APS/123-QED

A Theory for Shock Dynamics in Particle-Laden Thin Films

Junjie Zhou,1 B. Dupuy,1 A. L. Bertozzi,2 and A. E. Hosoi11Department of Mechanical Engineering, Hatsopoulos Microfluids Laboratory,

Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 021392UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555

(Dated: February 8, 2005)

We present a theory to explain the emergence of a particle-rich ridge observed experimentally inthin film particle-laden flow on an incline. We derive a lubrication theory for this system whichis qualitatively compared to preliminary experimental data. The ridge formation arises from thecreation of two shocks due to the differential transport rates of fluid and particles. This parallelsrecent findings of double shocks in thermal-gravity driven flow [A. L. Bertozzi et. al., PRL, 81, 5169(1998), J. Sur et. al., PRL 90, 126105 (2003), A. Munch, PRL 91, 016105 (2003)]. However, herethe emergence of the shocks arises from a new mechanism involving the settling rates of the species.

PACS numbers: 47.15.Pn, 68.15.+e, 68.03.-g

Thin films and coating flows have long been studiedin a rich array of industrial, biological and environmen-tal contexts [1]. While extensive research has been con-ducted on clear fluids [2–6] and dry granular flows [7, 8],far fewer studies have been dedicated to particle-ladenfluid films. In applications such as erosion and debrisflow [9, 10], slurry transport, and mixing of pharmaceu-ticals, particles play a dominant role in the dynamics andcannot be neglected [11].

New and unexpected behavior is observed in these par-ticulate systems owing to the introduction of a new timescale. In particle-laden, gravity-driven films [12], onetime scale is associated with the viscous fingering thatoccurs at the contact line and a second is associated withparticle settling. At low inclination angles, if these twotime scales are widely disparate, the behavior of the sus-pension closely reflects that of clear fluids. However, newand interesting phenomena arise when the two time scalesare comparable, resulting in nonuniform particle distri-butions. The main purpose of this manuscript is to derivea theory to explain the new phenomena observed in thisregime.

In clear fluid experiments involving flow on an inclinedplane, the large scale dynamics is described by convec-tive theory in which the first order terms in the equationsof motion yield quantitative information about the speedand shape of the front. Recent results [13–15] show thattemperature gradient driven flow balanced by gravity canbe described by a scalar conservation law with a noncon-vex flux. In the present study we find similar dynamicsin particle-laden films. New phenomena are observed ex-perimentally in the limit of high particle concentrationand inclination angle, and this data is compared qualita-tively to predictions arising from shock theory.

The experimental apparatus consists of an acrylic sheetwith an adjustable inclination angle, β. A suspension ofviscous fluid and polydisperse glass beads (250-425 µm indiameter), flows from a reservoir (see figure 1). Profiles ofthe flowing layer at fixed y, as functions of x and time, aremeasured using a laser light sheet and a high resolutiondigital camera. Experiments were conducted using both

ReservoirAdjustable

gate

β

Lasersheet

x y

z

Camera

FIG. 1: Schematic of the experimental apparatus.

wetting (Dow Corning 200, 1000 cSt fluid) and partially-wetting (glycerol) fluids.

Three distinct regimes were observed experimentally(figure 2). At low inclination angles and concentrations,sedimentation perpendicular to the plate occurs rapidlyand particles are deposited in a uniform particulate frontbehind the advancing free surface. Clear fluid flows overthe particulate matter and eventually fingers; the dy-namics is precisely that observed in numerous clear fluidexperiments ([e.g. 2, 16, 17]). At high inclination an-gles and particle concentrations, the evolution deviatesstrikingly from the clear fluid case and a large particle-rich ridge appears at the advancing free surface front.The ridge observed in these dense suspensions may growto several times the thickness of the trailing film. Be-tween these two regimes, a well-mixed transition regionappears, characterized by regular fingers containing par-ticles. Intuitively we expect the large ridge to arise if theparticle settling velocity along the plate is faster thanthat of the propagating front.

To derive a model valid in the large-ridge region ofparameter space, we consider the governing equations ofmotion for the mixture

∇ ·Π + ρ(φ)g = 0, ∇ · j = 0, (1)

where j is the volume averaged flux of the mixture, g is

2

55

50

45

40

35

30

25

20

155045403530252015

clear fingerstransitionridge

angl

e

1.2

1

0.8

0.6

0.4

0.2

0

0 2 4 6 8

concentration

FIG. 2: (Top) Photographs of typical fingering instabilitiesin each regime. (Bottom-L) Phase diagram for 250-425 µmbeads in glycerol showing three regimes. (Bottom-R) Pro-files of an advancing silicon oil front illustrating the wideningridge. Initial concentration φ0 = 45% and inclination angleβ = 42◦. Both axes are made dimensionless using H, thethickness of the film at the gate. Profiles are plotted every 30sec.

the acceleration of gravity, and Π = −pI + µ(φ)(∇j +(∇j)T ) is the stress tensor for the mixture. Here p ispressure, ρ(φ) and µ(φ) are the effective density and vis-cosity respectively, and I is the identity matrix. We con-sider low Reynolds number flows and neglect inertia ofboth the particulate and the fluid phase. The volumeflux is related to the velocity by jp = φvp, jf = (1−φ)vf

and j = jp + jf , where subscripts p and f refer to particleand fluid phase respectively and φ is the local particleconcentration (by volume).

Particles move relative to the fluid phase via settling.This relative velocity is given by the Stokes settling ve-locity modified by two effects: (1) the presence of otherparticles in the dense suspension which hinders settlingand (2) the presence of the no-slip boundary conditionat the wall. To model the first, we adopt the Richard-son & Zaki correction [18], valid for high concentrations,f(φ) = (1−φ)m, where m is empirically determined to be≈ 5.1. To capture the wall effect, we modify the Stokessettling velocity by a function that vanishes near the walland approaches unity in the bulk flow [22]. Thus

vR =29

(ρp − ρf ) a2

µff(φ)w(h)g

where vR denotes particle velocity relative to the fluid

phase, a is a characteristic particle radius, µf is theviscosity of the fluid phase, ρf and ρp are the den-sity of the fluid and particulate phase respectively, andw(h) ≡ Ah2√

1+(Ah2)2. Here h is the local film thickness and

A is a parameter that determines the sharpness of thetransition from the no-slip wall to the bulk flow.

The effective mixture density ρ(φ) = φρp + (1− φ)ρf ,and effective mixture viscosity [19]

µ(φ) = µf (1− φ/φmax)−2 (2)

are both functions of the local concentration. The vis-cosity diverges as the concentration approaches φmax andthe mixture becomes increasingly more solid-like.

At the substrate, we enforce a no-slip boundary con-dition. At the free surface, the normal stress balance isgiven by n ·Π · n = γκ and the tangential stress balanceby t · Π · n = 0 where n and t are respectively the unitoutward normal and unit tangential vectors to the freesurface, γ is the surface tension of the mixture (here as-sumed to be independent of particle concentration) andκ is the curvature of the free surface.

We begin by analyzing the 2D evolution of the filmprior to the onset of fingering, neglecting variations inthe y direction. Consider the following rescaling, mo-tivated by the scaling adopted in gravity-driven clearfluid films [20]: x = xL ≡ x

(`2H/ sinβ

)1/3, z = Hz,p = P p ≡ µf LU

H2 p, t = tT ≡ (3µf/γ)(`2L/H2 sinβ)t,and (jx, jz) = U(jx, εjz). Here ` =

√γ/ρfg is the cap-

illary length, H is the thickness of the film far behindthe moving front, dimensionless quantities are denotedby a tilde, and a characteristic velocity is defined asU = L/T . Equations (1) are rescaled using the aboverelations and we retain only the lowest order terms inε ≡ H/L = (3Ca)1/3 where Ca = µfU/γ is the capillarynumber. If εPe � 1, where Pe = UH/D is the Pecletnumber and D is the diffusion coefficient for the partic-ulate phase, particles diffuse rapidly across the depth ofthe film creating a uniform distribution in z; hence φmay be approximated as φ(x, t) [23]. Following a stan-dard long wavelength approach for thin films [e.g. 4], wesolve for the velocities and pressures and depth-averagethe equations bearing in mind that there are two massconservation conditions, one for the mixture and one forthe particles. Dropping the tildes, this yields two coupledequations for the evolution of φ and h:

∂(ρ(φ)h)∂t

+{

ρ(φ)µ(φ)

h3hxxx −D(β)[

ρ(φ)µ(φ)

h3(ρ(φ)h)x −58

ρ(φ)µ(φ)

h4(ρ(φ))x

]+

ρ(φ)2

µ(φ)h3

}x

= 0 (3)

∂(φh)∂t

+{

φ

µ(φ)h3hxxx −D(β)

[φ

µ(φ)h3(ρ(φ)h)x −

58

φ

µ(φ)h4(ρ(φ))x

]+

φρ(φ)µ(φ)

h3 +23Vsφhf(φ)w(h)

}x

= 0 (4)

3

where Vs ≡ ρp−ρf

ρf

a2

H2 , D(β) ≡ (3Ca)1/3 cot(β), sub-scripts denote partial derivatives, and we have absorbedthe extra factor of (1 − φ) into f by defining f(φ) ≡(1− φ)f(φ).

To determine macroscopic properties of the flow, it isinstructive to investigate the underlying convective dy-namics, described by the system of two conservation laws:

∂(ρ(φ)h)∂t

+{

ρ(φ)2

µ(φ)h3

}x

= 0 (5)

∂(φh)∂t

+{

φρ(φ)µ(φ)

h3 +23Vsφhf(φ)w(h)

}x

= 0. (6)

Consider as an initial condition a jump in the height from1 to the precursor thickness b, and a constant concentra-tion of φ0. That is, at time zero, h = 1 for x ≤ 0, h = bfor x > 0, and φ(x) = φ0. The above system has the formof a 2× 2 system of scalar conservation laws

∂u

∂t+ [F (u, v)]x = 0,

∂v

∂t+ [G(u, v)]x = 0, (7)

where u = ρh and v = φh. The initial conditiongives us a Riemann problem i.e. both of the systemvariables u and v have a jump at the origin at timezero. Classical solutions of this problem, when they ex-ist, are well-known in the theory of shock-waves [21].An intermediate state (ui, vi) emerges separating theleft state (ul, vl) = (ρfρ(φ0), φ0), from the right state(ur, vr) = (ρfρ(φ0)b, φ0b). The values ui and vi deter-mine the density of particles, φi, and and the fluid thick-ness, hi, in the ridge (see figure 4-Left).

A general 2× 2 system of conservation laws (7) is hy-

perbolic if the Jacobian matrix J(u, v) =(

Fu Fv

Gu Gv

)has

two real distinct eigenvalues λ1(u, v) and λ2(u, v), witheigenvectors r1, r2. The problem is called genuinely non-linear if ri · ∇λi 6= 0, where the gradient is with respectto the variables u and v. For a large range of relevantphysical parameters in (5-6), both conditions are satis-fied provided we include the wall function w(h). Settingw(h) = 1 results in a change in hyperbolicity of the equa-tions (5-6) at small h.

Under the assumptions of hyperbolicity and genuinenonlinearity, there is a methodology for determining theintermediate values ui and vi in terms of the left and rightstates. Thus, for a fixed concentration in the reservoirand upstream film thickness, given the particle concen-tration and thickness of the precursor film, we can findthe height, concentration and propagation speed of theridge. A one parameter family of shock solutions, con-necting (ui, vi) to (ul, vl) or (ur, vr) can be determined byrecognizing that the weak form of (7) yields two Rankine-Hugoniot conditions for the shock speed both of whichhave to be satisfied. For the 1-shock, connecting (ul, vl)to (ui, vi), we have

s1 =F (ui, vi)− F (ul, vl)

ui − ul=

G(ui, vi)−G(ul, vl)vi − vl

(8)

0.4

0.36

0.32

0.28

0 50 100 200150 250 300

conc

entr

atio

n

distance

1.5

0.5

1

0

heig

ht

reduced model full model

FIG. 3: Full and reduced model numerical profiles for φ0 =30%. (Top) Comparison of height profiles of the full system(3-4) with the reduced system (5-6). (Bottom) Comparison ofconcentration profiles of the full system (3-4) with the reducedsystem (5-6). Profiles are plotted at 40 sec intervals. Thelarge spike at the leading edge of the full system correspondsto a capillary ridge due to surface tension.

0.9

0.8

0.7

0.6

0.5

0.40 0.05 0.1 0.15 0.2

precursor film (b)

shoc

k sp

eed

FIG. 4: (Left) Schematic diagram of the connection problem.(Right) Shock speeds calculated via the connection problem asa function of precursor thickness for φ0 = 15% (filled symbols)and φ0 = 30% (open symbols).

which gives two equations in three unknowns. Analo-gous formulas hold for the 2-shock connecting (ui, vi) to(ur, vr). This gives a total of four equations and four un-knowns, namely s1, s2, ui, and vi. Generically there willbe a locally unique solution, however these equations arealso accompanied by entropy conditions which restrictwhether a 1-shock or 2-shock is an admissible connec-tion. In the case of the 1-shock, the entropy conditionis λ1l < s1 < λ1i and λ2l, λ2i > s1. For the 2-shockwe require λ2l < s2 < λ2i and λ1l, λ1i < s2 (see [21]for details). In the results presented here, the entropycondition is satisfied for both the 1- and 2- shocks andthe shock connection problem can be solved numerically,providing an implicit formula for determining the inter-mediate state (ui, vi) that emerges from the Riemanninitial data.

Numerical simulations show that the reduced model(5-6), captures the the large scale dynamics of the fullequations, (3-4), including the shock speeds s1 and s2

and ridge height hi and ridge particle concentration φi.The numerical methods used are an explicit upwind for(5-6) and backward Euler with spatial centered differ-ences for (3-4). Figure 3 shows a sample calculation withparameters m = 5, φmax = 0.67, φ0 = 0.3, b = 0.1,β = 90◦, and A = 0.5. The large spike at the leadingedge of the full system solution corresponds to the cap-

4

Full Model Reduced Model Shock Theory

hi 1.15 1.162 1.1619φi 0.359 0.362 0.36199s1 0.435 0.434 0.4556s2 0.498 0.493 0.4956

TABLE I: Comparison of shock speeds and intermediatestates in model simulations and shock theory. Data shownfor φ0 = 0.30, b = 0.1. Quantitative experimental data is notpresented here owing to the sensitive dependence on precursorfilm as discussed in the text.

illary ridge due to surface tension, which is neglected inthe the reduced system. In both solutions, an interme-diate state appears connecting the left and right statesvia shocks. The trailing shock in figure 3 is smeared outdue to numerical diffusion. Note that while the capillaryridge of the full system propagates with a fixed width, theparticle-rich region between the two shocks continues tobroaden as seen in the experiment. The full lubricationtheory, reduced model and shock connection problem allconsistently predict the formation of a large particle-richridge at high inclination angles and high concentrationsas observed in the experimental data. The numericallyobserved values in the full and reduced model are com-pared with the analytic calculations from the connectionproblem in table I. All three models display a sensitivedependence on the precursor thickness making quantita-tive comparison with experimental data challenging.

This sensitivity is illustrated in the shock speeds shown

in figure 4-Right in which s1 and s2 are shown to con-verge as b decreases. We also observe hi to increase as bdecreases. This presents a marked contrast to the clearfluid case in which the front speed does not depend in asingular way on the precursor thickness. Moreover, thelogarithmic dependendence of the capillary ridge on b forthe clear fluid problem may be complicated by the sensi-tivity of the underlying particle transport problem to theprecursor thickness. The strong influence of the precursoron the basic flow structure (without surface tension ef-fects) underscores the need for more detailed experimen-tal measurements. Specifically, quantitative comparisonbetween theory and experiment requires precise measure-ments of the precursor thickness and shock separationover time. Both of these are subjects of current study.

Shock theory for systems of conservation laws is wellknown in gas dynamics but has never before been ap-plied to the physics of contact lines. This new theorysuggests many avenues for further study including thedependence of the dynamics on contact line physics, andthe extension of the theory to the transverse direction toaddress the development of fingers, which appear to besomewhat suppressed in the experiment by the formationof the particle-rich ridge.

Acknowledgments

We thank Daniel Blair, Arshad Kudrolli, Peter Mucha,Christian Ratsch and Michael Shearer for useful com-ments. This work is supported by NSF grants CCF-0323672, CCF-0321917, DMS-0244498, DMS-0243591,and ONR grant N000140410078.

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