavenumber asymptoticsFor completeness we begin by considering the inviscid case ν = 0 The real part ofthe three roots is found to behave like

σ(1)R sim

(FGprime0 minus 1)(Fmicroh minusmicrou)+ FG0(1minus F)microφ2F2(FGprime0 minus 1)

(528)

σ(2)R sim

minus(FGprime0 + 1)(Fmicroh +microu)+ FG0(1+ F)microφ2F2(FGprime0 + 1)

(529)

σ(3)R sim

G0(1minusGprime0)microφF2(Gprime0)2 minus 1

(530)

in the high wavenumber limit (see sect A1) meaning that the growth rate σM tends toa constant value as k minusrarr infin From the definition (517) and the assumption (35)that large particles experience greater frictional forces it follows that microφ lt 0 Usingthis and the fact that Gprime0 lt 1 by (514) it can be seen that that σ (3)R gt 0 for F lt Fcwhere the critical Froude number is defined by (527) The growth rate σM is thereforealways positive for Flt Fc meaning perturbations grow in time and the base state isunstable as k minusrarrinfin (figure 9a) For F gt Fc the third root σ (3)R is now stable in theasymptotic limit and one must investigate the sign of the other roots σ (1)R and σ

(2)R

For the parameters given in table 1 at least one of (528) or (529) is positive for allregions of (h0 φ0) space meaning that σM gt 0 and the base state is again unstable(figure 9b)

At the critical Froude number F = Fc the third root σ (3)R and either σ (1)R or σ (2)Rbecome infinite depending on the sign of Gprime0 In these singular cases the leading-ordergrowth rate is instead given by the distinguished limit

σ(plusmn)R simplusmn 1

2 |G0Gprime0(1minusGprime0)microφ|12k12 (531)



for large k The positive root in (531) will give unbounded growth rates proportionalto k12 (figure 10) which is the same as the one-dimensional result found inWoodhouse et al (2012) and means that the inviscid model is ill posed in thesense of Joseph amp Saut (1990)

54 Viscous high wavenumber asymptoticsReturning to the viscous equations (finite Rgt 0) the leading-order behaviour of thethree roots are

σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim

R2G0(1minusGprime0)(F2(Gprime0)

2 minus 1)microφF4(Gprime0)2

1k2 (534)

for k 1 (see sect A2) In this high wavenumber limit the first two of these are stablefor all Froude numbers whereas the third root is stable for F gt Fc and unstable forFltFc To see this note that Gprime0 lt 1 microφ lt 0 and F2(Gprime0)

2 may be written as (FFc)2

Whilst the sign is consistent with the inviscid case (530) the third root (534) doesnot tend to a constant value as k minusrarr infin and instead decays to zero like 1k2

(figure 9a) This is significant because it means that the growth rate σM(k) will takeits maximum value σMax = max (σM) at a finite wavenumber k = kM correspondingto the most unstable mode Furthermore for finite values of R gt 0 the base stateis always stable in the high wavenumber limit when F gt Fc This leads to a cutoffwavenumber k= kc and all perturbations with kgt kc are stable (figures 9b and 11b)Neither of these features are present in the inviscid regime where the maximumgrowth rates are given by the asymptotic limits (528) (529) or (530) It is possibleto derive analytical expressions for the cutoff wavenumber kc details of which aregiven in sect A3

At the critical Froude number the expression (534) reduces to zero (since F2(Gprime0)2=

1) and a new asymptotic analysis leads to

σ(3)R sim 2R3G2

0(Gprime0)

2(1minusGprime0)2micro2

φ

1k4 (535)

in the high wavenumber limit The expression (535) is strictly positive meaning theroot is unstable for k 1 Intuitively this result agrees with the inviscid analysis forthe critical regime since these are in some sense the most unstable cases Howeverthe growth rate crucially remains bounded for all values of k when viscous termsare included and in fact decays like 1k4 as kminusrarrinfin (figure 10) This ensures thatthe model is well-posed even in the previously problematic critical regime Theseimportant results are highlighted in figure 11(a) which shows the maximum growthrates σMax as a function of steady uniform Froude number F For the inviscid equationsσMax minusrarrinfin as F minusrarr Fc whereas the viscous curves remain bounded for all Froudenumbers

As already noted the linear stability calculations shown on figures 9 and 10 arecomputed using a fixed coefficient in the effective viscosity ν = 0001 m32 sminus1 andqualitatively similar behaviour is found for all positive choices of ν The specific valuedoes not affect whether the equations are well posed but it does have an influence



2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50

FIGURE 11 (a) Plots of the maximum growth rate σMax against steady-state Froudenumber F The inviscid values are infinite at F=Fc whereas σMax remains bounded for allFroude numbers when viscosity is included (b) The cutoff wavenumber kc as a functionof F which only exists for the viscous regime providing that FgtFc Larger values of thecoefficient ν lead to smaller σMax and kc In both plots the depth-averaged concentrationis fixed at φ0 = 08 and F is varied by changing the steady-state thickness h0

on the stability of steady uniform flows Increasing ν and consequently reducing theReynolds number R leads to lower maximum growth rates σMax (figure 11a) andtherefore has a stabilising effect on the base state The most unstable mode kM andcutoff wavenumber kc (when it exists) decrease with increasing ν (figure 11b)

6 Two-dimensional numerical simulations for a propagating frontHaving eliminated the possibility of unbounded growth rates as a source of ill

posedness the capability of the new governing equations to model physical systemsmay be tested One such system is segregation-induced fingering instabilities thatdevelop as a front of bidisperse material propagates down an inclined plane (seefigures 2 3 movie 1 and Pouliquen et al 1997 Pouliquen amp Vallance 1999Woodhouse et al 2012 for example) In an initially homogeneous mixture largeparticles are rapidly segregated to the surface where the higher velocity shears themto the front Here they are overrun resegregated upwards and recirculated by thebulk to form a coarse-rich flow head This head experiences enhanced frictionalforces and may break down into a series of lsquofinger-likersquo structures Each finger isbounded by static or slowly moving levees of coarse material that channelise thefiner more mobile interior This phenomenon is important because it is an exampleof a segregation-mobility feedback effect with the instability being suppressed inexperiments using monodisperse material (see figure 5) Woodhouse et al (2012) were



able to produce numerical simulations showing spontaneous leveed finger formationand elongation but their results were grid dependent due to the ill posedness discussedin the previous sections

61 Generalised equationsMany granular flow configurations have little variation in the lateral directionallowing them to be modelled by one-dimensional (depth-averaged) equationsgoverning the evolution in time and downslope direction (eg Gray amp Edwards2014 Edwards amp Gray 2015) However the instability that occurs as a bidispersemixture propagates down a plane is predominantly in the transverse direction meaningthat the lateral dependence needs to be explicitly accounted for in the model Thegoverning equations must therefore be extended to two spatial dimensions Bakeret al (2016) examined a similar problem for monodisperse material and generalisedthe one-dimensional depth-averaged micro(I)-rheology of Gray amp Edwards (2014) to twodimensions Using their work as a base and introducing a cross-slope coordinate yand depth-averaged velocity v the two-dimensional segregation-mobility feedbackequations (reverting to dimensional variables) are

parthpartt+nabla middot (hu)= 0 (61)

part

partt(hu)+nabla middot (huotimes u)+nabla

(12

gh2 cos ζ)= ghS+nabla middot (νh32D) (62)

part

partt(hφ)+nabla middot (huφ)minusnabla middot (huG(φ))= 0 (63)

where u= (u v) is the depth-averaged velocity vector nabla = (partpartx partparty) denotes thegradient operator in (x y) space lsquomiddotrsquo is the dot product and otimes the dyadic productAssuming no basal topography the source terms S= (Sx Sy) are given by

Sx = cos ζ(

tan ζ minusmicrobu|u|) Sy =minus cos ζ

(microb

v

|u|) (64ab)

where |u| = (u2 + v2)12 is the magnitude of the velocity The basal friction lawmicrob(h Fr φ) is defined by (34) and (37) using the more general Froude numberdefinition


(65)

The coefficient ν gt 0 remains a free parameter for slope angles in the range (310)and the depth-integrated strain-rate tensor is

D = 12(nablau+ (nablau)T) (66)

The generalised large particle transport equation (63) can be derived in a similarmanner to the one-dimensional case (215) providing assumptions are made about thevertical variation of the transverse velocity Whilst the downslope velocity magnitudesare typically much larger than their lateral counterparts the thinness of the flow meansthat material responds to shear comparably in both directions and thus the shear



profile is taken to be the same (see Woodhouse et al 2012) The transport functionG then remains unchanged and is given by (217) with α = 17

Note that for simplicity the linear stability analysis conducted in sect 5 is for theone-dimensional equations and does not strictly apply to their generalised form(61)ndash(63) One cannot guarantee that these will also be well posed without furthercalculations Woodhouse et al (2012) carried out a full two-dimensional stabilityanalysis for their inviscid equations and found that the most unstable mode wasusually in the downslope direction In particular the unbounded growth rates wereonly apparent in the large kx (downslope wavenumber) limit The equivalent analysishas been carried out for the viscous equations (see appendix B) and unstable modesare only found for non-zero downslope perturbations In all cases the growth ratesremain bounded for all wavenumbers meaning the generalised equations are wellposed

62 Numerical solutionsThe system of coupled partial differential equations (PDEs) (61)ndash(63) is now solvednumerically using the shock-capturing central scheme of Kurganov amp Tadmor (2000)This is second order in space and has a semi-discrete formulation allowing it tobe combined with a time stepper of choice In this case a RungendashKuttandashChebyshevadaptive step method (Medovikov 1998) is employed The scheme is well suited tothis particular problem because it is easily generalised to multiple spatial dimensionsand is capable of handling convectionndashdiffusion equations It has previously been usedto solve similar systems governing granular flows for example a two-dimensionalbreaking size-segregation wave (Johnson et al 2012) granular roll waves (Gray ampEdwards 2014 Razis et al 2014) and erosionndashdeposition waves (Edwards amp Gray2015) To calculate the numerical fluxes the scheme requires the specification of aflux limiter Here the weighted essentially non-oscillatory (WENO) limiter detailedin Noelle (2000) is chosen In order to utilise the numerical method of Kurganov ampTadmor (2000) the governing equations must be written in conservative form Thisintroduces numerical singularities in both the convective and diffusive fluxes as hminusrarr0 To get around this potential problem a minimum flow thickness smaller than oneparticle diameter hmin= 10minus4 m is introduced and the fluxes are set to zero wheneverhlt hmin

For numerical simulations of a front of granular material advancing down aninclined plane initial conditions of an empty slope

h(x y 0)= 0 u(x y 0)= 0 v(x y 0)= 0 φ(x y 0)= 0 (67aminusd)

are prescribed Simulations are carried out on a computational domain Lx times Ly

discretized over Nx timesNy grid cells giving a spatial resolution of NxLx times NyLy

points per metre Periodic boundary conditions are specified in the y direction andat the upstream boundary (x= 0) the inflow conditions

h(0 y t)= (1minus eminus10t)h0 u(0 y t)= u0 v(0 y t)= 0 φ(0 y t)= φ0 + δφp(y)(68aminusd)

are enforced These correspond to the steady uniform flow of sect 4 with the thicknessprefactor (1 minus eminus10t) used to ensure a smooth transition from an empty slope Notethat the simulations represent a continuous inflow (akin to figures 3a 4a) as opposed



to a finite release of material (figures 3b 4b) To introduce transverse variation intothe system the inflow concentration is perturbed with a periodic function

φp(y)= sin (2πyLy) (69)

of magnitude δ = 8 times 10minus5 Simulations are halted before material reaches thedownstream boundary and so since the equations degenerate when h= 0 no boundaryconditions are required at this end of the domain

Figures 12ndash14 and supplementary movie 3 show the resulting contours of theflow height h depth-averaged concentration φ and speed |u| It can be seen that thegoverning equations are able to qualitatively reproduce many of the key phenomenapresent in the experimental set-up The concentration plots (figure 12) show that alarge-particle-rich front quickly develops and subsequently breaks up into a seriesof fingers These are bounded by coarse-grained lateral levees and elongate astime progresses As the fingers emerge from the uniform front the large particlesare shouldered aside into slow moving coarse-particle-rich levees and the largeparticle regions at the finger tips are dramatically reduced in size In practice abreaking size-segregation wave will form just behind the front (Thornton amp Gray2008 Gray amp Ancey 2009 Johnson et al 2012 Gajjar et al 2016) which in thedepth-averaged segregation theory (Gray amp Kokelaar 2010ab) is replaced by adepth-averaged concentration shock The existence of a breaking size-segregationwave in the physical experiments as well as diffusion makes the transition at thelarge-rich front much more diffuse than predicted in the simulations

Note that the coarse-rich front that forms before the onset of finger formation is nota steady travelling wave solution to the system of equations because large particlescontinue to accrue at the flow front (Gray amp Kokelaar 2010a) A steadily travellingbase state must include a mechanism to counteract this accumulation which Gray ampAncey (2009) achieved by depositing the coarser grains arriving at the front onto theunderlying substrate In the absence of basal deposition the large particles may beremoved from the frontal region through lateral advection towards the levees andJohnson et al (2012) showed that this gave rise to a steady travelling segregationprofile on the flow centreline for a specified flow thickness and velocity field We nowbelieve that a single leveed finger can propagate steadily downslope and anticipatethat such a travelling wave solution to equations (61)ndash(63) would be an appropriatebase state although this has not been confirmed This behaviour is in direct contrastto the pure small limit (φ = 1) Assuming no transverse variation the monodisperseequations admit a steady travelling front solution (Gray amp Edwards 2014) which canbe found analytically (Gray amp Ancey 2009) or numerically (Pouliquen 1999b) and isunaffected by the new viscous terms Two-dimensional computations suggest that thisbase state does not break down into fingers which is consistent with the monodisperseexperiments (figure 5) although roll waves may form and travel to the flow front

From figure 13 it can be seen that once the fingers form the material travellingdown the centre of the channels is moving approximately twice as fast as the steadyuniform flow behind This can be explained by examining the coarse-rich laterallevees where the flow approaches zero velocity at the margins between adjacentfingers To account for these reduced flux sections the central regions must increasetheir flux through a combination of flow thickening and increased speed in orderto conserve material from the constant inflow There is an important discrepancy inthe slow moving zones between experiments and numerics In the laboratory someof the material in the levees is completely stationary but this is not achieved at any



05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050

FIGURE 12 Numerical solutions of the system of PDEs (61)ndash(63) showing the depthaveraged concentration φ at times t = 19 s (non-dimensional t = 198) t = 33 s (t =343) t = 51 s (t = 530) t = 62 s (t = 645) and t = 75 s (t = 780) A large-rich regionquickly develops at the flow front (t = 19 s) starts to become unstable (t = 51 s) anddevelops into fingers (t = 62 s) which elongate and coarsen over time It is clear thatlateral levees bounding the fingers consist predominantly of large particles The colourscheme has been chosen to mimic experiments with turquoise representing the regionwhere the chute is empty hlt hmin Parameters are h0= 2 mm φ0= 08 u0= 0208 msminus1F = 157 ν = 001 m32 sminus1 NxLx = NyLy = 750 mminus1 Note that the axes limits Lx =15 m Ly = 05 m correspond to non-dimensional values of Lx = 750 and Ly = 250respectively Supplementary movie 3 available online

point in the simulations Whilst this is not a major problem for the fingers that formfrom a continuous source (figures 3a 4a) as the inflow is cutoff and the flow wanesthe static regions become more significant and lead to distinct fingers separated bygrain-free regions (figures 3b 4b) In order to bring material to rest and prevent



05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050

FIGURE 13 Numerical solutions of the system of PDEs (61)ndash(63) showing the speed|u| = (u2 + v2)12 at times t = 19 s t = 33 s t = 51 s t = 62 s and t = 75 s Oncethe flow breaks up into fingers the lateral levees are close to stationary with areas ofmuch faster flow down the central channelised regions representing the more mobileinterior Parameters are the same as in figure 12 meaning that a dimensional velocity of|u|= 0208 m sminus1 would correspond to a non-dimensional value of | ˆu|= 1 Supplementarymovie 3 available online

merging and coarsening of fingers in the final stages of flow the extended frictionlaw of Pouliquen amp Forterre (2002) needs to be generalised to bidisperse materialand implemented

The flow thickness plots (figure 14) show that the region immediately behindthe flow front is elevated above the steady uniform inflow depth This is consistentwith observations of bulbous flow fronts in both experiments and geophysical events(Johnson et al 2012 Kokelaar et al 2014) There are also regions at the rear ofthe fingers that are significantly higher than the rest of the material Comparing with



05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050

FIGURE 14 Numerical solutions of the system of PDEs (61)ndash(63) showing the flowthickness h Simulations are shown at times t= 19 s t= 33 s t= 51 s t= 62 s andt= 75 s The propagating front breaks up into a series of fingers going to zero thicknessat the boundaries A region of thicker flow follows behind the main front Parametersused are the same as figures 12 and 13 meaning a dimensional thickness of h= 2 mmrepresents h= 1 in non-dimensional terms Supplementary movie 3 available online

figures 12 and 13 these correspond to the large particle islands that are also seen insmall-scale experiments For the continuous inflow regime simulated here they moveslowly downstream which matches the experimental erosion processes (figure 3a)Whilst there are no quantitative comparisons made at this stage typical time scalesfor the onset of finger formation are roughly consistent The channel widths do notcorrespond to the imposed inflow perturbation meaning they are set by the systemitself and are of the correct order of magnitude (around 5 cm for the simulationsshown) Further discussion about the finger characteristics follows in the next section



0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050

FIGURE 15 Numerical simulations of depth-averaged concentration φ at time t=75 s fordifferent grid resolutions and domain widths Ly Black dotted lines denote the maximumand minimum front position as defined by (610) and (611) respectively The final resultsare not identical but the width and downslope extent of the fingers is similar for all runsOther parameters used are the same as in figures 12ndash14

63 Finger characteristics numerical versus physical viscosityAs well as showing that the new equations remove the possibility of unboundedgrowth rates it also needs to be checked that the resulting simulations are gridconvergent Both the experiments and the non-linear equations exhibit high sensitivityto the initial conditions which is highlighted numerically by the apparent randomnature of the resulting fingers Even though these arise through the integrationof deterministic PDEs numerical round-off error subsequently magnified by theunderlying instability of the equations is sufficient to break the symmetry of theinflow (68) and (69) Changing the grid resolution will change these conditions andone can therefore not expect to obtain identical results when running simulationswith different numbers of grid points Attention is instead given to the wavelengthof fingers (cross-slope distance between two adjacent channels) and their elongation(downslope distance between front and rear of leveed region) Woodhouse et al(2012) showed that for the inviscid equations the wavelength got progressivelysmaller and the fingers more elongated with decreasing mesh size suggesting thatnumerical viscosity was a controlling factor for channel characteristics

Computations are carried out for different grid resolutions and domain widths Lyand example results are shown on figure 15 The depth-averaged concentration φis plotted at time t = 75 s As expected the results are not perfectly identical forvarying numbers of grid points with the position and shape of the fingers changingslightly between runs This is actually a desirable property since no two experimentsare identical and so it is good that the numerics exhibit a similar degree of sensitivityHowever like the experiments the numerical finger width and their elongation staysapproximately the same which is in direct contrast to Woodhouse et al (2012) At agiven time it is straightforward to calculate the mean finger wavelength Λ by dividingthe domain width Ly by the number of fingers and this is shown on figure 16(a) forthe different computations There is some variation when too few cells are used butthe wavelength converges at sufficiently high resolutions and is independent of the



1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4

FIGURE 16 The effect of changing the grid resolution on (a) the mean finger wavelengthΛ at time t = 75 s and (b) the front elongation xl (given by (610)ndash(612)) at timest= 25 s t= 50 s and t= 75 s Both tend to roughly constant values for large enoughnumbers of grid points

domain width Note that counting the number of fingers from plots such as figure 15is fairly intuitive since the individual channels are well defined at this time t= 75 sHowever running the simulations for longer can lead to merging events makingprecise definitions more difficult and meaning that Λ will also evolve over time

To estimate the elongation of the fingers a maximum and minimum front positionx+f and xminusf respectively are defined for each time as

x+f (t)=max x h(x y t) lt hmin (610)

xminusf (t)=min x |φ(x y t)minus φ0|gt φ02 (611)

where (611) is chosen to capture the sharp transition between the mixed inflow andpure large region at the back of the fingers The lines where x= x+f (t) and x= xminusf (t)are shown for reference on figure 15 These values can be used to calculate the lengthor elongation of the fingers as

xl(t)= x+f (t)minus xminusf (t) (612)

Figure 16(b) shows the variation of xl with changing numbers of grid points atdifferent times t Again there is slight variation at low resolutions but it eventuallyconverges to roughly the same value at all times This confirms that the numericalviscosity is no longer controlling the finger properties which is major progress inmodelling spontaneous finger formation

The wavelength and downslope extent of the fingers is now being set by theequations themselves in particular the newly introduced physical viscosity All thecomputations thus far were carried out with a coefficient in the effective viscosityν = 001 m32 sminus1 This may be unphysically high but was chosen to ensure that itwould outweigh any numerical diffusion in checking grid convergence Since it isbeing treated as a free parameter in this paper simulations were also conducted withvalues of ν ranging from 0005 to 005 to investigate what effect this has on the



0025 00500

02

04(a) (b)

0025 00500

025

050

FIGURE 17 The effect of changing the viscosity coefficient ν on (a) the mean fingerwavelength Λ at a given time t= 75 s and (b) the front elongation xl The grid resolutionis fixed at NxLx =NyLy = 500 mminus1

25

t (s)

x (m

)

50 750

05

10

15

FIGURE 18 Plots of minimum and maximum front position xminusf and x+f and finger lengthxl as functions of time for different values of ν All travel at approximately constantvelocities until the onset of finger formation which occurs at earlier times for smallerviscosities

physical characteristics of the fingers The results are shown on figure 17 where itcan be seen that increasing ν typically leads to fewer fingers (larger wavelength) thatare less elongated (smaller xl) One interpretation is that higher material viscositiessuppress finger formation This is investigated further on figure 18 where the frontpositions xminusf x+f and finger lengths xl are plotted as functions of time t for differentviscosities During the initial uniform propagation both the minimum and maximumfront positions move at constant velocities with x+f travelling faster than xminusf leadingto a steady growth in xl At the onset of finger formation there is an increasein the speed of x+f as the more mobile interior breaks through the resistive frontInterestingly the maximum extent of the flow then travels faster than the prescribedinflow which could have important implications from a hazards mitigation perspectiveAt the same time there is a corresponding deceleration of xminusf as the slow moving largeparticle islands form and combined with the acceleration of the finger tips this leads



to a sudden increase in the rate of elongation of the frontal region This transitionpoint happens at earlier times for lower values of ν which is consistent with thelinear stability results in sect 5 where higher viscosities lead to slower maximum growthrates It is also consistent with the idea that the frontal break down is closely tied tothe region of steady uniform flow immediately behind

7 Conclusions

Polydispersity in granular flows can significantly alter the overall flow characteristicswith particle size segregation leading to the formation of regions with different sizedgrains If these grains also have different frictional properties this then feeds backon the bulk flow and can produce rich behaviour that is not seen in monodisperseflows such as self-organisation into coarse-grained levees and segregation-inducedfingering instabilities This paper has presented a fully coupled depth-averaged modelfor such segregation-mobility feedback effects The evolving particle distributionis governed by a large particle transport equation which is derived from a fullsegregation equation by assuming perfect vertical inverse grading and a shear profilethrough the avalanche It is shown that using a physically motivated Bagnold profileleads to qualitatively similar transport functions to previous models (Gray amp Kokelaar2010ab Woodhouse et al 2012) providing the correct shear parameter is chosenThis Bagnold assumption is also consistent with that used by Gray amp Edwards(2014) to incorporate higher-order terms into their depth-averaged micro(I)-rheology formonodisperse flows These second-order terms are generalised for the bidisperseregime considered here giving a viscous model with coupling achieved via aconcentration-weighted effective basal friction

Linear stability calculations of the steady uniform base state highlight thesignificance of the higher-order terms which were not present in Woodhouse et al(2012) The growth rates now remain bounded everywhere whereas for the inviscidequations there is a critical Froude number that gives rise to unbounded growthrates in the high wavenumber asymptotic limit This means that the introductionof viscosity regularises the problem and ensures that the governing equations aremathematically well posed Perhaps this is not such a surprising result but theadvantage of the particular viscosity formulation used in this paper is that thestructure of the higher-order terms is physically motivated with the parameterscompletely determined by the micro(I)-rheology and no additional fitting parametersrequired at least in the monodisperse regime

The system is then generalised to two spatial dimensions in an analogous wayto Baker et al (2016) (for monodisperse flows) and numerical simulations of apropagating flow front are presented The equations are able to predict the formationof a coarse-rich flow front that develops instabilities and breaks up into a series offinger-like structures elongating over time The lateral boundaries of these fingersconsist of large particle levees where material is travelling significantly slower thanthe finer-grained interior which itself moves faster than the steady uniform supplyThere is also a noticeable speed-up at the tip of the flow as the fingers first emergeAt the rear of the levees are clusters of coarse grains that migrate slowly downslopeconsistent with the erosion processes observed during continuous inflow experimentsUnlike the physical system there are no positions in the simulation domain wherethe velocities reach precisely zero This is not problematic for the onset of fingerformation considered in this paper but is expected to become more significant whenexamining the final run-out of flows after the supply of particles has been stopped



In this case the static regions are important in preventing lateral levee spreading andforming grain-free regions between distinct fingers To bring material to rest andlock in the structure of the channel walls the extended friction law of Pouliquen ampForterre (2002) appears to be necessary

The ill posedness of the inviscid model (Woodhouse et al 2012) manifesteditself as grid-dependent simulations with increasing numbers of computational cellsleading to larger numbers of fingers that spread over greater downslope distancesThis suggests that the finger characteristics were being determined by numericalviscosity Solutions of the new equations do not suffer from the same problemwith both the finger wavelength and their elongation converging for sufficiently highresolutions The sensitivity of the system means that there is still some variety inthe exact shape and position of the channels but this is not necessarily undesirablesince the laboratory experiments are also highly sensitive to the initial conditionsThe channels in consecutive experimental runs will always form in slightly differentplaces although the characteristic width and time scales will be similar

This paper is only a preliminary investigation into finger formation focusing onestablishing the well posedness and grid convergence of the system as opposed todetailed analysis of finger growth rates and wavelengths Indeed such analysis iscomplicated by a number of factors including the sensitivity described above thetemporal evolution of the base state (which is uniform in y but evolving in x andt) nonlinear coarsening and the interactions between instabilities of the front andinstabilities of the steady uniform flow behind The linear stability analysis of sect 5is presented in one spatial dimension to simplify and aid visualisation and whilstthe two-dimensional version (appendix B) of a uniform base state is important forchecking well posedness in the general case it is difficult to relate this to thenumerical computations of a non-uniform base state

Despite these difficulties the qualitative properties of the fingers in the simulationsare now controlled by the physical viscosity via the effective coefficient ν Increasingthis leads to larger wavelength fingers that do not spread as far in the downslopedirection Higher viscosity flows also take longer to break down into leveed channelswhich is consistent with linear stability calculations of the steady uniform base stateThe value of ν is currently treated as a free parameter since fingers only form at slopeangles where the depth-averaged micro(I)-rheology of Gray amp Edwards (2014) needs tobe regularised Choosing ν so that the numerical simulations match experimental datamay provide insight into precisely how to achieve this regularisation It may also bepossible to calibrate the coefficient of viscosity from other bidisperse experiments forexample using the cutoff frequency for the roll-wave instability

Acknowledgements

This research was supported by NERC grants NEE0032061 and NEK0030111 aswell as EPSRC grants EPI0191891 EPK00428X1 and EPM0224471 JLB wouldalso like to acknowledge support from a NERC doctoral training award JMNTG isa Royal Society Wolfson Research Merit Award holder (WM150058) and an EPSRCEstablished Career Fellow (EPM0224471)

Supplementary movies

Supplementary movies are available at httpsdoiorg101017jfm2016673



Appendix A Details of one-dimensional linear stability analysisThis appendix provides derivations of the asymptotic results presented in sectsect 53

and 54 as well as analytical expressions for the cutoff wavenumber for instabilityin the viscous regime Firstly the characteristic polynomial (522) is given again forcompleteness

f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)

It is useful to expand the coefficients in powers of the wavenumber k

f0 = f02k2 + if03k3 + f04k4 (A 2)f1 = if11k+ f12k2 + if13k3 (A 3)

f2 = f20 + if21k+ f22k2 (A 4)

where

f02 =minus 1F2(1minusGprime0)(microu minusmicroh) f03 = 1

F2(1minusGprime0)(1minus F2) f04 =minus 1

R(1minusGprime0)

(A 5aminusc)

f11 = 1F2((2minusGprime0)microu minusmicroh +G0microφ) f12 = 1

F2minus 3+ 2Gprime0 f13 = 1

R(2minusGprime0)

(A 6aminusc)

f20 = microu

F2 f21 = 3minusGprime0 f22 = 1

R (A 7aminusc)

A1 Inviscid asymptoticsFirstly consider the inviscid case so that f04= f13= f22= 0 An asymptotic expansionis sought of the form

σ sim σ0kp + σ1kq + σ2kr + σ3ks + middot middot middot (A 8)

for k 1 where the exponents p gt q gt r gt s middot middot middot are to be determined The termsσ0 σ1 σ2 σ3 are all strictly O(1) and are calculated in increasing order until anon-zero real part is found This then determines the leading-order growth rate of theroot Substituting the ansatz (A 8) into the characteristic polynomial (A 1) gives thedominant balance p= 1 and the leading-order behaviour at O(k3) is

if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)

with associated solutions

σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)

These are all purely imaginary so the next order in the expansion is considered bysetting q= 0 which gives at O(k2) the linear equation

( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)



Rearranging for σ1 and substituting in the expressions (A 10) gives the real values

σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)

The growth rate of all three roots therefore tends to a constant as k minusrarr infinwith the value being determined by (A 12)ndash(A 14) At the critical Froude numberF = Fc = 1|Gprime0| two of the above roots become infinite because the coefficient ofσ1 in (A 11) is zero (meaning the denominator in (A 12)ndash(A 14) degenerates) In thiscase the alternative dominant balance q= 12 is chosen which gives at O(k2)

( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)

Substituting in σ0 =minusi(1minusGprime0) which is the same for both critical roots gives

σ 21 = iG0Gprime0(1minusGprime0)microφ (A 16)

and the corresponding real parts

σ(plusmn)1R=plusmn 1

2 |G0Gprime0(1minusGprime0)microφ|12 (A 17)

In this critical regime these two roots scale with k12 in the large wavenumber limitand the positive choice in (A 17) grows unboundedly By definition (Joseph amp Saut1990) this means that the inviscid equations are ill posed at the critical Froudenumber

A2 Viscous asymptoticsRepeating the same process for the viscous equations two possible dominant balancesare found for the leading-order behaviour Choosing p= 2 gives at O(k6)

f22σ20 + σ 3

0 = 0 (A 18)

which has non-zero real solution

σ(1)0 =minus

1R (A 19)

The growth rate of the first root is therefore negative and decays like k2 for large kThe other roots are found by letting p = 1 giving leading-order behaviour at O(k4)determined by the quadratic

f04 + if13σ0 + f22σ20 = 0 (A 20)

which has purely imaginary solutions

σ(2)0 =minusi σ

(3)0 =minusi(1minusGprime0) (A 21ab)



The next-order dominant balance is achieved by setting q= 0 which gives at O(k3)the linear equation

( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)

Substituting in for σ0 = σ (2)0 from (A 21) gives the real solution

σ(2)1 =minus

RF2 (A 23)

Hence the second root is also stable in the large wavenumber limit with growth ratestending to the negative constant (A 23) However using σ0= σ (3)0 gives σ1= 0 In thiscase the alternative scaling q=minus1 is chosen and the next-order behaviour is at O(k2)and given by

( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)

which leads to the imaginary solution

σ(3)1 =

iRG0(1minusGprime0)microφF2Gprime0

(A 25)

The next order is at O(k) and achieved by setting r = minus2 This gives the linearequation

( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)

The third growth rate therefore decays to zero like O(kminus2) for k 1 It may bestable or unstable depending on the sign of (A 27) but remains bounded for allwavenumbers Note that (A 27) degenerates to zero at F = Fc In this case one mustinstead choose r=minus3 giving at O(1)

(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)

with associated imaginary solution

σ(3)2 =

iR2G0(1minusGprime0)(Gprime0microh minus (Gprime0)2microu minusG0microφ)microφ

F4(Gprime0)3 (A 29)

The next highest order is achieved by setting s=minus4 which gives at O(kminus1)

( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)

This is positive meaning the third root will be unstable at the critical Froude numberin the large wavenumber limit However the choice s=minus4 means that it will decayto zero according to kminus4 crucially remaining bounded for all values of k The viscousequations therefore remain well posed



A3 Viscous cutoff wavenumberThe cutoff wavenumber for instability k= kc occurs when the growth rate is purelyimaginary say σ = iσc Substituting into the characteristic polynomial (A 1) andequating real and imaginary parts gives

( f20 + f22k2c)σ

2c + ( f11kc + f13k3

c)σc minus ( f02k2c + f04k4

c)= 0 (A 32)σ 3

c + f21kcσc minus f12k2cσc minus f03k3

c = 0 (A 33)

Equation (A 32) is a quadratic in σc which can be solved to give the relation

σplusmnc (kc)=Cplusmn(kc)kc (A 34)

where the functions Cplusmn(kc) are defined as

Cplusmn(kc)= minus( f11 + f13k2c)plusmn

radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)

Substituting (A 34) into the imaginary parts equation (A 33) and neglecting the trivialroot kc = 0 gives the cubic equation

C3 + f21C2 minus f12Cminus f03 = 0 (A 36)

with associated real solutions

C1 = 1Fminus 1 C2 =minus 1

Fminus 1 C3 =Gprime0 minus 1 (A 37aminusc)

Now equation (A 35) can be rearranged squared and factorised to give(k2

c +f20

f22

)((C2f22 +Cf13 minus f04)k2

c + (C2f20 +Cf21 minus f02))= 0 (A 38)

Two solutions of (A 38) for kc are given by

kc1plusmn =plusmnradicminus f20

f22=plusmn 1

F

radicminusmicrouR (A 39)

which are independent of the value of C However since microu gt 0 and the cutoffwavenumber must be real these can immediately be discarded The other roots aregiven by

kc =plusmnradicminus(C2f20 +Cf11 minus f02)

C2f22 +Cf13 minus f02 (A 40)

When C = C3 the denominator in (A 40) is zero meaning there are no additionalroots in this case Substituting in the other values of C given by (A 37) leads to thesolutions

kc2plusmn =plusmn1F

radicR((FGprime0 minus 1)(Fmicroh minusmicrou)+ FG0(1minus F)microφ)

FGprime0 minus 1 (A 41)



kc3plusmn =plusmn1F

radicR(minus(FGprime0 + 1)(Fmicroh +microu)+ FG0(1+ F)microφ)

FGprime0 + 1 (A 42)

The roots kc2minus and kc3minus are not permitted since they are negative whenever they arereal This leaves two possibilities for the cutoff wavenumber kc2+ and kc3+ For theparameters used in this paper they are both imaginary when F lt Fc but at leastone of them is real for F gt Fc meaning a cutoff of the growth rates Note thesimilar structure between (A 40) (A 41) and the asymptotic behaviour of the inviscidequations (A 12) (A 13) highlighting the link between both models

Appendix B Details of two-dimensional linear stability analysisThis appendix provides details of the linear stability analysis of the two-dimensional

viscous equations used in sect 6 in order to check the well posedness of the full systemThe methodology is similar to that of appendix A and so only the key results arepresented

Firstly the two-dimensional governing equations admit (dimensionless) steady-statesolutions (h u v φ) = (1 1 0 φ0) Perturbing about this base state linearising andseeking normal modes of the form

(h1 u1 v1 φ1)= (HU V Φ)eσ tei(kxx+kyy) (B 1)

for real kx ky and complex σ(kx ky) now leads to the quartic characteristic polynomial

f (σ )equiv f0 + f1σ + f2σ2 + f3σ

3 + σ 4 = 0 (B 2)

which admits solutions σ (1) σ (2) σ (3) σ (4) with corresponding real parts σ (1)R σ(2)R σ

(3)R

σ(4)R The growth rate σM is then found by taking the maximum of these four values as

in the one-dimensional case To establish well posedness this must remain bounded inthe asymptotic limit |k|= (k2

x + k2y)

121 and so it is useful to expand the coefficientsof (B 2) in terms of the downslope wavenumber kx

f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)

and then also in terms of the transverse wavenumber ky

f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)

f30 = f300 + f302k2y f31 = f310 f32 = f320 (B 4dminusf )



Written out explicitly for reference these coefficients are

f012 = microu

F4(1minusGprime0) f014 = (1minusGprime0)

2F2R f020 = tan ζ

F4(1minusGprime0)(microh minusmicrou)

f022 = (1minusGprime0)2F2R

(microh minus 2microu minus 2Rminus tan ζ ) f024 =minus (1minusGprime0)2R2

f030 = (1minusGprime0)F4

((microh minusmicrou)F2 + (1minus F2) tan ζ ) f032 = (1minusGprime0)2F2R

(2minus 3F2)


(2R(F2 minus 1)minus 2 tan ζ +microh minusmicrou) f042 =minus (1minusGprime0)R2


(1minus 3F2) f060 =minus (1minusGprime0)2R2

(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ

F4(minusmicroh + (2minusGprime0)microu +G0microφ)

f112 = 12F2R

((2minusGprime0)(tan ζ + 2Rminus 2microu)minusmicroh +G0microφ) f114 = (2minusGprime0)2R2

f120 = 1F4((2minusGprime0)F

2microh + (2Gprime0 minus 3)F2microu minusGF2microφ + tan ζ (1+ (2Gprime0 minus 3)F2))

f122 = 12F2R

(2minus 3(3minusGprime0)F2) f132 = (2minusGprime0)

R2

f130 = 12F2R

((2minusGprime0)(2R+ 2 tan ζ +microu)minusmicroh +G0microφ + 2F2R(3Gprime0 minus 2))

f140 = 12F2R

(1+ 3F2(2Gprime0 minus 3)) f150 = (2minusGprime0)2R2

(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2

f210 = 1F2((3minusGprime0)(microu + tan ζ )minusmicroh +G0microφ) f212 = 3(3minusGprime0)

2R

f220 = 12F2R

(microu + 2 tan ζ + 2R(1+ 3F2(Gprime0 minus 2))) f222 = 1R2

f230 = 3(3minusGprime0)2R

f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3

2R f310 = 4minusGprime0 f320 = 3

2R

(B 5d)

B1 High downslope wavenumber asymptoticsFirst consider the high downslope wavenumber limit kx 1 for a fixed cross-slopewavenumber ky gt 0 This closely resembles the one-dimensional analysis conducted insect A2 and is a viscous analogue of the results presented in Woodhouse et al (2012)An asymptotic expansion is sought of the form

σ sim σ0kpx + σ1kq

x + σ2krx + σ3ks

x + middot middot middot (B 6)

for kx 1 and exponents pgt qgt rgt sgt middot middot middot to be determined alongside the strictlyO(1) values σ0 σ1 σ2 σ3 Omitting the details for simplicity the leading-order



growth rates of the four roots are found to be

σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)

for kx 1 Note the similarities to the one-dimensional case with the asymptoticbehaviour being independent of the transverse wavenumber ky and remaining boundedfor all kx The extra spatial dimension simply introduces an extra root here referredto as σ (2)R which is stable in the asymptotic limit At the critical Froude number σ (2)Rin (B 7) degenerates to zero and the leading-order growth rate in this case is insteadgiven by

σ(c)R sim

(2R3G2

0(Gprime0)


φminus R2G0(Gprime0)

2(1minusGprime0)microφk2y

) 1k4

x

(B 8)

for kx 1 This is always positive meaning that the root is unstable but remainsbounded even in this critical regime decaying like 1k4

x as kx minusrarrinfin Consequentlywhen considering the asymptotic behaviour (B 7) (B 8) of all of the roots it can beseen that the maximum growth rate σ is bounded above for all parameter values

B2 High cross-slope wavenumber asymptoticsNow consider the alternative asymptotic behaviour of high cross-slope wavenumbersky 1 for a fixed downslope perturbation kx gt 0 It is interesting to begin with thespecial case of purely cross-slope disturbances kx= 0 In this regime the characteristicpolynomial (B 2) can be solved explicitly to give the four roots

σ (0) = 0 σ (1) =minus(microu

F2+ k2

y

2R

)

σ (plusmn) =minus(F2k2

y + R tan ζ )plusmnradic(F2k2

y + R tan ζ )2 minus 4F2R2k2y

2F2R

(B 9)

The real part of σ (1) and σ (plusmn) is negative for all values of ky meaning these roots arestable Consequently for kx = 0 the maximum growth rate is given by σM equiv 0 andhence the base state is neutrally stable One requires non-zero downslope perturbationsfor disturbances to grow in time so the subsequent calculations assume kx gt 0 andseek an asymptotic expansion of the form

σ sim σ0kpy + σ1kq

y + middot middot middot (B 10)

Following the same procedure as in appendix A the leading-order growth rates of thefour roots is given by

σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus

R2G0(1minusGprime0)microφk2x

R2 + F4(Gprime0)2k2x

1k2

y

(B 11)

for ky 1 The first three of these are stable and analogous to the downslope results(B 7) whereas the fourth root is unstable for non-zero kx in this asymptotic limit For afixed downslope wavenumber the growth rate remains bounded as kyminusrarrinfin (decayingto zero) and hence so too does the maximum growth rate σM



B3 High wavenumber asymptoticsIn order to completely establish the well posedness of the system it also needs to bechecked that the growth rates remain bounded when both the downslope and cross-slope wavenumbers tend to infinity ie kx 1 and ky 1 The analysis conducted insectsect B1 and B2 ensures it is sufficient to consider the curves

kx = k ky =mka (B 12ab)

for k 1 where the constant mgt 0 and exponent agt 0 In this case an asymptoticexpansion is sought of the form

σ sim σ0kp + σ1kq + σ2kr + middot middot middot (B 13)

where σ0 σ1 σ2 and the parameters pgt qgt rgt middot middot middot may now depend on a For0lt alt 1 the leading-order growth rates of the first three roots is given by

σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)

for k 1 which are all stable The growth rate of the final root is more complicatedto calculate but note that the first two terms in the asymptotic expansion (B 13) aregiven by the imaginary expressions

σ (4) simminusi(1minusGprime0)k+iRG0(1minusGprime0)microφ

F2Gprime0

1k+ middot middot middot (B 15)

for k 1 and hence the exponent of k in the leading-order real growth rate musttherefore be less than minus1 This growth rate will therefore decay to zero in the largewavenumber asymptotic limit remaining bounded even if the root is unstable Similarbehaviour is found for other values of the parameter a with the first three roots fora= 1 having leading-order growth rates

σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)

for k 1 and the expansion of the fourth root begins with the imaginary terms


F2Gprime0(1+m2)


For the final case agt 1 it can be shown that the roots have growth rates

σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)

for k 1 alongside the asymptotic behaviour


F2Gprime0m2

1k2aminus1

+ middot middot middot (B 19)

for the fourth root and hence the real growth must necessarily remain bounded for allwavenumbers This concludes the two-dimensional linear stability analysis where all



possible variants of the asymptotic limit |k| = (k2x + k2

y)12 minusrarrinfin have been covered

Whilst the full algebraic expressions have sometimes been omitted for simplicity ithas been shown that in all cases the growth rates remain bounded and therefore thatthe two-dimensional governing equations are well posed

All of the above stability analysis has been validated numerically using time-dependent solutions of the fully nonlinear equations as described in sect 62

REFERENCES

ABRAMOWITZ M amp STEGUN I 1970 Handbook of Mathematical Functions 9th edn p 337Dover

ARANSON I S MALLOGGI F amp CLEMENT E 2006 Transverse instability in granular flows downan incline Phys Rev E 73 050302(R)

BAKER J L BARKER T amp GRAY J M N T 2016 A two-dimensional depth-averaged micro(I)-rheology for dense granular avalanches J Fluid Mech 787 367ndash395

BARKER T SCHAEFFER D G BOHORQUEZ P amp GRAY J M N T 2015 Well-posed andill-posed behaviour of the micro(I)-rheology for granular flow J Fluid Mech 779 794ndash818

BRIDGWATER J 1976 Fundamental powder mixing mechanisms Powder Technol 15 215ndash236BRIDGWATER J FOO W amp STEPHENS D 1985 Particle mixing and segregation in failure zones ndash

theory and experiment Powder Technol 41 147ndash158COURANT R amp HILBERT D 1962 Methods of Mathematical Physics vol II InterscienceDA CRUZ F EMAM S PROCHNOW M ROUX J-N amp CHEVOIR F M C 2005 Rheophysics of

dense granular materials discrete simulation of plane shear flows Phys Rev E 72 021309DEBOEUF S LAJEUNESSE E DAUCHOT O amp ANDREOTTI B 2006 Flow rule self channelization

and levees in unconfined granular flows Phys Rev Lett 97 158303DOLGUNIN V N amp UKOLOV A A 1995 Segregation modelling of particle rapid gravity flow

Powder Technol 83 95ndash103EDWARDS A N amp GRAY J M N T 2015 Erosionndashdeposition waves in shallow granular free-

surface flows J Fluid Mech 762 35ndash67FEacuteLIX G amp THOMAS N 2004 Relation between dry granular flow regimes and morphology of

deposits formation of leveacutees in pyroclastic deposits Earth Planet Sci Lett 221 197ndash213FORTERRE Y amp POULIQUEN O 2003 Long-surface-wave instability in dense granular flows

J Fluid Mech 486 21ndash50GAJJAR P amp GRAY J M N T 2014 Asymmetric flux models for particle-size segregation in

granular avalanches J Fluid Mech 757 297ndash329GAJJAR P VAN DER VAART K THORNTON A R JOHNSON C G ANCEY C amp GRAY J

M N T 2016 Asymmetric breaking size-segregation waves in dense granular free-surfaceJ Fluid Mech 794 460ndash505

GDR MIDI 2004 On dense granular flows Eur Phys J E 14 (4) 341ndash365GOLICK L A amp DANIELS K E 2009 Mixing and segregation rates in sheared granular materials

Phys Rev E 80 (4) 042301GOUJON C DALLOZ-DUBRUJEAUD B amp THOMAS N 2007 Bidisperse granular avalanches on

inclined planes a rich variety of behaviours Eur Phys J E 23 199ndash215GOUJON C THOMAS N amp DALLOZ-DUBRUJEAUD B 2003 Monodisperse dry granular flows on

inclined planes role of roughness Eur Phys J E 11 147ndash157GRAY J M N T amp ANCEY C 2009 Segregation recirculation and deposition of coarse particles

near two-dimensional avalanche fronts J Fluid Mech 629 387ndash423GRAY J M N T amp ANCEY C 2011 Multi-component particle size-segregation in shallow granular

avalanches J Fluid Mech 678 535ndash588GRAY J M N T amp ANCEY C 2015 Particle-size and -density segregation in granular free-surface

flows J Fluid Mech 779 622ndash668GRAY J M N T amp CHUGUNOV V A 2006 Particle-size segregation and diffusive remixing in

shallow granular avalanches J Fluid Mech 569 365ndash398



GRAY J M N T amp EDWARDS A N 2014 A depth-averaged micro(I)-rheology for shallow granularfree-surface flows J Fluid Mech 755 503ndash534

GRAY J M N T GAJJAR P amp KOKELAAR B P 2015 Particle-size segregation in dense granularavalanches C R Physique 16 73ndash85

GRAY J M N T amp HUTTER K 1997 Pattern formation in granular avalanches Contin MechThermodyn 9 341ndash345

GRAY J M N T amp KOKELAAR B P 2010a Large particle segregation transport and accumulationin granular free-surface flows J Fluid Mech 652 105ndash137

GRAY J M N T amp KOKELAAR B P 2010b Large particle segregation transport and accumulationin granular free-surface flows ndash Erratum J Fluid Mech 657 539

GRAY J M N T TAI Y C amp NOELLE S 2003 Shock waves dead-zones and particle-freeregions in rapid granular free-surface flows J Fluid Mech 491 161ndash181

GRAY J M N T amp THORNTON A R 2005 A theory for particle size segregation in shallowgranular free-surface flows Proc R Soc Lond A 461 1447ndash1473

GRAY J M N T WIELAND M amp HUTTER K 1999 Free surface flow of cohesionless granularavalanches over complex basal topography Proc R Soc Lond A 455 1841ndash1874

GRIGORIAN S S EGLIT M E amp IAKIMOV I L 1967 New state and solution of the problemof the motion of snow avalance Snow Avalanches Glaciers Tr Vysokogornogo Geofizich Inst12 104ndash113

HILL K M KHARKAR D V GILCHRIST J F MCCARTHY J J amp OTTINO J M 1999Segregation driven organization in chaotic granular flows Proc Natl Acad Sci USA 9611701ndash11706

HOGG A J amp PRITCHARD D 2004 The effects of hydraulic resistance on dam-break and othershallow inertial flows J Fluid Mech 501 179ndash212

IVERSON R M 1997 The physics of debris-flows Rev Geophys 35 245ndash296IVERSON R M 2003 The debris-flow rheology myth In Debris-flow Hazards Mitigation Mechanics

Prediction and Assessment (ed D Rickenmann amp C L Chen) pp 303ndash314 MillpressJOHNSON C G amp GRAY J M N T 2011 Granular jets and hydraulic jumps on an inclined

plane J Fluid Mech 675 87ndash116JOHNSON C G KOKELAAR B P IVERSON R M LOGAN M LAHUSEN R G amp GRAY

J M N T 2012 Grain-size segregation and levee formation in geophysical mass flowsJ Geophys Res 117 F01032

JOP P FORTERRE Y amp POULIQUEN O 2005 Crucial role of sidewalls in granular surface flowsconsequences for the rheology J Fluid Mech 541 167ndash192

JOP P FORTERRE Y amp POULIQUEN O 2006 A constitutive relation for dense granular flowsNature 44 727ndash730

JOSEPH D D amp SAUT J C 1990 Short-wave instabilities and ill-posed initial-value problemsTheor Comput Fluid Dyn 1 191ndash227

KOKELAAR B GRAHAM R GRAY J M N T amp VALLANCE J 2014 Fine-grained linings ofleveed channels facilitate runout of granular flows Earth Planet Sci Lett 385 172ndash180

KURGANOV A amp TADMOR E 2000 New high-resolution central schemes for nonlinear conservationlaws and convection-diffusion equations J Comput Phys 160 (1) 241ndash282

LUBE G CRONIN S J PLATZ T FREUNDT A PROCTER J N HENDERSON C amp SHERIDANM F 2007 Flow and deposition of pyroclastic granular flows a type example from the 1975Ngauruhoe eruption New Zealand J Volcanol Geotherm Res 161 (3) 165ndash186

MAKSE H A HAVLIN S KING P R amp STANLEY H E 1997 Spontaneous stratification ingranular mixtures Nature 386 379ndash382

MALLOGGI F LANUZA J ANDREOTTI B amp CLEacuteMENT E 2006 Erosion waves transverseinstabilities and fingering Eur Phys Lett 75 (5) 825

MARKS B ROGNON P amp EINAV I 2012 Grainsize dynamics of polydisperse granular segregationdown inclined planes J Fluid Mech 690 499ndash511

MEDOVIKOV A 1998 High order explicit methods for parabolic equations BIT 38 (2) 372ndash390



MIDDLETON G V 1970 Experimental studies related to problems of flysch sedimentation In FlyschSedimentology in North America (ed J Lajoie) pp 253ndash272 Business and Economics ScienceLtd

NOELLE S 2000 The MoT-ICE A new high-resolution wave-propagation algorithm formultidimensional systems of conservation laws based on Feyrsquos method of transport J ComputPhys 164 (2) 283ndash334

PIERSON T C 1986 Flow behavior of channelized debris flows Mount St Helens Washington InHillslope Processes (ed A D Abrahams) pp 269ndash296 Allen and Unwin

POULIQUEN O 1999a Scaling laws in granular flows down rough inclined planes Phys Fluids 11(3) 542ndash548

POULIQUEN O 1999b On the shape of granular fronts down rough inclined planes Phys Fluids11 (7) 1956ndash1958

POULIQUEN O DELOUR J amp SAVAGE S B 1997 Fingering in granular flows Nature 386816ndash817

POULIQUEN O amp FORTERRE Y 2002 Friction law for dense granular flows application to themotion of a mass down a rough inclined plane J Fluid Mech 453 133ndash151

POULIQUEN O amp VALLANCE J W 1999 Segregation induced instabilities of granular fronts Chaos9 (3) 621ndash630

RAZIS D EDWARDS A GRAY J M N T amp VAN DER WEELE K 2014 Arrested coarseningof granular roll waves Phys Fluids 26 297ndash329

SAVAGE S B amp HUTTER K 1989 The motion of a finite mass of granular material down a roughincline J Fluid Mech 199 177ndash215

SAVAGE S B amp LUN C K K 1988 Particle size segregation in inclined chute flow of drycohesionless granular solids J Fluid Mech 189 311ndash335

STARON L amp PHILLIPS J C 2014 Segregation time-scale in bi-disperse granular flows PhysFluids 26 033302

THORNTON A R amp GRAY J M N T 2008 Breaking size-segregation waves and particlerecirculation in granular avalanches J Fluid Mech 596 261ndash284

TUNUGUNTLA D R BOKHOVE O amp THORNTON A R 2014 A mixture theory for size anddensity segregation in shallow granular free-surface flows J Fluid Mech 749 99ndash112

VAN DER VAART K GAJJAR P EPELY-CHAUVIN G ANDREINI N GRAY J M N T ampANCEY C 2015 Underlying asymmetry within particle size segregation Phys Rev Lett 114238001

VALLANCE J W amp SAVAGE S B 2000 Particle segregation in granular flows down chutesIn IUTAM Symposium on Segregation in Granular Materials (ed A D Rosato amp D LBlackmore) Kluwer

WIEDERSEINER S ANDREINI N EPELY-CHAUVIN G MOSER G MONNEREAU M GRAY JM N T amp ANCEY C 2011 Experimental investigation into segregating granular flows downchutes Phys Fluids 23 013301

WOODHOUSE M J THORNTON A R JOHNSON C G KOKELAAR B P amp GRAY J MN T 2012 Segregation-induced fingering instabilities in granular free-surface flows J FluidMech 709 543ndash580

ZURIGUEL I GRAY J M N T PEIXINHO J amp MULLIN T 2006 Pattern selection by a granularwave in a rotating drum Phys Rev E 73 061302


JFM_809_2016_originalpdf

Segregation-induced finger formation in granular free-surface flows

Introduction

A depth-averaged model for particle size-segregation

Segregation-mobility coupling

Steady uniform flows

Linear stability analysis

Non-dimensionalisation

Linearised equations and the characteristic polynomial

Inviscid high wavenumber asymptotics

Viscous high wavenumber asymptotics

Two-dimensional numerical simulations for a propagating front

Generalised equations

Numerical solutions

Finger characteristics numerical versus physical viscosity

Conclusions

Acknowledgements

Appendix A Details of one-dimensional linear stability analysis

Inviscid asymptotics

Viscous asymptotics

Viscous cutoff wavenumber

Appendix B Details of two-dimensional linear stability analysis

High downslope wavenumber asymptotics

High cross-slope wavenumber asymptotics

High wavenumber asymptotics

References

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TooltipField

J Fluid Mech (2016) vol 809 pp 168ndash212 ccopy Cambridge University Press 2016This is an Open Access article distributed under the terms of the Creative Commons Attributionlicence (httpcreativecommonsorglicensesby40) which permits unrestricted re-use distribution andreproduction in any medium provided the original work is properly citeddoi101017jfm2016673

168

Segregation-induced finger formation ingranular free-surface flows

J L Baker1dagger C G Johnson1 and J M N T Gray1dagger1School of Mathematics and Manchester Centre for Nonlinear Dynamics University of Manchester

Oxford Road Manchester M13 9PL UK

(Received 19 April 2016 revised 30 September 2016 accepted 12 October 2016)

Geophysical granular flows such as landslides pyroclastic flows and snow avalanchesconsist of particles with varying surface roughnesses or shapes that have a tendency tosegregate during flow due to size differences Such segregation leads to the formationof regions with different frictional properties which in turn can feed back on the bulkflow This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects The full segregation equation for dense granular flows isintegrated through the avalanche thickness by assuming inversely graded layers withlarge particles above fines and a Bagnold shear profile The resulting large particletransport equation is then coupled to depth-averaged equations for conservation ofmass and momentum with the feedback arising through a basal friction law that iscomposition dependent implying greater friction where there are more large particlesThe new system of equations includes viscous terms in the momentum balance whichare derived from the micro(I)-rheology for dense granular flows and represent a singularperturbation to previous models Linear stability calculations of the steady uniformbase state demonstrate the significance of these higher-order terms which ensurethat unlike the inviscid equations the growth rates remain bounded everywhere Thenew system is therefore mathematically well posed Two-dimensional simulations ofbidisperse material propagating down an inclined plane show the development of anunstable large-rich flow front which subsequently breaks into a series of finger-likestructures each bounded by coarse-grained lateral levees The key properties of thefingers are independent of the grid resolution and are controlled by the physicalviscosity This process of segregation-induced finger formation is observed inlaboratory experiments and numerical computations are in qualitative agreement

Key words fingering instability geophysical and geological flows granular media

1 IntroductionThe process of particle size segregation whereby mixtures of different sized

particles separate into distinct grain-size classes during flow can be very pronounced

dagger Email addresses for correspondence jamesbakeralumnimanchesteracuknicograymanchesteracuk











~10 m







~5 cm






~5 cm

~5 cm

(a)

(b)






Erodingmaterial


Staticmaterial

Grain-free regions

(a) (b)






~5 cm










partφ

partt+ part

partx(φu)+ part


partz(Q(φ))= part

partz

(Dpartφ

partz

) (21)



z

x

0





partspartt+ us



Q(φ)+Dpartφ





part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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ikona

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TooltipField










~10 m







~5 cm






~5 cm

~5 cm

(a)

(b)






Erodingmaterial


Staticmaterial

Grain-free regions

(a) (b)






~5 cm










partφ

partt+ part

partx(φu)+ part


partz(Q(φ))= part

partz

(Dpartφ

partz

) (21)



z

x

0





partspartt+ us



Q(φ)+Dpartφ





part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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89

ikona

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183

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195

196

197

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199

200

201

203

207

TooltipField


~5 cm






~5 cm

~5 cm

(a)

(b)






Erodingmaterial


Staticmaterial

Grain-free regions

(a) (b)






~5 cm










partφ

partt+ part

partx(φu)+ part


partz(Q(φ))= part

partz

(Dpartφ

partz

) (21)



z

x

0





partspartt+ us



Q(φ)+Dpartφ





part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField


~5 cm

~5 cm

(a)

(b)






Erodingmaterial


Staticmaterial

Grain-free regions

(a) (b)






~5 cm










partφ

partt+ part

partx(φu)+ part


partz(Q(φ))= part

partz

(Dpartφ

partz

) (21)



z

x

0





partspartt+ us



Q(φ)+Dpartφ





part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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199

200

201

203

207

TooltipField



Erodingmaterial


Staticmaterial

Grain-free regions

(a) (b)






~5 cm










partφ

partt+ part

partx(φu)+ part


partz(Q(φ))= part

partz

(Dpartφ

partz

) (21)



z

x

0





partspartt+ us



Q(φ)+Dpartφ





part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

ikona

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

203

207

TooltipField


~5 cm










partφ

partt+ part

partx(φu)+ part


partz(Q(φ))= part

partz

(Dpartφ

partz

) (21)



z

x

0





partspartt+ us



Q(φ)+Dpartφ





part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField







partφ

partt+ part

partx(φu)+ part


partz(Q(φ))= part

partz

(Dpartφ

partz

) (21)



z

x

0





partspartt+ us



Q(φ)+Dpartφ





part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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TooltipField


z

x

0





partspartt+ us



Q(φ)+Dpartφ





part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField



part

partt(hφ)+ part

partx(hφu)minus

[φ

(partzpartt+ u

partzpartxminusw

)]s

b

=[

Q(φ)+Dpartφ

partz

]s

b

(25)

where

φ = 1h

int s

bφ dz φu= 1

h

int s

bφu dz (26ab)


part

partt(hφ)+ part

partx(hφu)= 0 (27)


u(x t)= 1h

int s

bu(x z t) dz (28)


φ =


(29)


u(x z t)= u(x t)f (z) (210)


0f (z) dz= 1 (211)








φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField





φu= 1h

int l

bu dz= u

int φ

0f (z) dz (214)


part

partt(hφ)+ part



where


0f (z) dz (216)








05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField


05 10 15 200

02

04

06

08

10LinearBagnold

02 04 06 08 100

005

010

015

020

025(a) (b)






parthpartt+ part

partx(hu)= 0 (31)

part

partt(hu)+ part

partx(χhu2)+ part

partx

(12


partx

(νh32 part u

partx

) (32)





S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField



S= cos ζ(


) (33)



b(h Fr) (34)

where


b(h Fr) (35)



(36)



1 +microN

2 minusmicroN1








ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField


ζ = 270 ζ S1 = 200 ζ S





νN = 2LNradicg9βN

sin ζradiccos ζ

(microN


1





ζ S1 lt ζ lt ζ

L1 lt ζ

S2 lt ζ

L2 (310)










where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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200

201

203

207

TooltipField







where


(43)


AF2 + Bh0F+Ch20 = 0 (44)












φlowast0 =microL


1 minusmicroS1 (48)


F= h0


2A

) (49)



1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

08

09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField


1

23 4 5 6 7

1 2 3 4 50

01

02

03

04

05

06

07

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09

10






lowast0




(410)






u0t (51aminusd)




parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

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0

025

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0

025

050

0

025

050

0

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05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

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0

025

050

0

025

050

0

025

050

0

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025

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05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

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0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField



parthpartt+ part

partx(hu)= 0 (52)

F2h(part upartt+ u

part upartx

)+ h

parthpartx= h


)+ F2

Rpart

partx

(h32 part u

partx

) (53)

h(partφ

partt+ u

partφ

partx

)minus part




b(h u) (55)



1 +microN

2 minusmicroN1

(γN h32)u+ 1 (56)


R= u0radic

h0

ν (57)





parth1

partt+ parth1

partx+ part u1

partx= 0 (511)

F2

(part u1

partt+ part u1

partx

)+ parth1


Rpart2u1

partx2 (512)



partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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186

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189

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197

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TooltipField


partφ1

parttminusG0

(parth1

partx+ part u1

partx

)+ (1minusGprime0)

partφ1

partx= 0 (513)



7(1minus 2φ0) (514ab)


microh = partmicrob

parth

∣∣∣∣(11φ0)

=minus32

(φ0Γ

S + (1minus φ0)ΓL)lt 0 (515)

microu = partmicrob

part u

∣∣∣∣(11φ0)


L gt 0 (516)


partφ

∣∣∣∣(11φ0)


b(1 1) lt 0 (517)


Γ N = γN (microN

2 minusmicroN1 )

(1+ γN )2 (518)




AW = σW (520)


A=

minusik minusik 0

minusmicroh

F2minus ik

F2minusmicrou

F2minus ikminus k2

Rminusmicroφ

F2


(521)


f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (522)



(3)R ) (523)




3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField


3 6 90

005

010

InviscidViscous

0 3 6 9

minus004

0

004

008

(a)

(b)





3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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191

192

193

194

195

196

197

198

199

200

201

203

207

TooltipField


3 6 90

01

02

InviscidViscous




dxdt= λ (524)


λ(1) = u+ 1F


F

radich (525ab)







(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 02 04

025

050






05x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

h (mm)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 35 70

025

050





0 05

x (m)

y (m

)y

(m)

y (m

)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField






(527)



σ(1)R sim


(528)

σ(2)R sim


(529)

σ(3)R sim


(530)


(2)R









σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

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y (m

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(m)

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1000 20000

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02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField




σ(1)R simminus

1R

k2 (532)

σ(2)R simminus

RF2 (533)

σ(3)R sim



1k2 (534)







σ(3)R sim 2R3G2

0(Gprime0)


φ

1k4 (535)





2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























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y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

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050

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050






1000 20000

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025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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TooltipField


2 40

005

010

015(a)

(b)

InviscidViscous

2 40

25

50










part


(12


part



Sx = cos ζ(


(microb

v

|u|) (64ab)



(65)


























05

x (m)

y (m

)y

(m)

y (m

)y

(m)

y (m

)

10 150

025

050

0

025

050

0

025

050

0

025

050

0

0 05 10

025

050





05x (m)

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)y

(m)

y (m

)y

(m)

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)

10 150

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05x (m)

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)

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10 150

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)

10 15

025

0

025

0

025

050






1000 20000

01

02

1000 20000

025

050

(a) (b)2 4 0 2 4












0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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0025 00500

025

050


25

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)

50 750

05

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7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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0025 00500

02

04(a) (b)

0025 00500

025

050


25

t (s)

x (m

)

50 750

05

10

15






7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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7 Conclusions










Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField






Acknowledgements








f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField




f (σ )equiv f0 + f1σ + f2σ2 + σ 3 = 0 (A 1)



f2 = f20 + if21k+ f22k2 (A 4)

where



R(1minusGprime0)

(A 5aminusc)



R(2minusGprime0)

(A 6aminusc)

f20 = microu


R (A 7aminusc)




if03 + f12σ0 + if21σ20 + σ 3

0 = 0 (A 9)


σ(1)0 =minusi

(1minus 1

F

) σ

(2)0 =minusi

(1+ 1

F

) σ

(3)0 =minusi

(1minusGprime0

) (A 10aminusc)


( f02 + if11σ0 + f20σ20 )+ ( f12 + 2if21σ0 + 3σ 2

0 )σ1 = 0 (A 11)




σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField



σ(1)1 =


(A 12)

σ(2)1 =


(A 13)

σ(3)1 =


(A 14)


( f02 + if11σ0 + f20σ20 )+ (2if21 + 3σ0)σ

21 = 0 (A 15)








f22σ20 + σ 3

0 = 0 (A 18)


σ(1)0 =minus

1R (A 19)


f04 + if13σ0 + f22σ20 = 0 (A 20)


σ(2)0 =minusi σ





( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField



( f03 + f12σ0 + f21σ20 + σ 3

0 )+ ( f13 + 2f22σ0)σ1 = 0 (A 22)


σ(2)1 =minus

RF2 (A 23)


( f02 + if11σ0 + f20σ20 )+ (if13 + 2f22σ0)σ1 = 0 (A 24)


σ(3)1 =


(A 25)


( f12 + 2if21σ0 + 3σ 20 )σ1 + (if13 + 2f22σ0)σ2 = 0 (A 26)

with real solution

σ(3)2 =



(A 27)


(if11 + 2f20σ0 + f22σ1)σ1 + (if13 + 2f22σ0)σ2 = 0 (A 28)


σ(3)2 =


F4(Gprime0)3 (A 29)


( f12σ2 + 2if21σ0σ2 + if21σ21 + 3σ0σ

21 + 3σ 2

0 σ2)+ (if13 + 2f22σ0)σ3 = 0 (A 30)

with real solution

σ(3)3 = 2R3G2

0(Gprime0)


φ (A 31)





( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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TooltipField



( f20 + f22k2c)σ

2c + ( f11kc + f13k3


c)= 0 (A 32)σ 3


c = 0 (A 33)





radic( f11 + f13k2

c)2 + 4( f02 + f04k2

c)( f20 + f22k2c)

2( f20 + f22k2c)

(A 35)







c +f20

f22


c + (C2f20 +Cf21 minus f02))= 0 (A 38)



f22=plusmn 1

F




C2f22 +Cf13 minus f02 (A 40)


kc2plusmn =plusmn1F





kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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kc3plusmn =plusmn1F


FGprime0 + 1 (A 42)








3 + σ 4 = 0 (B 2)


(3)R



x + k2y)


f0 = if01kx + f02k2x + if03k3

x + f04k4x + if05k5

x + f06k6x

f1 = f10 + if11kx + f12k2x + if13k3

x + f14k4x + if15k5

x

f2 = f20 + if21kx + f22k2x + if23k3

x + f24k4x

f3 = f30 + if31kx + f32k2x

(B 3)


f01 = f012k2y + f014k4

y f02 = f020 + f022k2y + f024k4

y

f03 = f030 + f032k2y f04 = f040 + f042k2

y f05 = f050 f06 = f060

(B 4a)

f10 = f102k2y + f104k4

y f11 = f110 + f112k2y + f114k4

y

f12 = f120 + f122k2y f13 = f130 + f132k2

y f14 = f140 f15 = f150

(B 4b)

f20 = f200 + f202k2y + f204k4

y f21 = f210 + f212k2y

f22 = f220 + f222k2y f23 = f230 f24 = f240

(B 4c)





f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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f012 = microu


2F2R f020 = tan ζ






(2minus 3F2)





(B 5a)

f102 = microu

F4 f104 = 1

2F2R f110 = tan ζ


f112 = 12F2R




f122 = 12F2R


R2

f130 = 12F2R


f140 = 12F2R


(B 5b)

f200 = tan ζF4

microu f202 = 12F2R

(tan ζ + 2(R+microu)) f204 = 12R2


2R

f220 = 12F2R



f240 = 12R2

(B 5c)

f300 = 1F2(tan ζ +microu) f302 = 3


2R

(B 5d)



x + σ2krx + σ3ks






σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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3

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σ(1)R simminus

1R

k2x σ

(2)R simminus

12R

k2x

σ(3)R simminus

RF2 σ

(4)R sim



1k2

x

(B 7)


σ(c)R sim

(2R3G2

0(Gprime0)




) 1k4

x

(B 8)





F2+ k2

y

2R

)




2F2R

(B 9)





σ(1)R simminus

1R

k2y σ

(2)R simminus

12R

k2y

σ(3)R simminus

RF2 σ

(4)R simminus



1k2

y

(B 11)









σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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σ(1)R simminus

1R

k2 σ(2)R simminus

12R

k2 σ(3)R simminus

RF2 (B 14aminusc)



F2Gprime0



σ(1)R simminus

1+m2

Rk2 σ

(2)R simminus

1+m2

2Rk2 σ

(3)R simminus

RF2 (B 16aminusc)



F2Gprime0(1+m2)



σ(1)R simminus

m2

Rk2a σ

(2)R simminus

m2

2Rk2a σ

(3)R simminus

RF2 (B 18aminusc)



F2Gprime0m2

1k2aminus1









REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

animtiph

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REFERENCES










































































Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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Introduction











Numerical solutions


Conclusions

Acknowledgements



Viscous asymptotics






References

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issn 0022-1120 · 2016. 12. 6. · issn 0022-1120 25 december 2016 volume 809 volume 809 25 dec....

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