arbitrary oscillatory stokes flow past a porous sphere using brinkman model
TRANSCRIPT
Meccanica (2012) 47:1079–1095DOI 10.1007/s11012-011-9494-1
Arbitrary oscillatory Stokes flow past a porous sphere usingBrinkman model
Jai Prakash · G.P. Raja Sekhar
Received: 26 September 2008 / Accepted: 5 October 2011 / Published online: 8 November 2011© Springer Science+Business Media B.V. 2011
Abstract The present paper deals with the hydrody-namics of a porous sphere placed in an arbitrary oscil-latory Stokes flow. Unsteady Stokes equation is usedfor the flow outside the porous sphere and Brinkmanequation is used for the flow inside the porous sphere.Corresponding Faxén’s law for drag and torque is de-rived and compared with few existing results in somespecial cases. Examples like uniform flow, oscilla-tory shear flow and oscillating Stokeslet are discussed.Also, translational oscillation of a weakly permeablesphere is discussed.
Keywords Brinkman equation · Faxén’s law ·Oscillatory flow · Stokes flow
Nomenclaturea radius of the porous sphere [m]k permeability of the porous sphere [m2]r radial distanceve oscillatory velocity external to the porous
sphere [m/s]
Dedicated to Professor S.D. Nigam.
J. Prakash · G.P. Raja Sekhar (�)Department of Mathematics, Indian Instituteof Technology Kharagpur, Kharagpur 721 302, Indiae-mail: [email protected]
J. Prakashe-mail: [email protected]
J. Prakashe-mail: [email protected]
pe oscillatory pressure external to the poroussphere [N/m2]
Ve amplitude of the oscillatory velocity externalto the porous sphere [m/s]
P e amplitude of the oscillatory pressure externalto the porous sphere [N/m2]
Vi velocity internal to the porous sphere [m/s]P i pressure internal to the porous sphere [N/m2]p0 constant [N/m2]U magnitude of the far field uniform velocity
[m/s]Da Darcy numberV0 basic velocity [m/s]V∗ velocity due to the disturbance [m/s]A, B scalarsfn modified spherical Bessel function of first
kindgn modified spherical Bessel function of second
kindSn, Tn spherical harmonicsP m
n associated Legendre polynomial|V i
θ | magnitude of the internal tangential velocity
Greek symbolsθ inclinationϕ azimuth angleα slip coefficientλ dimensionless parameterω frequency of oscillation [s−1]� dimensionless frequency of oscillationρ density of the fluid [kg/m3]
1080 Meccanica (2012) 47:1079–1095
μ dynamic viscosity [kg/m−1/s−1]ν kinematic viscosity [m2/s]
Subscript/Superscripte external to the porous spherei internal to the porous sphere
1 Introduction
The problem of Stokes flow past porous particles usingBrinkman equation in the interior is studied by Higdonand Kojima [1]. Yu and Kaloni [2] obtained a carte-sian tensor solution of Brinkman equation and calcu-lated the hydrodynamic force on a porous sphere in auniform flow. It is well known in low Reynolds num-ber hydrodynamics that Faxén’s law can be used toevaluate the drag force and torque on a solid imperme-able particle subject to an arbitrary flow. A generaliza-tion of Faxén’s law is obtained by Keh and Chen [3],which is applicable to the surface of an impermeablesolid sphere with slip boundary condition. In case ofliquid droplet, Hetsroni and Haber [4, 5] obtained thecorresponding Faxén’s law, while Rallison [6] derivedthe expression in case of a stresslet. Masliyah et al. [7]solved the creeping flow of an incompressible Newto-nian fluid past a spherically symmetric composite par-ticle by using Brinkman equation for the flow field in-side the fluid-permeable surface layer and the Stokesequations for the flow field external to the particle.
Hydrodynamics of steady Stokes flow past poroussphere has been studied by many researchers. But, lim-ited attention is given in case of time dependent flowsthrough porous media or oscillatory flows throughporous media. The unsteady Stokes equation for themicroscopic flow in porous media subject to an os-cillatory pressure gradient has been studied by Chap-man and Higdon [8] where the media consists of peri-odic array of spheres ranging from dilute systems withisolated spheres to highly concentrated consolidatedmedia with overlapping spheres. Study on the hydro-dynamics of a rigid, weakly permeable sphere under-going translational oscillations in an incompressibleNewtonian fluid has been done by Looker and Carnie[9] where Darcy’s law is used inside the porous sphere.The Brinkman equation with continuity of velocityand normal stress at the porous interface is believedto be a successful model of steady Stokes flow in caseof highly permeable particles. In [9], it is mentioned
that the inclusion of an unsteady term in the Brinkmanequation enabled Poliak [10] to solve the oscillatoryproblem. Although, there currently seems no extensiveliterature or theoretical justification on this approach,the problem appears to be of interest due to several ap-plications mentioned here. Curcio [11] made a theoret-ical and experimental analysis of membrane bioreac-tors behavior in unsteady state conditions. The theoret-ical model is based on the coupling between transientNavier–Stokes equation with Brinkman equation con-taining the corresponding time derivative term. Thehydrodynamic problem is solved using continuity ofvelocity and stress components and the results are usedin order to solve the convection, diffusion and reactionsystem inside the fiber regions.
Another important area where oscillatory forcingplays a significant role is while understanding con-vective mass transfer in porous catalysts [12–15]. Incase of small, highly porous catalyst particles diffu-sion alone may not account for the nutrient transportand convective flow has a major role. It is evident thatunder steady state, convective flow within a porouscatalyst is not so important whereas oscillatory forc-ing at higher amplitude and/or lower frequencies en-hances the mass transfer. Ni et al. [16] observed thatoscillatory flow improves the performance of a bedpacked with spherical particles. Crittenden et al. [17]studied the influence of oscillatory flow on axial dis-persion in packed bed of spheres. They observed thatthe best reduction (up to 50%) in the axial dispersioncoefficient from the non-oscillation base value is atthe highest frequency considered and when the col-umn to particle size is the smallest. Hence, the presentstudy aims at understanding the hydrodynamics of os-cillatory Stokes flow past a porous sphere consider-ing Brinkman equation with the time derivative terminside porous region. Such an investigation not onlygives an idea of the hydrodynamic forces acting onthe surface of a porous sphere, but also, the corre-sponding calculations can be used in order to under-stand the mass transfer inside porous pellets under os-cillatory forcing. Another important application is onacoustic properties of granular materials. Umnova etal. [18] have considered oscillatory flow of viscous in-compressible flow around a spherical particle and usedcell model in order to estimate the hydrodynamic dragdue to oscillating flow in a stack of fixed identical rigidspheres. The present investigation also is useful in un-derstanding acoustic properties of porous materials.
Meccanica (2012) 47:1079–1095 1081
Recently, Vainshtein and Shapiro [19] gave a the-oretical investigation on the forces acting on a poroussphere oscillating in a viscous fluid. Linearized oscil-latory flow outside the porous sphere is governed byinhomogeneous Stokes equation and Darcy/Brinkmanequation that includes an unsteady term for the flowinside the porous sphere is used. They have obtaineda general expression for the drag acting on the poroussphere in case of Darcy equation for the flow inside theporous sphere. While, the corresponding expressionsfor the limiting values of low and high frequenciesare obtained when the flow inside the porous sphereis governed by Brinkman equation. It may be notedthat the continuity of velocity components, shear stresstogether with the continuity of pressure are justifiedwhen Brinkman equation is used inside the porous re-gion, while these boundary conditions need some at-tention when Darcy equation is used. Because, in thelatter case, a more appropriate Saffman condition isused and its applicability for oscillatory flows is dis-cussed in [9]. Moreover, in case of Darcy equation,the addition of time derivative term is redundant as ithas been shown by homogenization that the macro-scopic unsteady Stokes equations in a periodic porousmedium have the same form as for steady Stokes flow,i.e., Darcy’s law. In this paper, we consider an arbitraryoscillatory Stokes flow past a stationary porous sphereusing Brinkman equation in a viscous, incompressiblefluid. The representation of the velocity and the pres-sure fields for the Brinkman equation suggested byPadmavathi et al. [20], Raja Sekhar et al. [21] will beused. This enables us to derive Faxén’s law for dragand torque acting on the surface of the porous sphere.The present study differs from that of [19] in severalissues. The investigation is in case of any arbitrary os-cillatory flow past a Brinkman porous sphere. The pur-pose of considering arbitrary flow is to obtain generalexpressions of the flow quantities like velocity, force,etc. One can use this general frame work in order to getresults corresponding to a particular flow such as uni-form flow, shear flow, etc. The force and torque are ex-pressed in terms of Faxén’s law and these expressionsare more general. Several examples are discussed inorder to display the potential of the present method.
2 Mathematical formulation
Let us consider a porous sphere of radius a and perme-ability k in an arbitrary oscillatory flow of a viscous in-
compressible fluid. Let us assume that the flow insidethe porous sphere is described by Brinkman equation,and that the flow outside the porous sphere is an un-steady Stokes flow. Consequently, the flow inside theporous sphere (r < a) is governed by the unsteadyBrinkman equation and continuity equation:
ρ∂vi
∂t= −∇pi + μ∇2vi − μ
kvi , (1)
∇ · vi = 0, (2)
and the flow outside the porous sphere (r > a) is de-scribed by the unsteady Stokes and continuity equa-tions:
ρ∂ve
∂t= −∇pe + μ∇2ve, (3)
∇ · ve = 0. (4)
Since we are interested in the study of oscillatory flowwith the frequency ω, we set the velocity and pressurefields as v = Ve−iωt and p = P e−iωt . Thus, the gov-erning equations transform to
−iρωVi = −∇P i + μ∇2Vi − μ
kVi , (5)
∇ · Vi = 0, (6)
for the flow inside the porous sphere and
−iρωVe = −∇P e + μ∇2Ve, (7)
∇ · Ve = 0, (8)
for the flow outside the porous sphere, where μ is thefluid viscosity, Vi and P i are the velocity and pres-sure fields inside the porous sphere, and Ve and P e
are those of the flow outside the porous sphere.Now, the physical quantities are non-dimensionali-
zed by using the transformation
X = Xa
, V = VU
, P = P
μUa
,
where U is a characteristic velocity. Therefore, thenon-dimensional equations for the flow inside theporous region (r < 1) take the form
(∇2 − λ2i )V
i = ∇P i, (9)
∇ · Vi = 0, (10)
1082 Meccanica (2012) 47:1079–1095
where λ2i = a2
k− iωa2
ν, k is the permeability of porous
sphere, a is the radius of porous sphere, ω is the fre-quency of oscillation and ν is the kinematic viscosity.The equations for the flow in the liquid region (r > 1)
become
(∇2 − λ2e)V
e = ∇P e, (11)
∇ · Ve = 0, (12)
where λ2e = − iωa2
ν. The parameter λi can be defined
in terms of Darcy number Da = k
a2 as λ2i = 1
Da+ λ2
e .Note that we have omitted the symbol ∼ from (9)–(12).
2.1 Boundary conditions
It is well known that the boundary conditions to beused at a porous–liquid interface are non-trivial. How-ever, when Darcy’s law is used for the flow inside theporous region and Stokes equation for the free flowregion, continuity of pressure, continuity of normalvelocity and Saffman slip condition [22] or Beavers–Joseph condition is used [23–25]. It may be notedthat in case of Brinkman equation it is customaryto use continuity of velocity components togetherwith the continuity of stress components and theseboundary conditions are accepted by a large commu-nity [20, 25–30]. A very recent note by Nield [31]gives a useful historical background to the Beavers–Joseph boundary condition at the interface of a porousmedium and a clear fluid together with a discussionon using the Brinkman equation. Ochoa-Tapia andWhitaker [32, 33] introduced a jump condition thattells that the tangential velocity gradients are discon-tinuous at a porous–liquid interface. This condition ispopularly known as stress jump condition. However,keeping the complexity of the system of equations thatare involved in the present formulation, we use the fol-lowing boundary conditions on the boundary betweenthe porous and fluid regions, i.e., on r = 1:
(i) continuity of velocity components:
V er = V i
r , V eθ = V i
θ , V eϕ = V i
ϕ, (13)
(ii) continuity of stress components:
T err = T i
rr , T erθ = T i
rθ , T erϕ = T i
rϕ. (14)
The above boundary conditions are well accepted inthe research fraternity when Stokes-Brinkman cou-pling is involved (see [19, 20, 34]).
3 Method of solution
It may be noted that Padmavathi et al. [20] proposed arepresentation of the velocity and pressure fields cor-responding to Brinkman equation of the type (11) and(12) and is given by
Vj = CurlCurl(Aj X) + Curl(Bj X), (15)
P j = p0 + ∂
∂r[r(∇2 − λ2
j )Aj ], (16)
where X is the position vector of the current point, p0
is a constant, and Aj and Bj are unknown scalar func-tions satisfying the equations
∇2(∇2 − λ2j )A
j = 0, (∇2 − λ2j )B
j = 0. (17)
The superscript j can be i or e depending on whetherthe flow is internal or external to the porous sphere.Raja Sekhar et al. [35] have shown that this representa-tion indeed is a complete general solution of Brinkmanequations, in the sense that it is obtained by integrat-ing the equations. One may also see [21], where thecompleteness of the above representation is shown viaan alternate method. Hence, the representation shownabove is an exact solution of the Brinkman equations.The velocity representation given in (15) admits thefollowing component form in (r, θ,φ) spherical polarcoordinates given by
Vr = −1
rLA, (18)
Vθ = 1
r
∂
∂θ
∂
∂r(rA) + csc θ
∂B
∂ϕ, (19)
Vϕ = 1
r sin θ
∂
∂ϕ
∂
∂r(rA) − ∂B
∂θ, (20)
where L = 1sin θ
∂∂θ
(sin θ) ∂∂θ
+ csc2 θ ∂2
∂ϕ2 , is the trans-verse part of the Laplacian in spherical coordinates(r, θ,ϕ) system. Moreover the corresponding stresscomponents are given by
Trr = −P + 2μ∂Vr
∂r, (21)
Trθ = μ
[1
r
∂Vr
∂θ− Vθ
r+ ∂Vθ
∂r
], (22)
Trϕ = μ
[1
r sin θ
∂Vr
∂ϕ− Vϕ
r+ ∂Vϕ
∂r
]. (23)
Meccanica (2012) 47:1079–1095 1083
Now let us obtain the boundary conditions in termsof the scalars A and B . By invoking the continuity ofvelocity components, we get from (18) and (19), onr = 1
Ae = Ai,∂Ae
∂r= ∂Ai
∂r, Be = Bi. (24)
Continuity of tangential stress, considering the factthat the velocity components are continuous, gives onr = 1
∂V eθ
∂r= ∂V i
θ
∂r. (25)
However, keeping the relations (19) and (24) in view,the above condition gives
∂2Ae
∂r2= ∂2Ai
∂r2,
∂Be
∂r= ∂Bi
∂r. (26)
Now, let us consider the continuity of the normal stressthat can be expressed with the help of (21) and (18) as
−P e − 2μ∂
∂r
(1
rLAe
)= −P i − 2μ
∂
∂r
(1
rLAi
).
(27)
Keeping the first of the relations (24) in view, theabove condition reduces to
P e = P i,
which is nothing but the continuity of pressure. It maybe noted that the continuity of normal stress, togetherwith the continuity of velocity components reduce tothe continuity of pressure. The explicit expressionscorresponding to these pressure fields inside and out-side the porous sphere, given in (16) reduce to the fol-lowing relation in terms of the corresponding scalarsAe and Ai , when the continuity of pressure is enforced
∂3Ae
∂r3− ∂3Ai
∂r3= (λ2
e − λ2i )
∂
∂r(rAi). (28)
Hence the boundary value problem given in (9)–(12),together with the boundary conditions (13) and (14)is reduced to solving scalar equations as given in (17)with the corresponding boundary conditions (24)–(28).
Let us now assume that the velocity field V0 of thebasic flow, i.e., of the unperturbed flow in the absenceof any boundaries is given by
V0 = CurlCurl(A0X) + Curl(B0X), (29)
A0 =∞∑
n=1
[αnr
n + βnfn(λer)]Sn(θ,ϕ), (30)
B0 =∞∑
n=1
γnfn(λer)Tn(θ,ϕ), (31)
where Sn(θ,ϕ) and Tn(θ,ϕ) are spherical harmonicsgiven by
Sn(θ,ϕ) =n∑
m=0
P mn (ξ)(Anm cosmϕ + Bnm sinmϕ),
ξ = cos θ, (32)
Tn(θ,ϕ) =n∑
m=0
P mn (ξ)(Cnm cosmϕ + Dnm sinmϕ),
(33)
with P mn , as the associated Legendre polynomials and
αn, βn, γn are known constants and (r, θ,ϕ) are spher-ical coordinates with respect to the origin chosen at thecenter of the sphere r = 1. Here, fn represent modifiedspherical Bessel functions of the first kind. In addition,the scalar functions A0 and B0 satisfy (17). It may benoted that the scalars A, B represent the flow fieldand the problem can be solved in terms of these scalarfunctions.
On the other hand, if the basic flow with the veloc-ity field V0 is perturbed by the presence of a stationaryporous sphere with radius r = 1, then the velocity fieldVe of the resulting flow outside the porous sphere isgiven by Ve = V0 + V∗, where V∗ is the velocity dueto the disturbance flow such that V∗ → 0 as r → ∞.Hence, the resulting flow in the exterior region (r > 1)
is given by
Ae =∞∑
n=1
[αnr
n + α′n
rn+1+ βnfn(λer)
+ β ′ngn(λer)
]Sn(θ,ϕ), (34)
Be =∞∑
n=1
[γnfn(λer) + γ ′
ngn(λer)]Tn(θ,ϕ), (35)
where gn are modified spherical Bessel functions ofsecond kind and α′
n, β ′n, γ ′
n are unknown constants. Forthe flow inside the porous sphere, the solution must befinite at origin, hence the corresponding scalars Ai and
1084 Meccanica (2012) 47:1079–1095
Bi can be expressed as
Ai =∞∑
n=1
[δnr
n + δ′nfn(λir)
]Sn(θ,ϕ), (36)
Bi =∞∑
n=1
σnfn(λir)Tn(θ,ϕ), (37)
where δn, δ′n, σn are unknown constants that will be
determined from the boundary conditions.The unknowns are determined using the boundary
conditions and are obtained as
α′n = −λeEnβn + λeFnβ
′n
(2n + 1)λifn(λi),
β ′n = −(
(2n + 1)(n + 1)(λ2e − λ2
i )λifn(λi)αn
+ βn[Gn + λe(nλ2e + (n + 1)λ2
i )Hn])/Zn,
γ ′n = [λefn+1(λe)fn(λi) − λifn(λe)fn+1(λi)]γn
Nn
,
δn = (λifn+1(λi)αn + Mnβn + [λifn+1(λi)
+ (2n + 1)fn(λi)]α′n + Nnβ
′n
)/(λifn+1(λi)
),
δ′n = λefn+1(λe)βn − (2n + 1)α′
n − λegn+1(λe)β′n
λifn+1(λi),
σn = λe[fn(λe)gn+1(λe) + fn+1(λe)gn(λe)]γn
Nn
,
En = λefn+1(λi)fn(λe) − λifn(λi)fn+1(λe),
Fn = λefn+1(λi)gn(λe) + λifn(λi)gn+1(λe),
Gn = (2n + 1)(n + 1)(λ2e − λ2
i )λifn(λi)fn(λe),
Hn = λefn+1(λi)fn(λe) − λifn(λi)fn+1(λe),
Mn = λifn(λe)fn+1(λi) − λefn+1(λe)fn(λi),
Nn = λign(λe)fn+1(λi) + λegn+1(λe)fn(λi),
Zn = (2n + 1)(n + 1)(λ2e − λ2
i )λifn(λi)gn(λe)
+ λe(nλ2e + (n + 1)λ2
i ){λefn+1(λi)gn(λe)
+ λifn(λi)gn+1(λe)}.
4 Faxén’s law for porous sphere in oscillatory flow
Faxén derived expressions for the drag and torque ex-erted by an exterior steady Stokes flow on a rigidsphere [36]. This enables us to express the drag forceand torque in terms of basic flow. Faxén’s law in terms
of singularity solutions for fluid-fluid, fluid-solid andsolid-solid dispersions is given by Kim and Lu [37].By using the singularity method, similar results for un-steady Stokes flow are obtained [36, 38, 39]. Next, weobtain Faxén’s law for arbitrary oscillatory Stokes flowpast a Brinkman sphere.
It is well known that the drag D exerted by an ex-terior flow on a spherical surface r = 1, as well as thetorque T, are given by
D =∫ 2π
ϕ=0
∫ π
θ=0
[T e
rr er + T erθ eθ
+ T erϕ eϕ
]r2 sin θ dθ dϕ
∣∣∣∣∣r=1
, (38)
T =∫ 2π
ϕ=0
∫ π
θ=0
[rT e
rθ eϕ − rT erϕ eθ
]r2 sin θ dθ dϕ
∣∣∣∣∣r=1
,
(39)
where er , eθ , eϕ are the unit vectors corresponding tothe spherical coordinates (r, θ,ϕ), and T e
rr , T erθ and
T erϕ are the components of the stress tensor.
Using the corresponding stress components in theabove formulae, we obtain the following expressionsfor drag and torque:
D = 4π
3λ2
e
[18λif1(λi)Pα1 + Qβ1]3λif1(λi)Z1
× (A11 i + B11 j + A10k), (40)
T = 8π
3λe
λif2(λi)[f2(λe)g1(λe) + f1(λe)g2(λe)]γ1
λig1(λe)f2(λi) + λeg2(λe)f1(λi)
× (C11 i + D11 j + C10k), (41)
where
P = λ2eλ
2i f2(λi)g1(λe) + λeλ
3i f1(λi)g2(λe),
Q = {λ2ef2(λi)f1(λe) − λeλif1(λi)f2(λe)
+ 6λif1(λi)f1(λe)}Z1
− {G1 + λe(λ2e + 2λ2
i )H1}{λ2ef2(λi)g1(λe)
+ λeλif1(λi)g2(λe) + 6λif1(λi)g1(λe)},G1 = 6(λ2
e − λ2i )λif1(λi)f1(λe),
H1 = λef2(λi)f1(λe) − λif1(λi)f2(λe),
Z1 = 6(λ2e − λ2
i )λif1(λi)g1(λe) + λe(λ2e
+ 2λ2i ){λef2(λi)g1(λe) + λif1(λi)g2(λe)}.
Meccanica (2012) 47:1079–1095 1085
Here the coefficients A10,A11,B11,C10,C11 and D11
are due to the spherical harmonics given in (32) and(33) and are dictated by the basic flow. The modifiedspherical Bessel functions are given by
f1(�) = � cosh� − sinh�
�2,
f2(�) = (�2 + 3) sinh� − 3� cosh�
�3,
g1(�) = e−�(� + 1)
�2, g2(�) = e−�(�2 + 3� + 3)
�3.
The expressions given in (40) and (41) can be ex-pressed in terms of the basic flow as follows
D = 4πλ2e
(P
Z1
)[V0]0
− 4π
Z1
(P − Q
6λeλif1(λi)
)[∇2V0]0, (42)
T = 4πλif2(λi){f1(λe)g2(λe) + f2(λe)g1(λe)}
λeg2(λe)f1(λi) + λig1(λe)f2(λi)
× [∇ × V0]0, (43)
where V0 is the velocity field corresponding to the ba-sic flow, the notation [ ]0 means the evaluation at theorigin r = 0. The above expressions correspond to theFaxén’s law in case of arbitrary oscillatory Stokes flowpast a porous Brinkman sphere.
In the limiting case of Da → 0, the above expres-sions reduce to
D = 2π
(λ2
e + 3λe + 3
)[V0]0
− 2π
(1 + 3
λe
+ 3
λ2e
− 3eλe
λ2e
)[∇2V0]0, (44)
T = 4π
[eλe
λe + 1
][∇ × V0]0. (45)
These formulae, which give the drag and torque incase of an arbitrary oscillatory Stokes flow past a rigidsphere, have been obtained by Pozrikidis [39].
Now in case of steady Stokes flow (λe → 0), resultsobtained in (42) and (43) reduce to
D = (12πλ2f1(λ)[V0]0 + 2π[(λ2 + 6)f1(λ)
− 2λf0(λ)][∇2V0]0)/((2λ2 + 3)f1(λ)
+ 2λf0(λ)), (46)
T = 4π[λf0(λ) − 3f1(λ)]
λf0(λ)[∇ × V0]0, (47)
where λ = 1√Da
. The expressions obtained above fordrag and torque acting on the surface of porous spherein a steady Stokes flow have been obtained by Padma-vathi et al. [20].
It may be noted that the Faxén’s law given in (42)and (43) is valid for any arbitrary basic flow. Hence,in order to realize the corresponding expressions forvarious inputs of basic flow, here, we consider someexamples.
4.1 Uniform oscillatory flow past a porous sphere
If the basic flow is uniform along z-axis, then the cor-responding expressions for A0 and B0 in nondimen-sional form are
A0 = 1
2r cos θ, B0 = 0.
Comparing the above (A0,B0) with the general ex-pressions given in (30) and (31), we have α1 = 1
2 ,β1 = 0 and γ1 = 0. Hence, the corresponding expres-sions for drag and torque given in (40) and (41) reduceto
D = 4πλ2e
P
Z1k, (48)
T = 0. (49)
In the low frequency limit (λe 1), the modifiedspherical Bessel functions behave like
fn(λe) ∼ λen
(2n + 1)! ; gn(λe) ∼ (2n − 1)!λe
(n+1),
and hence, the corresponding expression for drag re-duces to
D = 4πλ2e
λi2(λe
2 + 120)
(240λe2 + 120λi
2 + λe4 + 2λi
2λe2)
k.
(50)
In the high frequency limit λe n(n+1)2 , we have
fn(λe) ∼ eλe
2λe
; gn(λe) ∼ e−λe
λe
.
Hence, the corresponding expression for drag inthis limit reduces to
D = 4πλ2e
[λi
2λe
λe3 + 6λeλi + 2λi
2λe − 6λi2
]k. (51)
1086 Meccanica (2012) 47:1079–1095
In the limit λi → ∞ (Da → 0), the formula (48)reduces to
D = 6π
(1 + λe + λ2
e
3
)k. (52)
The expression (52) obtained independently by Boussi-nesq and Basset [40, 41] includes the Stokes drag,force due to added mass and Basset force. The addedmass force is a result of the fluid surrounding the par-ticle being accelerated. It has a tendency to keep theparticle from being accelerated in any direction andthe Basset force is the force associated with past move-ments of the particle.
4.2 Flow due to an oscillating Stokeslet
The flow due to an oscillatory point force located atthe point y in free space, whose strength is given bythe real or imaginary part of b exp−iωt , where b is aconstant vector, is called oscillating Stokeslet. The ve-locity and pressure of such an oscillatory Stokeslet interms of the corresponding tensor notation in R
3, isgiven by
vj (x) = 1
8πμGλe
2
jk (x − y)bk,
p(x) = 1
8π�
λe2
k (x − y)bk, j, k = 1,2,3.
(53)
The components of the fundamental oscillatory Stokestensor Gλe
2and those of its associated pressure vec-
tor �λe2, which determine the fundamental solution
(Gλe2,�λe
2) of the oscillatory Stokes system are given
by
Gλe2
jk (x − y) = δjk
|x − y|A1(λe|x − y|)
+ (xj − yj )(xk − yk)
|x − y|3 A2(λe|x − y|),
and Πλe
2
j (x − y) = 2xj − yj
|x − y|3 ,
(54)
where
A1(R) = 2e−R(1 + R−1 + R−2) − 2R−2,
A2(R) = −2e−R(1 + 3R−1 + 3R−2) + 6R−2,
R = λer.
(55)
It can be seen from the expressions (42) and (43)that in order to compute Faxén’s law correspondingto a given basic velocity V0, it is enough to compute[V0]0, [∇2V0]0 and [∇ × V0]0. Hence, to compute theFaxén’s law corresponding to an oscillatory Stokeslet,we consider the location of the oscillatory Stokesletto be at (0,0, c) where c > a and the strength beingb1/8πμ with axis along positive x-axis. The velocitycomponents of such an oscillatory Stokeslet in carte-sian form are given by
u = b1
8πμ
(A1(λer)
r+ x2
r3A2(λer)
),
v = b1
8πμ
(xy
r3A2(λer)
),
w = b1
8πμ
(x(z − c)
r3A2(λer)
),
(56)
where r = √x2 + y2 + (z − c)2. Hence, we get
[V0]0 = b1
4πμ
((1
c+ 1
λec2+ 1
λe2c3
)e−λec
− 1
λe2c3
)i,
[∇2V0]0 = b1
4πμ
(λe
2
c+ λe
c2+ 1
c3
)e−λec i,
[∇ × V0]0 = b1
4πμ
(1
c2+ λe
c
)e−λec j.
(57)
The corresponding Faxén’s law when the basic flowis due to an oscillatory Stokeslet is obtained as
D = b1d10
((1
c+ 1
λec2+ 1
λe2c3
)e−λec
− 1
λe2c3
)i + b1d20
(λe
2
c+ λe
c2+ 1
c3
)e−λec i,
(58)
T = b1t10
(1
c2+ λe
c
)e−λec j, (59)
where
d10 = λe2 P
Z1,
d20 = − 1
Z1
(P − Q
6λeλif1(λi)
),
t10 = λif2(λi){f1(λe)g2(λe) + f2(λe)g1(λe)}λeg2(λe)f1(λi) + λig1(λe)f2(λi)
.
(60)
Meccanica (2012) 47:1079–1095 1087
We discuss now the low frequency limit and highfrequency limit of the oscillatory Stokeslet.
Low frequency limit:In order to consider the case of low frequency oscil-lations (small values of the frequency parameter λe),we expand Gλe
2in a Taylor series with respect to λe as
follows (see [38])
Gλ2e = G(0) + λeG(1) + λ2
eG(2) + · · · (61)
where
G(0)jk (x − y) = δjk
r+ (xj − yj )(xk − yk)
r3,
G(1)jk (x − y) = −4
3δjk,
G(2)jk (x − y) = r2
4
(3δjk
r− (xj − yj )(xk − yk)
r3
).
(62)
It may be noted that G(0)jk is nothing but the steady
Stokeslet and G(1)jk represents uniform flow.
Also, in this low frequency limit, the coefficientsgiven in (60), behave like
d10 ∼ λe2 λi
2(λe2 + 120)
(240λe2 + 120λi
2 + λe4 + 2λi
2λe2)
,
d20 ∼ −1
2
λi2(λe
2 + 120)
(240λe2 + 120λi
2 + λe4 + 2λi
2λe2)
,
t10 ∼ λi2
120
(λe2 + 120)
(λi2 + 120)
.
(63)
Hence, the corresponding Faxén’s law reduces to
D = b1
2
λi2(λe
2 + 120)
240λe2 + 120λi
2 + λe4 + 2λi
2λe2
×[
λe2
2c− 4λe
3
3+ 3λe
4c
4− 1
c3
]i, (64)
T = b1
2
λi2
120
(λe2 + 120)
(λi2 + 120)
(2
c2− λe
2)
j.
The total drag in this case is the superposition of thedrag due to steady Stokeslet and that of a uniform flowwith an additional perturbation term.
The case of high frequency limit:In the high frequency limit (λe → ∞) the unsteady
Stokeslet has the following asymptotic form (see [38])
Gλ2
e
jk(x − y) = 2
λe2
(−δjk
r3+ 3
(xj − yj )(xk − yk)
r5
).
(65)
It may be noted that this corresponds to the steady po-tential dipole, which gives the velocity field of an ir-rotational flow. Hence, we see that [∇2V0]0 = 0 and[∇ × V0]0 = 0. The only contribution to the Faxén’slaw comes from
[V0]0 = − b1
4πμλe2c3
i. (66)
In this frequency limit, the coefficients given in (60),behave like
d10 ∼ λi2λe
λe3 + 6λeλi + 2λi
2λe − 6λi2,
d20 ∼ −(λi
2(λe3 + λiλe
2 − 3 eλeλi
))/(λe
(6λiλe
2 − 6λi3 + λiλe
3
+ λe4 + 2λeλi
3 + 2λi2λe
2)),t10 ∼ λie
λi
λe(λe + λi).
(67)
Hence, the corresponding Faxén’s law for drag andtorque reduces to
D = − b1
λe2c3
[λi
2λe
λe3 + 6λeλi + 2λi
2λe − 6λi2
]i,
T = 0.
(68)
The drag which is due to an irrotational flow vanishesat infinity. Vanishing of torque is consistent with theirrotational nature of the flow.
4.3 A porous sphere in a linear oscillatory shear flow
We consider the porous sphere in a linear oscilla-tory shear flow along z-axis. Therefore, we havethe far field basic velocity in dimensionless form asV0 = τ�xk, where the coordinates x, y, and z havebeen non-dimensionalized by the radius of poroussphere, the shear rate has been non-dimensionalizedby U/a, and the frequency is non-dimensionalizedby � = ωa2/ν. Here τ is the shear rate coefficient.
1088 Meccanica (2012) 47:1079–1095
We can see that [V0]0 = 0 and [∇2V0]0 = 0 and[∇ × V0]0 = τ� j. Hence we have
D = 0, (69)
T = 4πτ�
×[λif2(λi){f1(λe)g2(λe) + f2(λe)g1(λe)}
λeg2(λe)f1(λi) + λig1(λe)f2(λi)
]j.
(70)
4.4 A porous sphere in a quadratic oscillatory shearflow
We consider the porous sphere in oscillatory shearflow along z-axis. Hence, we have the far field basicvelocity in dimensionless form as V0 = τ�(x −x2)k.We can see that [V0]0 = 0, [∇2V0]0 = 2τ� k and[∇ × V0]0 = τ� j. Hence
D = −8πτ�
Z1
[P − Q
6λeλif1(λi)
]k, (71)
T = 4πτ�
×[λif2(λi){f1(λe)g2(λe) + f2(λe)g1(λe)}
λeg2(λe)f1(λi) + λig1(λe)f2(λi)
]j.
(72)
It may be noted that linear shear gives zero drag,while quadratic shear produces a non-zero drag. Thetorque in both the cases remain same. Also, the corre-sponding expressions due to low/high frequency limitcan be obtained using the coefficients in (63)/(67).
5 Weakly permeable sphere
In this section, we consider the hydrodynamics of os-cillating weakly permeable sphere in an incompress-ible Newtonian fluid. Looker et al. [9] have shownthat for very low permeability, the continuity of ve-locity at a porous–liquid interface reduces to no pene-tration for the flow in the liquid region. O’Neill et al.[42] have also used Saffman condition [22] togetherwith the no penetration condition while discussing theslow motion of a solid sphere in presence of a natu-rally permeable surface. This means that the porousparticle is so weakly permeable that the external flowslips before it can penetrate the interface. Also, whilediscussing the motion of a permeable sphere at finite
but small Reynolds number, Feng et al. [43] com-puted a general expression for the drag acting on theporous sphere using Darcy equation inside the porousregion. In the limiting case of low but finite permeabil-ity, they have obtained the corresponding drag mul-tiplier. In this limit, the no-penetration condition to-gether with Saffman condition is satisfied. Such a nopenetration condition together with a slip condition isgenerally used at rough impermeable surfaces wherethe amount of slip depends on the characteristics of thesurface [44, 45]. Hence, we consider that the externalflow is governed by the continuity and unsteady Stokesequations, and assume that there is no flow inside theweakly permeable sphere. The corresponding bound-ary conditions at the interface differ from those in theprevious sections. It may be noted that the velocity andpressure fields outside the porous sphere obey the gov-erning equations given in (11) and (12) and hence thecorresponding representation in terms of A and B isvalid as in (34) and (35).
5.1 Boundary conditions
For the reasons mentioned above, we use the follow-ing boundary conditions on the surface of the weaklypermeable sphere
(i) No-penetration condition: V er = 0
(ii) Saffman boundary condition for the tangentialcomponents of the velocity field:
V eθ =
√Da
α
∂V eθ
∂r, V e
ϕ =√
Da
α
∂V eϕ
∂r,
where α is the dimensionless slip coefficient. Theunknown coefficients are determined in terms of theknown coefficients αn, βn and γn, and are given by
α′n = (
Unαn + Vnβn
)/({α(2n + 1)
+ l(2n − λ2e + 1)}gn(λe) − λe(α + l)gn+1(λe)
),
β ′n = −αn + βnfn(λe) + α′
n
gn(λe),
γ ′n = γn{(α − nl)fn(λe) − lλefn+1(λe)}
(nl − α)gn(λe) − lλegn+1(λe),
Un = lλ2egn(λe) + λe(α + l)gn+1(λe),
Vn = λe(α + l){fn(λe)gn+1(λe) + fn+1(λe)gn(λe)},where l = √
Da.
Meccanica (2012) 47:1079–1095 1089
The corresponding Faxén’s law for the drag andtorque acting on the weakly permeable sphere is ob-tained in the form
D = −4πλ2e(A11 i + B11 j + A10k)α′
1, (73)
T = 8π
3λe(l − α)
(f1(λe)g2(λe) + f2(λe)g1(λe)
(l − α)g1(λe) − lλeg2(λe)
)
× (C11 i + D11 j + C10k)γ1. (74)
In addition, if we express the drag and torque in termsof basic flow, we get the formulae
D = 2πλ2eR[V0]0 − 2π[R − 3S][∇2V0]0, (75)
T = 4π(l − α)
(f1(λe)g2(λe) + f2(λe)g1(λe)
(l − α)g1(λe) − lλeg2(λe)
)
× [∇ × V0]0, (76)
where
R = lλ2eg1(λe) + λe(α + l)g2(λe)
(lλ2e − 3l − 3α)g1(λe) + λe(α + l)g2(λe)
,
S = (α + l){f1(λe)g2(λe) + f2(λe)g1(λe)}(lλ2
e − 3l − 3α)g1(λe) + λe(α + l)g2(λe).
If we take l → 0 (solid sphere) in these formulae,the expressions for drag and torque reduce to
D = 2π(λ2e + 3λe + 3)[V0]0
− 2π
(1 + 3
λe
+ 3
λ2e
− 3eλe
λ2e
)[∇2V0]0, (77)
T = 4π
[eλe
λe + 1
][∇ × V0]0, (78)
that are due to (Pozrikidis [39]).Now, expanding the formulae (75) and (76) up to
O(l2), we obtain
D = 2π
(λ2
e + 3λe + 3 − 3l
α(λe + 1)2 + O(l2)
)[V0]0
− 2π
(1 + 3
λe
+ 3
λ2e
− 3l
αλ2e
(λe + 1)2
− 3eλe
λ2e
{1 − l
α(λe + 1)
}+ O(l2)
)[∇2V0]0,
(79)
T = 4πeλe
(1 − λe + λ2
e + l
α(2λ3
e + 2λ2e + 3λe − 2)
+ O(l2)
)[∇ × V0]0. (80)
In case of uniform oscillatory flow past a weakly per-meable sphere, the expression for drag given in (73)reduces to
D = 2πλ2e
×[
lλ2eg1(λe) + λe(α + l)g2(λe)
(lλ2e − 3l − 3α)g1(λe) + λe(α + l)g2(λe)
]k.
(81)
In the case of weakly permeable sphere oscillat-ing with uniform translational velocity in an incom-pressible Newtonian fluid (which is at rest at in-finity), the pressure gradient at infinity is given byp∞ = −λ2
eU∞, where, U∞ corresponds to the mag-
nitude of uniform velocity at far field and hence thecorresponding contribution to the drag force wouldbe: −λ2
eVp , where Vp = 43π is the volume of the rigid
sphere with unit radius. Therefore, the correspondingexpression for the drag takes the form
D = 2πλ2e
[lλ2
eg1(λe) + λe(α + l)g2(λe)
(lλ2e − 3l − 3α)g1(λe) + λe(α + l)g2(λe)
− 2
3
]k, (82)
which after expansion up to O(l2) reduces to
D = 6π
[1 + λe + λ2
e
9− l
α(λe + 1)2 + O(l2)
]k. (83)
This formula agrees with that of Looker and Carnie[9]. It is to be noted that the expression for unsteadyforce exerted on the surface of the porous sphere inuniform oscillatory Stokes flow can be obtained bytaking the inverse Fourier transform of (83) as ob-tained by Looker and Carnie [9]. Similarly, the ex-pression for force exerted on the surface of the poroussphere in an arbitrary unsteady Stokes flow can be ob-tained by taking the inverse Fourier transform of (79),but the corresponding inversion appears to be compli-cated analytically.
1090 Meccanica (2012) 47:1079–1095
Fig. 1 Variation of drag with frequency in case of uniform flow
Fig. 2 Variation of drag with Darcy number in case of uniform flow
6 Results and discussion
The hydrodynamic force and torque acting on a poroussphere due to arbitrary oscillatory Stokes flow are de-rived and expressed in the form of Faxén’s law. Theseexpressions are verified against some of the existingresults in the literature, due to various limiting cases.Also, in order to display the potential of the present
method while handling arbitrary given basic flow, ex-amples like uniform flow, oscillatory shear flow andoscillating Stokeslet are discussed. The correspondingFaxén’s law in case of a weakly permeable sphere isalso derived.
Figure 1 shows the variation in magnitude of dragin case of uniform flow for several values of the Darcynumber (which is proportional to permeability) on fre-
Meccanica (2012) 47:1079–1095 1091
Fig. 3 Variation of interface velocity with Darcy number in case of uniform flow
Fig. 4 Variation of interface velocity with frequency in case of uniform flow
quency parameter. We consider the frequency of oscil-lation between 1 and 10 MHz as considered by Lookerand Carnie [9] and a2/ν = 10−6 s. It is seen that dragdecreases monotonically with increasing Darcy num-ber (Da). Increase in Darcy number reduces the re-sistance offered by the porous sphere resulting in re-duction of drag (Fig. 2). As frequency increases themagnitude of drag becomes larger, due to more resis-
tance. These results are consistent with the observa-tions made by Vainshtein et al. [19] that the Stokescorrection factor increases with increasing frequencyparameter due to enhanced energy dissipation. Varia-tion of interface velocity with Darcy number and fre-quency is shown in Figs. 3 and 4. It is seen in Fig. 3that for solid sphere the interface velocity is zero inorder to satisfy the no–slip condition and increases
1092 Meccanica (2012) 47:1079–1095
Fig. 5 Variation of internal tangential velocity with r/a in case of uniform flow at different frequencies
Fig. 6 Variation of internal tangential velocity with r/a in case of uniform flow at different Darcy number
with increasing Da. For a fixed Darcy number, theinterface velocity increases with frequency parame-ter (Fig. 4). Variation in magnitude of internal tan-gential velocity component against radial coordinateis shown with frequency and Darcy number, respec-tively (Figs. 5 and 6). It may be seen that for a fixedfrequency or Darcy number the magnitude of the tan-gential velocity is less towards the center of the poroussphere and increases as we move towards the surface
from inside. Increasing frequency and Darcy numberresults in an increase in tangential velocity with theradial distance. This is due to larger resistance en-countered by the flow while moving towards the cen-ter.
The magnitude of drag in case of oscillatory quad-ratic shear flow is plotted on frequency (�) for fixedshear rate, τ = 2.2 (Fig. 7). The qualitative behaviorwith Darcy number is same as in the case of uniform
Meccanica (2012) 47:1079–1095 1093
Fig. 7 Variation of drag with frequency in case of oscillatory quadratic shear flow
Fig. 8 Variation of torque with frequency in case of oscillatory linear/quadratic shear flow
flow. The magnitude of torque in case of oscillatorylinear/quadratic shear flow is plotted with frequency(Fig. 8) for fixed shear rate, τ = 2.2. The magnitude oftorque decreases with increase in permeability. Dragexpressions obtained in case of porous sphere and thatof weakly permeable sphere are compared for low per-meability (Fig. 9). As expected, at very small Darcynumber porous sphere behaves almost like a weakly
permeable sphere and hence experiences almost equalresistance.
7 Conclusion
The objective of the paper is to obtain Faxén’s lawfor an arbitrary oscillatory Stokes flow past a porous
1094 Meccanica (2012) 47:1079–1095
Fig. 9 Variation of drag with frequency for Brinkman sphere and weakly permeable sphere
sphere. We have assumed that the internal flow is gov-erned by the Brinkman equation and that the externalflow is governed by the continuity and unsteady Stokesequations. The solenoidal decomposition of velocityfield is used to obtain the velocity and pressure fields.Examples like uniform flow, oscillatory shear flow andoscillating Stokeslet are discussed. Also comparisonbetween porous sphere and weakly permeable spherein oscillatory flow has been done. The present modelcan easily be extended to study flow through an assem-blage of porous particles as done by Prakash et al. [46]in case of steady flow. However, this could be a futureinvestigation.
Acknowledgements One of the authors (JP) would like torecognize the support of Council of Scientific and Industrial Re-search (CSIR), India. The other author (GPRS) acknowledgesthe support of Alexander von Humboldt Foundation for the fel-lowship.
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