On hydrodynamic permeability of a membrane built up by porous deformed spheroidal particles

ISSN 1061�933X, Colloid Journal, 2013, Vol. 75, No. 5, pp. 611–622. © Pleiades Publishing, Ltd., 2013.

611

1 INTRODUCTION

The flow of fluids through a swarm of porous parti�cles has numerous applications in different branchesof engineering and sciences. The problem of uniformflow past and through the porous media has attractedseveral important applications, notably in the flow ofoil through porous rocks, extractions of energy fromthe geothermal regions, the filtration of solids fromliquids, flow of liquids through ion exchange beds,drug permeation through human skin, chemical reac�tors for economical separation or purification of mix�tures, the study of transport of nutrients from synovialfluid to cartilages in synovial joints, the study of dis�persion of cholesterol and other fat substances fromarteries to endothelium and so on. In addition toabove, the fluid flows through a porous media havebeen used with great success to predict flow character�istics in many physical problems. Due to vast applica�tions, several conceptual models have been developedfor describing fluid flow through porous media. Foreffective use of a porous medium in above areas, thestructure of porous layer should be viewed from allangles e.g. it is not necessary that the particles always

1 The article is published in the original.

have a smooth homogeneous surface but also have arough surface or a surface covered by porous shell. Forthe medium of high porosity, the sum suggested byBrinkman [1] is more suitable for describing the flowthrough the porous medium. He evaluated the viscousforce exerted by a flowing fluid on a dense swarm ofparticles by modifying Darcy’s equation for porousmedium, which is commonly known as Brinkmanequation.

The problem of flow through a swarm of particlesshall become complex, if we consider the solution ofthe flow field over the entire swarm by taking exactpositions of particles. In order to avoid the above com�plication, it is sufficient to obtain the analyticalexpression by considering the effect of the neighboringparticles on the flow field around a single particle ofthe swarm, which can be used to develop relativelysimple and reliable models for heat and mass transfer.This has lead to the development of particle�in�cellmodels. The cell model technique involves the con�cept that a random assemblage of particles can bedivided into a number of identical cells, one particleenveloped by each cell. Furthermore, the volume offluid cell is so chosen that the solid volume fraction inthe cell equals the solid volume fraction of the assem�

On Hydrodynamic Permeability of a Membrane Built up by Porous Deformed Spheroidal Particles1

Pramod Kumar Yadava, Satya Deob, Manoj Kumar Yadavc, and Anatoly Filippovd

a Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Allahabad�211004, Indiab Department of Mathematics, University of Allahabad, Patna�80000, Allahabad�211002, India

c Department of Mathematics, National Institute of Technology Patna, Patna�800005, Indiad Department of Higher Mathematics, Gubkin Russian State University of Oil and Gas,

119991 Leninsky Prosp., 65�1, Moscow, Russiae�mail: [email protected]; [email protected]

Received January 09, 2013

Abstract—This paper concerns the slow viscous flow of an incompressible fluid past a swarm of identicallyoriented porous deformed spheroidal particles, using particle�in�cell method. The Brinkman’s equation inthe porous region and the Stokes equation for clear fluid region in their stream function formulations areused. Explicit expressions are investigated for both the inside and outside flow fields to the first order in a smallparameter characterizing the deformation. The flow through the porous oblate spheroid is considered as theparticular case of the porous deformed spheroid. The hydrodynamic drag force experienced by a porousoblate spheroid and permeability of a membrane built up by porous oblate spheroids having parallel axis areevaluated. The dependence of the hydrodynamic drag force and the hydrodynamic permeability on particlevolume fraction, deformation parameter and viscosity of porous fluid are also discussed. Four known bound�ary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’sand Cunningham’s (Mehta–Morse’s condition). Some previous results for hydrodynamic drag force andhydrodynamic permeability have been verified. The model suggested can be used for evaluation of changinghydrodynamic permeability of a membrane under applying unidirectional loading in pressure�driven pro�cesses (reverse osmosis, nano�, ultra� and microfiltration).

DOI: 10.1134/S1061933X13050165

612

COLLOID JOURNAL Vol. 75 No. 5 2013

PRAMOD KUMAR YADAV et al.

blage. Thus, the entire disturbance due to each particleis confined within the cell of fluid with which it is asso�ciated.

Stokes flow in spheroidal particle�in�cell modelswith Happel and Kuwabara boundary conditions hasbeen studied by Dassios et al. [2]. Flow through bedsof porous particles was studied by Davis and Stone [3]and they evaluated the overall bed permeability ofswarm by using cell model. Datta and Deo [4] havesolved the problem of Stokes flow with slip and Kuwa�bara boundary conditions and evaluated the drag forceexperienced by a rigid oblate spheroid in a cell. Thedrag force experienced by a swarm of porous deformedoblate spheroidal particles was evaluated by Deo andYadav [5] by using Kuwabara boundary condition.Deo [6] had studied the problem of slow flow past aslightly deformed porous sphere by using Happel for�mulation and evaluated the drag force experienced bya porous oblate spheroid in a cell. Deo and Gupta [7]have also studied the problem of Stokes flow past aswarm of porous approximately spheroidal particleswith Kuwabara boundary condition and evaluatedsome important results for drag force. Stokes flow pasta porous spheroid embedded in another porousmedium were discussed by Yadav and Deo [8]. Theflow through the porous oblate spheroid embedded inanother porous medium is considered as the particularexample of the deformed porous sphere embedded inanother porous medium.

Epstein and Masliyah [9] proposed a useful gener�alization of the sphere�in�cell model by considering aspheroid�in�cell model for swarms of spheroidal parti�cles. However, they had to solve the creeping flowproblem numerically. Filippov et al. [10] evaluated thehydrodynamic permeability of membrane of porousspherical particles using Mehta�Morse condition onthe cell surface. Many early authors on convection inporous media used various types of extended Darcymodels e.g. Boutros, et al. [11], were discussed Lie�group method of solution for steady two dimensionalboundary�layer stagnation�point flow towards aheated stretching sheet placed in a porous medium,Radiation effect on forced convective flow and heattransfer over a porous plate in a porous medium wasstudied by Mukhopadhyay and Layek [12]. Happel[13, 14] proposed cell models in which both particleand outer envelope are spherical/cylindrical. Hesolved the problem when the inner sphere/cylinder issolid with respective boundary conditions on the cellsurface. The Happel model assumes uniform velocitycondition and no tangential stress at the cell surface.The merit of this formulation is that, it leads to an axi�ally symmetric flow that has a simple analytical solu�tion in closed form, and thus can be used for heat andmass transfer calculations. Analytical solutions of par�ticle�in�cell models discussed above are always practi�cally useful to many industrial problems, but solutionsof creeping flow for the above models have not beenfound in case of complex geometry. Kuwabara [15]

proposed a cell model in which he used the nil vorticitycondition on the cell surface to investigate the flowthrough swarm of spherical/cylindrical particles.However, Kuwabara formulation requires a smallexchange of mechanical energy with the environment.The mechanical power given by the sphere to the fluidis not all consumed by viscous dissipation in the fluidlayer.

Apart from this, Kvashnin [16] and Mehta–Morse[17] gave their respective boundary conditions for theouter cell surface. Kvashnin [16] proposed the condi�tion that the tangential component of velocity reachesa minimum at the cell surface with respect to radialdistance, signifying the symmetry on the cell. How�ever, Mehta–Morse [17] used Cunningham’s [18]approach by assuming the tangential velocity as acomponent of the fluid velocity, signifying the homo�geneity of the flow on the cell boundary. The impor�tance of the Mehta–Morse [17] boundary condition isthat since we are interested in the flow behavior on alarge scale, we shall average the flow variables on thesmall scale over a cell volume to obtain large scalebehavior. Creeping flow past a slightly deformedsphere was studied by Palaniappan [19] andRamkissoon [20] by using slip boundary condition atthe porous surface.

A Cartesian�tensor solution of Brinkman’s equa�tion (Brinkman [1]) for the porous medium was inves�tigated by Qin and Kaloni [21] and then using thissolution they evaluated the hydrodynamic drag forceexperienced by the porous sphere. Creeping flow pasta porous approximate sphere is studied by Sriniv�ascharya [22]. Uchida [23] proposed a cell model for asedimenting swarm of particles, considering sphericalparticle surrounded by a fluid envelope with cubicouter boundary. Pal and Mondal were discussed Radi�ation effects on combined convection over a verticalflat plate embedded in a porous medium of variableporosity [24].

Vasin and Filippov [25] evaluated the hydrody�namic permeability of membrane of porous sphericalparticles using Mehta–Morse condition on the cellsurface. Recently, Vasin et al. [26, 27] compared allfour cell models to evaluate the permeability of mem�brane of porous spherical particles with a permeableshell and discussed the effect of different parameterson the hydrodynamic permeability of the membranefor all the four above mentioned boundary conditions.Using the stream function formulation, Zlatanovski[28], obtained a convergent series solution to theBrinkman’s equation for the problem of creeping flowpast a porous prolate spheroidal particle and reducedthe solution for flow past a porous spherical particle asa limiting case of the spheroidal particle.

This paper concerns the problem of the slow vis�cous flow through a swarm of clusters of porous sphe�roidal particles. As boundary conditions, continuity ofvelocity, continuity of normal stress and shearing stressat the porous and fluid interface are employed. On the


ON HYDRODYNAMIC PERMEABILITY OF A MEMBRANE BUILT UP 613

hypothetical surface, uniform velocity and conse�quently each of four known boundary conditions areused alternately (proposed by Happel, Kuwabara,Kvashnin and Cunningham/Mehta–Morse). Thedrag force experienced by each porous oblate spheroi�dal particle in a cell is evaluated. As a particular case,the drag force experienced by a porous oblate spheroidin an unbounded medium, the solid oblate spheroid incell and the solid oblate spheroid in an unboundedmedium are also investigated. The earlier resultsreported for the drag force by Davis and Stone [3] forthe drag force experienced by a porous sphere in a cell,Happel [13] for a solid sphere in a cell and Qin andKaloni [21] for a porous sphere in an unboundedmedium have been then deduced. The dependence ofdimensionless hydrodynamic permeability on the par�ticle volume fraction γ, deformation parameter ε andviscosity of porous fluid are presented graphically anddiscussed.

STATEMENT AND MATHEMATICAL FORMULATION OF THE PROBLEM

In the mathematical model, we will represent a dis�perse system (membrane) by a periodic net of identicalporous deformed spheroidal particles each of them isenveloped by a concentric spheroid, named as cell sur�face (Fig. 1). Let us assume that these porous de�formed spheroidal particles are identical and their axesare parallel and hence the porous medium shall be�

come homogeneous and isotropic. The problems ofsuch kind are arising when extended in a plane porousmedium (for example thin membrane) which is con�sisted from spherical particles undergoes deformableloading in the same direction. Actually all membranesduring pressure�driven processes are working undersuch loading. Therefore our goal here is to take intoaccount how is the membrane permeability changedunder dynamic loading. Let us now consider that po�rous deformed spheroidal particles are stationary andsteady axisymmetric flow has been established around

and through it by uniform velocity directedalong the positive z�axis.

Let the surface of a spheroid which departs but alittle in shape from a sphere be

(1)

The equation of correspondingly outer cell surface,i.e., hypothetical surface be

(2)

Further, assuming that the coefficients βm is suffi�ciently small, so that squares and higher powers maybe neglected i.e.

(3)

where, n may be positive or negative.

( )U=U U� ��

pSr a=� �

1(1 ( )), (1 ).m mr a G a d= + β ζ = − ε� � �

HS

2(1 ( )), (1 ).m mr b G b d= + β ζ = − ε� �

�

( ) 1 ( ),n

m mr n Ga

≈ + β ζ

Cell surface

Porous deformedspheroid

d2 (1 – ε)

θ

d2

Vθz

U~

d1 (1 – ε)

Vr

U~

d1

Approaching fluid

Fig. 1. Physical model and co�ordinate system of the problem.

614



The flow in the outside region of the porousdeformed spheroid is governed by the Stokes equation[29] with continuity condition:

(4)

For the region inside the porous deformed spher�oid, we assume that the flow is governed by the Brink�man [1] equation and the continuity condition:

(5)

where o and i are superscript, which mark flow outsidethe porous deformed spheroid and inside the porousdeformed spheroid, respectively; over a symbols marks

dimensional values; and are velocity andpressure in the outside and inside of the porous region;

and are viscosities in the outside and inside of the

porous region; is the hydrodynamic resistance of theporous region, which is inversely proportional to thehydrodynamic permeability.

SOLUTION OF THE PROBLEM

By using the following dimensionless variables

(6)

The system of governing Eqs. (4), (5) in dimensionlessform become:

(7)

(8)

The stream function formulation of the above equa�tions (7) and (8) in spherical polar coordinates reduces to the following fourth order partial differen�tial equations, respectively as

(9)

(10)

where the operator

(11)

, 0.o o o op∇ = μ Δ ∇ ⋅ =v v� � �

� � � �

, 0,i i i i ip k∇ = μ Δ − ∇ ⋅ =v v v� � � �

� � � � �

,ov� iv� ,op� ip�

oµ�

iµ�

k�

2

2

2

2

,

2

, , , , ,

, , , ,

, , ,

.

1

o i

ob

i

o

oo i

b

pb r a a pa a p

sU ap s sa R

kkU a

Rk U a

r ∇ = ∇ Δ = Δ =

μ μ= = = λ =

λ μ

μλ = = =

μ

μ ψ= ψ =

= =γ

v

�

��

� �

� �

� � �

�

� ��

�

�

��

�

��

�

� �

��

�

��

�

o

oo o

v

and

, 11 ,0

o o

o

pr

⎧∇ = Δ⎪ ⎛ ⎞≤ ≤⎨ ⎜ ⎟γ⎝ ⎠⎪∇ ⋅ =⎩

v

v

( )2 2

0 ,1 .

0

i i i

i

p sr

⎧∇ = λ Δ −⎪≤⎨

⎪∇ ⋅ =⎩

v v

v

( , , )r θ ϕ

2 2( ) 0,oψ = E E

2 2 2( ) 0,is− ψ =E E

22 22

2 2 2

(1 ), cos .

r r

− ζ∂ ∂= + ζ = θ∂ ∂ζ

E

Furthermore, the non�vanishing velocity components and tangential and normal stresses [30] re�

spectively are given by

(12)

(13)

(14)

(15)

(16)

Also, the pressure may be obtained in both regions byintegrating the following relations, respectively:

(17)

(18)

In the case of axi�symmetric incompressible creepingflow, the regular solution of Stokes equation (9) on theaxis of symmetry can be expressed [29] as

(19)

The complete regular solution on symmetry axis ofBrinkman’s equation (10) may be expressed as

(20)

where are the stream functions, and being the Gegenbauer function of first kind and

related to the Legendre function of first kind bythe relation

(21)

In particular,

(22)

( , ,0)r θv v

1 1, ,2 sinsin

rr rr

θ

∂ψ ∂ψ= − =

∂θ θ ∂θ

v v

2 2 2

2 2 2

(1 )1 2( , ) ,sin

o o oor r

r r rr rζ

⎡ ⎤∂ ψ ∂ψ − ζ ∂ ψσ θ = − −⎢ ⎥θ ∂∂ ∂ζ⎣ ⎦

2 ,o

o o rrr p

r

∂σ = − +

∂

v

2 2 22

2 2 2

(1 )2( , ) ,sin

i i iir r

r r rr rζ

⎡ ⎤∂ ψ ∂ψ − ζ ∂ ψλσ θ = − −⎢ ⎥θ ∂∂ ∂ζ⎣ ⎦

22 .i

i i rrr p

r

∂σ = − + λ

∂

v

2 2

21 1( ), ( ),

sinsin

o oo op p

r rr

∂ ∂∂ ∂= − ψ = ψ

∂ ∂θ ∂θ θ∂θE E

22 2

02

22 2

0

( ) ,sin

1 ( ) .sin

ii i

r

ii i

ps

r r

ps

r r rθ

∂ λ ∂= − ψ −

∂ ∂θθ

∂ λ ∂= ψ −

∂θ θ∂

v

v

E

E

1 3 2

2

( , )

[ ] ( ).

o

n n n nn n n n n

n

r

A r B r C r D r G∞

− + − + +

=

ψ ζ =

= + + + ζ∑

1

2

( , )

* * * *[ ( ) ( )] ( ),

i

n nn n n n n n n

n

r

A r B r C y sr D y sr G∞

− +

−

=

ψ ζ =

= + + + ζ∑

sψ ' cosζ = θ

( )nG ζ

( )nP ζ

2( ) ( )( ) , 2.

(2 1)n n

nP P

G nn

−

ζ − ζζ = ≥

−

2 2 22 4

1 1( ) (1 ), ( ) (1 )(5 1).2 8

G Gζ = − ζ ζ = − ζ ζ −



Also, we have used the symbols

(23)

where, are modified Bessel functions of firstkind and of order [31].

In particulars, we have

(24)

(25)

Since, are singular at for hence we

may take for The same is true for term

multiplied by Therefore, we may now takethe stream function outside the porous deformedspheroidal particle as

(26)

while, in the interior of porous deformed spheroid wemay take it as

(27)

It may be noted here that the coefficients

and only contribute to the flow past a poroussphere and consequently we expect that all other coef�ficients in (26) and (27) are of order Therefore, ex�cept where these coefficients enter, we may take thesurface to be perfect sphere instead of either their ex�act forms (1) or (2).

Boundary Conditions

To match the solution at the porous body – liquidinterface ( ), we use along with the con�tinuity of velocity components, continuity of normalstress and continuity of shear stress as follows:

(28)

(29)

(30)

1( ) ( ), ,2 2

nry sr s sr nν

± ±ν

π= ν = −I

( )sr±ν

Iν

1 21( ) sinh( ), ( ) cosh( ) sinh( ).y sr sr y sr s sr srr

= = −

( )1

2

( ) cosh( ),

1( ) sinh( ) cosh( ).

y sr sr

y sr sr srr

−

−

=

= σ −

( )ny sr−

0r = 2,n ≥

* 0nC = 2.n ≥

*,nB 2.n ≥

2 422 2 2 2

1 3 2

( )

[ ] ( ),

o

n n n nn n n n n

n

ba r c r d r G

r

A r B r C r D r G∞

− + − + +

⎡ ⎤ψ = + + + ζ +⎢ ⎥⎣ ⎦

+ + + + ζ∑

22 2 2 2* *[ ( )] ( )

* *[ ( )] ( ).

i

nn n n n

n

a r d y sr G

A r D y sr G∞

ψ = + ζ +

+ + ζ∑

2,a 2,b 2,c

2,d 2*a 2

*d

.mβ

1 ( )m mr G= + β ζ

, ,o i o ir r θ θ= =v v v v

22 2 ,o i

o ir rp pr r

∂ ∂− + = − + λ

∂ ∂

v v

( , ) ( , ).i or rr rζ ζσ θ = σ θ

The continuity of the radial components of fluid ve�

locity on the outer cell implies:

(31)

According to Happel [13] the tangential stress vanish�es on the outer cell surface:

(32)

According to Kuwabara [15] the curl of velocity ( )vanishes on the outer cell surface:

(33)

According to Kvashnin [16] a symmetry condition isintroduced on the outer cell surface:

(34)

Mehta–Morse’s [17] assumes homogeneity of theflow on the outer cell surface:

(35)

Substituting the values of

and from above in Eqs. (28)–(31) and usingthe following identities

(36)

(37)

(38)

1(1 ( ))m mr G= + β ζγ

cos .or = θv

2 2 2

2 2 2

(1 )2( , ) 0 . . 0,

1(1 ( )).

o o oor

m m

rr rr r

r G

ζ

∂ ψ ∂ψ − ζ ∂ ψσ θ = − − =

∂∂ ∂ζ

= + β ζγ

i e

at

ov�

2 2 2

2 2 2

(1 )( ) 0 . . 0,

1(1 ( )).

o oo

m m

r r

r G

∂ ψ − ζ ∂ ψ= + =

∂ ∂ζ

= + β ζγ

v�rot i e

10, (1 ( )).o

m mr Grθ∂= = + β ζ

∂ γ

v

at

1sin , (1 ( )).om mr Gθ = − θ = + β ζ

γv at

,op ,ip 0,rv ,irv ,o

θv ,iθv ( , ),o

r rζσ θ

( , )ir rζσ θ

21

1' ( ) ( ) ( ),n n nny sr s y sr y sr

r−

−= +

21

2

2

1' ( ) ( ) ( ),

( 1)''( ) ( ),

n n n

n n

ny sr s y sr y srr

n ny sr s y sr

r

− − + −

−= +

−⎡ ⎤= +⎢ ⎥⎣ ⎦

2

2

( 1)'' ( ) ( ),n nn n

y sr s y srr

− −

−⎡ ⎤= +⎢ ⎥⎣ ⎦

2 2

2

( 2)( 3)( ) ( ) ( )

2(2 1)(2 3)

( 1) ( 1)( 2)( ) ( ),

(2 1)(2 3) 2(2 1)(2 1)

m m

m m

m mG G G

m m

m m m mG G

m m m m

−

+

− −ζ ζ = − ζ +

− −

− + ++ ζ − ζ

+ − − +

1 3

1 1

( 2)( ) ( ) ( )

(2 1)(2 3)

( 1)1 ( ) ( ),(2 1)(2 3) (2 1)(2 1)

m m

m m

mP G P

m m

mP P

m m m m

−

− +

−ζ ζ = ζ +

− −

++ ζ − ζ

+ − + −

616



we have five equations involving six arbitrary con�stants. Thus the values of arbitrary constants men�tioned above depends on the selection of the modelused, i.e., on the boundary condition (32), (33), (34)or (35). Solving the resulting equations by taking eachmodel respectively, we find the values of all non�van�

ishing coefficients

and which correspond to Therefore, we have determined the explicit expressionsfor the stream function for the flow inside and outside ofthe porous deformed sphere S, respectively as

(39)

(40)

APPLICATION TO A POROUS OBLATE SPHEROID

We consider an approximate porous oblate spher�oid as a particular case of the preceding analysis. Let usconsider that porous oblate spheroid is stationary andthe steady axi�symmetric flow has been established

around it by a uniform velocity directed inthe positive z�axis. The Cartesian equation of an oblatespheroid can be taken as

(41)

whose equatorial radius is in which ε is so small thatsquares and higher powers of it may be neglected. Itspolar equation can be written in the form as

(42)

Here, it may be mentioned that for the shape ofspheroid will be oblate, whereas, for the shapewill become prolate. Upon comparison with equation(1), we are led to the values Since

are all become zero and further us�ing (42), we find from equations (39) and (40) that the

2,a 2,b 2,c 2,d 2*,a 2

*,d ,nA ,nB ,nC ,nD*nA *

nD 2,n m= − ,m 2.m +

2 422 2 2 2

2 3 5 42 2 2

1 3 22

2 12 2

1 42 2 2

( )

[ ]

( ) [ ]

( ) [

] ( ),

o

m m m mm n m m

m m m mm m m m m

m mm m m

m mm m m

ba r c r d r G

r

A r B r C r D r

G A r B r C r D r

G A r B r

C r D r G

− − + − + − +

− − −

− + − + +

−

+ − −

+ +

− + +

+ + +

⎡ ⎤ψ = + + + ζ +⎢ ⎥⎣ ⎦

+ + + + ×

× ζ + + + + ×

× ζ + + +

+ + ζ

22 2 2 2

22 2 2 2

22 2 2 2

* *[ ( )] ( )

* *[ ( )] ( )

* *[ ( )] ( )

* *[ ( )] ( ).

i

mm m m m

mm m m m

mm m m m

a r d y sr G

A r D y sr G

A r D y sr G

A r D y sr G

−

− − − −

+

+ + + +

ψ = + ζ +

+ + ζ +

+ + ζ +

+ + ζ

( )U=U U� ��

2 2 2

2 2 21 1

1,(1 )

x y zd d

++ =

− ε

1,d

2(1 2 ( )).r a G= + ε ζ� �

0,ε >

0ε <

2,m = 2 .mβ = ε 0,A

0,B 0,C 0,D 0*,A 0

*D

stream functions around and through the porous ob�late spheroid are

(43)

(44)

RESULTS AND DISCUSSION

The drag force experienced by the porous oblatespheroid in a cell can be evaluated by using the simpleelegant formula [29] as

(45)

Here, since and

(46)

Inserting these values in (45) and integrating, we get

. (47)

Here, it is interesting to note that only Stokes coeffi�cients and of the stream function contribute tothe drag force.

Hydrodynamic permeability of a membrane isdefined as the ratio of the uniform flow rate to the

cell gradient pressure [23]:

(48)

where, being the volume of the cell.

Substituting the value of from equation (47) and

the value of from above equation in (48), we get

(49)

where, being the dimension�

less hydrodynamic permeability of a membrane. The

ratio of drag force to the

Stokes force will be

(50)

2 422 2 2 2

2 1 42 2 2 2 2

4 3 1 64 4 4 4 4

( )

[ ] ( )

[ ] ( ),

o ba r c r d r G

r

A r B r C r D r G

A r B r C r D r G

−

− −

⎡ ⎤ψ = + + + ζ +⎢ ⎥⎣ ⎦

+ + + + ζ +

+ + + + ζ

22 2 2 2

2 42 2 2 2 4 4 4 4

* *[ ( )] ( )

* * * *[ ( )] ( ) [ ( )] ( ).

i a r d y sr G

A r D y sr G A r D y sr G

ψ = + ζ +

+ + ζ + + ζ

23

2

0

.o

oF Ua rdr

π

⎛ ⎞ψ∂= πμ ϖ θ⎜ ⎟∂ ϖ⎝ ⎠∫� �

� �

E

sinrϖ = θ

( ){ }{ }

2 22 2 2 2 2

44 4 43

12 ( ) 5( ) ( )

15 9 ( ) .

o c C d D r Gr

C D r Gr

⎡ψ = − + − + ζ +⎢⎣⎤+ − ζ ⎥⎦

E

1 2 24 [(1 ) ]oF d U c C= πμ − ε +� �

�

2c 2C

11L�

U�

F V� �

11 ,ULF V

=

�

�

� �

22

43

V d b= π� �

F�

V�

2 21 1

11 1132 2

(1 )1 ,{(1 ) }3 o o

d dL L

c C− ε

= =− ε +γ μ μ

�

� �

11 32 2

(1 )1{(1 ) }3

Lc C− ε

=− ε +γ

Ω 1 2 24 [(1 ) ]oF d U c C= πμ − ε +� �

�

16 oSF d U= π μ� �

�

2 22[(1 ) ].3

c CΩ = − ε +



The variation of natural logarithm of dimensionless

hydrodynamic permeability, of a membrane with

parameter (penetrability) is presented in Fig. 2 forall four models at and It is evi�dent from the figure that, when the penetrability de�

creases (i.e. increases) the membrane hydrodynam�ic permeability decreases.

Figure 3 shows the dependence of dimensionlesshydrodynamic permeability of membrane on viscosityratio at and The dimension�less hydrodynamic permeability of membrane de�creases with viscosity ratio, i.e., the flow throughmembrane becomes difficult with the increase of innerviscosity.

The dependence of dimensionless hydrodynamic

permeability, of a membrane on γ is presented inFig. 4 for all four models at and This shows that the hydrodynamic permeability of amembrane decreases with increasing volume fraction

γ. The rate of decrease of is higher for low values ofparticle volume fraction γ. Like the case of sphericaland cylindrical geometry, all four cell models agree onlow particle volume fraction. The nature of the Figs. 2,3 and 4 agrees with previously established results of Va�sin et al. [26].

Figure 5 shows the dependence of dimensionlesshydrodynamic permeability of membrane on defor�mation parameter for all models at and

The dimensionless hydrodynamic permeabilityof a membrane decreases with increasing deformationparameter ε.

11,L

0s0.8,γ = 1λ = 0.05.ε =

0s

λ 0.8,γ = 5s = 0.1.ε =

11,L4,s = 2λ = 0.1.ε =

11L

ε 0.6,γ = 5s =

2.λ =

POROUS OBLATE SPHEROID IN AN UNBOUNDED MEDIUM

When i.e., the physical problemscorresponds to the porous oblate spheroid in an un�bounded medium. In this case, the values of dimen�sionless hydrodynamic drag force and drag force ratio

respectively comes out as:

(51)

(52)

where

b → ∞� 0,γ →

,Ω

14 ,oF d UX= πμ� �

�

2 ,3

XΩ =

2 4 2

2 2

2 2 2 2 2

2 6

2 4 4

3 {2 [ 5(108 18( 12 )

(108 ) 442 5

( 108 18(12 5 ) 3(36 3

(6 ) )

8 3 )

s s s

X s s

s s s

s

s

= λ − + − + λ +

+ − + + λ−

ε − + + λ +

+

++ − λ +

–3.0

–4.0

–4.5

–5.0

80604020

lnLH

s0

–2.5

–3.5

1234

Fig. 2. Variation of natural logarithm of the dimensionlesshydrodynamic permeability, of a membrane built up aporous spheroidal particle, with the parameter s for theHappel (1), Mehta–Morse (2), Kvashnin (3) and Kuwa�bara (4) models; and

11L

0.8,γ = 1λ = 0 .0 5 .ε =

–3.5

–4.0

–4.5

–5.0

35252015105 30

1234

lnLH

λ

Fig. 3. Variation of natural logarithm of the dimensionlesshydrodynamic permeability, of a membrane built up bya porous spheroidal particle, with the parameter λ for theHappel (1), Mehta–Morse (2), Kvashnin (3) and Kuwa�bara (4) models; and

11L

0.8,γ = 5s = 0.1.ε =

12

10

0

4

0.50.40.30.20.1 0.6

1234

lnLH

γ

6

2

–2

Fig. 4. Variation of natural logarithm of the dimensionlesshydrodynamic permeability, of a membrane built upporous spheroidal particles with the parameter γ for theHappel (1), Mehta–Morse (2), Kvashnin (3) and Kuwa�bara (4) models; and

11,L

4,s = 2λ = 0.1.ε =

618



(53)

The effect of deformation parameter ε is discussedon the ratio of drag forces acting on porous oblatespheroid over a solid sphere (Stokes force). As evident,for all cases the quantity For higher values of

viscosity ratio λ, the drag force ratio Ω approaches toone ( ), i.e., as the inner viscosity increases, Ωapproaches to one (Fig. 6).

The drag force ratio Ω increases rapidly with in�creasing and for the growth become al�most steady (Fig. 7). An interesting observation is thatas deformation parameter increases, Ω decreases,signifying a relatively higher drag force on particles ofspherical geometry (Fig. 8).

SOLID OBLATE SPHEROID IN A CELL

When, then the physical problems corre�sponds to the solid oblate spheroid in a cell. In thiscase, the values of dimensionless hydrodynamic perme�ability of a membrane for all the models are given as:

The Happel’s model:

(54)

Kuwabara’s model:

(55)

Kvashnin’s model:

(56)

Mehta–Morse’s model:

(57)

SOLID OBLATE SPHEROID IN AN UNBOUNDED MEDIUM

When and i.e., then the po�rous oblate spheroid reduces to a solid oblate spheroidin an unbounded medium. In this case, the value of

Hydrodynamics drag force experienced by solid ob�late spheroid and dimensionless hydrodynamic dragforce ratio respectively are

(58)

(59)

Expression (58) agrees with the result of Palaniappan[19], Ramkissoon [20] and Datta and Deo [4] for theflow past a rigid spheroid in an unbounded clear fluid.

POROUS SPHERE IN A CELL

When, then porous oblate spheroid reducesto a perfect porous sphere in cell.

Therefore, the values of dimensionless hydrody�namic permeability of a membrane for all the modelsare given as:


2 2 6

4 2 2 2 2 4

2 2 2 4 4 2 6

2 2 2

4 4 2 2 2 6 2 2

4 (6 ) )]( cosh( ) sinh( ))sinh( )

4 (5( 1 )( 9 3( 3 ) 2 )

(9 3(6 ) (9 5 2 ) 2 ))

(sinh( )) (5(324 648 3( 108 24

7 ) 2 (6 ) ) ( 324 216(3 2 )

( 324 50

s s s s s s

s s s

s s s s

s s

s s s s

+ + λ − +

+ − + λ − − − + λ + λ + ε

+ − + λ + + + λ − λ ×

× + − λ − − +

+ λ + + λ − ε − + + λ +

+ −

+

−2 4 6 4 2 2 2 64 27 8 ) 2 (6 ) ))s s s s s− + λ + + λ ×

2 2 2

2 2 4

2 2 2 2 4 2

( cosh( ) sinh( )) )} 5{[54 9( 6 )

2 (6 ) (]

[ ]

cosh( ) sinh( ))

2 9 3( 3 ) 2 sinh( )} .

s s s s

s s s s s

s s s s

× − + + − + λ −

− + λ − +

+ − − − + λ + λ

1.Ω <

1Ω ≈

( 20)s s ≤ 20s >

ε

,s → ∞

4 2 2 2

3 5 3 4115( 1 ) (1 ) (2 2 ) ( 1 )

3 (3 2 )( 5( 2 2 ) ( 2 (5 3 (4 (3 2 ))).

) )L − + γ + γ + γ + γ − + ε

γ + γ − − − γ + γ + γ + − + γ + γ + γ + γ ε

= −

4 2

3112( 1 ) 5 [6 (3 )]} ( 1 )

9 5( 1 ){5 [6 (3 )]} { 5 [8 7 (3 (2 ))]}

{.

}{L

− + γ + γ + γ + γ − + ε

γ − − + γ + γ + γ + γ + − + γ + γ + γ + γ ε

= −

4 2

3 5115( 1 ) 16 [21 (15 8 )]} ( 1 )

.18 (4 ){ 5 [ 5 (6 (7 8 ))] 16( 5 ) [31 (78 (55 32 ))]

{

}L

− + γ + γ + γ + γ − + ε

−

γ + γ − γ − + γ + γ + γ − − + ε + γ + γ + γ + γ ε

=

211

3

3 3 4

5( 1 ) 4 (7 4 )]( 1 ).

18 (1 )( 5 )

[L

− + γ + γ + γ − + ε

−

γ + γ + γ + γ + γ − + ε

=

s → ∞ b → ∞� 0,γ →

F�

,Ω

( )16 1 ,5

oF Ud ε= πμ −� �

�

1 .5

Ω =ε

−

0,ε =



(60)

Kuwabara’s model:

(61)

Kvashnin’s model:

(62)


(63)

Expressions (60)–(63) agree with the results of Vasinet al. [26].

When then the value of drag force experi�enced by a porous sphere of radius for all models arecomes out as:


(64)

where,

(65)

This agrees with the result of Davis and Stone [3] forthe drag force experienced by a porous sphere in a cell. Kuwabara’s model:

(66)

4 2 2 3 211

5 2 5 2

5 6 4 2 2 3

{ { (6 ) ( 1 ) (1 )(2 2 )

54( 1 ) 3 [18 12 ( 3 6

8 6 )]}cosh( ) 3{ (2 )( 1 )

L s s s

s

s s s

= −λ + − + γ + γ + γ + γ +

+ − + γ + λ + γ + − + γ +

+ γ − γ + λ + − + γ ×

2 2 5

2 5 4 5 6

2 4

(1 )(2 2 ) 6(3 )( 1 )

[ 6(3 2 ) 2 (1 )

3 [ 1 2 ( 1 ( 2 ) )]]}

s

s

s

× + γ + γ + γ − + − + γ +

+ λ − + γ + − γ − γ + γ +

+ − + γ − + − + γ γ ×

2 2 3 5

2 2 5 2

5 2 2 5

sinh( )} {3 { [18( 1 )

(6 )(3 2 )]cosh( ) 3[2(3 )

( 1 ) (2 ) (3 2 )]sinh( )}},

s s s

s s s

s s

× λ γ − + γ +

+ λ + + γ − + ×

× − + γ + λ + + γ

4 2 2 311

2 3

2 2 6

{ [270 2 (6 )( 1 )

(5 6 3 )

9 [ 30 (5 12 2 )]]cosh( )

L s s s

s s

= − + λ + − + γ ×

× + γ + γ + γ +

+ λ − + − γ + γ +

2 4 2 2 3

2 3 2 4 6

[90(3 ) 6 (2 )( 1 )

(5 6 3 ) 3 [ 90 2 (5 6 )

s s s

s

+ + + λ + − + γ ×

× + γ + γ + γ + λ − + − γ + γ +

2 6 3

3 2 3 2

2

3 ( 5 12 2 )]]sinh( )} {45 [ [ 6

(6 )]cosh( ) [3 (2 )

2 (3 )]sinh( )]},

s s s s s

s s s s s

s s s

+ − − γ + γ λ γ − λ +

+ λ + − λ + −

− λ +

5 4 2 2 311

2 5 2 4

{ {54( 8 3 ) (6 )( 1 )

[16 (21 (15 8 ))]

18 [ 6(4 ) (4 ( 9 4( 1 ) ))]}

L s s s

s

= − + γ − λ + − + γ ×

× + γ + γ + γ −

− λ − + γ + + γ − + − + γ γ ×

2 5

2 5 2 5 6

4 5 6 4 2 2 3

cosh( ) 3{ 6(3 )( 8 3 )

2 [ 18(4 ) 3 ( 4 9 6 4 )

(8 9 3 4 )] (2 )( 1 )

s s

s

s s s

× + − + − + γ +

+ λ − + γ + − − γ − γ + γ +

+ − γ − γ + γ + λ + − + γ ×

2 2 3

5 2 2 5 2

2 5 2 5

[16 (21 (15 8 ))]}sinh( )} {18

{ [ 24 9 (6 )(4 )]cosh( ) [3

(2 )(4 ) (3 )( 8 3 )] sinh( )}},

s s

s s s

s s s

× + γ + γ + γ λ γ ×

× − + γ + λ + + γ − λ ×

× + + γ + + − + γ

5 4 2 2 411

2 5 2

4

{ {54(2 3 ) (6 )( 1 )

(4 (7 4 )) 18 [6 6

[ 1 (3 2( 2 ) ))]]}cosh( )

L s s s

s

s

= + γ − λ + − + γ ×

× + γ + γ − λ − γ + ×

× − + γ + − + γ γ +

2 5 4 2 2 4

2 5 4

5 6 2 4

3{ 6 (3 )(2 3 ) (2 )( 1 )

(4 (7 4 )) 2 [18 18 ( 2 3

3 2 ) 3 [1 (3 2( 3 ) )]]}

s s s

s

s

+ − + + γ + λ + − + γ ×

× + γ + γ + λ − γ + − + γ −

− γ + γ + + γ + − + γ γ ×

2 2 3 5 2 2 5

2 2 5

2 5

sinh( )} {18 [ [6 9 (6 )( 1 )]

cosh( ) [3 (2 )( 1 )

(3 )(2 3 )]sinh( )]}.

s s s s

s s

s s

× λ γ + γ + λ + − + γ ×

× − λ + − + γ +

+ + + γ

1,λ =

a�

2 2 5 5 2 2 5 5 1

4

4 [(3 2 30 )cosh( ) 3( 4 10 ) sinh( )],

aU s s s s s s sF

−

πμ + γ + γ − + γ + γ=

Δ

�

�

�

2 2 2 5 2 64

5 6 2 5

2 2 5 2 6

5 6 2 5 1

[2 3 3 2 3

42 30 90 ]cosh( )

[ 3 15 12 3

72 30 90 ] sinh( ).

s s s s

s s

s s s

s s s

−

− −

Δ = − γ + γ − γ + +

+ γ − γ + γ −

− − γ + γ − γ + +

+ γ − γ + γ

,[

a s U s s sF

s s s s s

π μ −=

− + γ − γ + γ − − γ + γ + + γ − γ − γ − γ + γ

�

� �

�

2

2 3 6 3 6 3 6 2 3 6

60 (sinh( ) cosh( ) )

2 ( 5 9 5 )+15( 1 4 2 )]cosh( ) 3[5 20 10 2 (3 5 2 )]sinh( )

620



Kvashnin’s model:

(67)


2 5 2 5

5 2 5 5

6 { [(15 (4 )]cosh( )

[15 (4 6 )]sinh( )} { [ 270

F s s s s

s s s

= γ + + γ −

− γ + + γ − γ +

�

4 3 2 3

2 3 5 6

( 1 ) (16 21 15 8 )

6 ( 4 10 21 20 )]cosh( )

s

s s

+ − + γ + γ + γ + γ +

+ − − γ − γ + γ +

5 2 3 5 6

4 3 5 6

3[90 (8 20 72 40 )

( 9 10 15 16 )]sinh( )},

s

s s

+ γ + + γ + γ − γ +

+ − γ + γ + γ − γ

2 5 2 5

5 2 5 5

6 { [15 ( 1 )]cosh( )

[ 15 (1 6 )]sinh( )} { [ 270

F s s s s

s s s

= γ + − + γ +

+ − γ + − γ − γ +

�

(68)

POROUS SPHERE IN AN UNBOUNDED MEDIUM

When, i.e., and then porousspheroid will reduce to a perfect porous sphere in anunbounded medium. In this case, the values of hydro�dynamic drag force and dimensionless hydrodynamicdrag force ratio are:

4 4 2

2 3 5 6

( 1 ) (4 7 4 )

6 (1 10 21 10 )]cosh( )

s

s s

+ − + γ + γ + γ +

+ + γ − γ + γ −

5 4 3 2

2 3 5 6

3[ 90 (�1 ) (3 9 8 )

(2 20 72 20 )] sinh( )}.

s

s s

− − γ + + γ γ + γ + γ +

+ + γ − γ + γ

b → ∞� 0,γ → 0,ε =

Ω

–1.5

–1.6

–1.7

–1.9

0.080.060.040.02 0.10

1234

lnLH

ε

–1.8

–2.0

–1.4

Fig. 5. Variation of natural logarithm of the dimensionlesshydrodynamic permeability, of a membrane built upporous spheroidal particles with the deformation parame�ter ε for the Happel (1), Mehta–Morse (2), Kvashnin (3)and Kuwabara (4) models; and

11,L

0.6,γ = 5s = 2 .λ =

0.9896

0.9892

0.9890

100604020

Ω

λ

0.9898

0.9894

1

2

3

80

0.9900

Fig. 6. Variation of the dimensionless drag force ratio with the parameter λ for different values of s: 1 – s = 5, 2 –10, 3 – 15; ε = 0.05.

Ω

0.990

0.985

0.980

100604020

Ω

s

0.995

1

2

3

80

Fig. 7. Variation of the dimensionless drag force ratio with the parameter s for different values of ε: 1 – ε = 0.01,2 – 0.03, 3 – 0.06 ; λ = 5.

Ω

0.97

0.95

0.94

0.250.150.100.05

Ω

ε

0.98

0.96

1

2

3

0.20

0.99

0.300

Fig. 8. Variation of the dimensionless drag force ratio with the parameter ε different values of λ: 1 – λ = 5, 2 –10, 3 – 15 ; s = 5.

Ω



(69)

(70)

where

(71)

When then the value of hydrodynamics dragforce experienced by the porous sphere of radius comes out as

(72)

and the value of dimensionless hydrodynamic dragforce ratio

(73)

These results agree with the previously establishedresults of Vasin and Filippov [27].

PERFECT SOLID SPHERE IN A CELL

When, and then the porous oblatespheroid reduces to a solid sphere in cell. In this case,the expressions for dimensionless hydrodynamic per�meability of a membrane for all the models are comeout as:

Happel’s model:

(74)

which agrees with the expression in Ref. [13];Kuwabara’s model:

(75)

which agrees with the expression in Ref. [15];Kvashnin’s model:

(76)

which agrees with the expression in Ref. [16].Mehta–Morse’s model:

(77)

which agrees with the expression in Ref. [17].

PERFECT SOLID SPHERE IN AN UNBOUNDED MEDIUM

When, i.e., and thenthe porous oblate spheroid reduces to a solid sphere inan unbounded medium. In this case, the value of dragforce experienced by the solid sphere comes out as

(78)

A well�known result for the drag reported earlier byStokes [32] for flow past a solid sphere in anunbounded medium.

CONCLUSIONS

The dependence of hydrodynamic permeability on characteristics of the internal structure of the po�rous spheroidal particle viscosity ratio of porousand fluid medium, particle volume fraction and de�formation parameter are presented graphically anddiscussed. From this analysis, we conclude that the de�pendencies of dimensionless hydrodynamic perme�ability on various parameters for all four cell mod�els are extremely close to each other except for theMehta–Morse boundary condition for which the val�ue of hydrodynamic permeability is significantly lowerthan the other models, signifying its non�suitability tomodel the hydrodynamic permeability of a mem�brane. So only three models suggested here can beused for evaluation of changing hydrodynamic perme�ability of a membrane under applying unidirectionalloading in pressure�driven processes (reverse osmosis,nano�, ultra� and microfiltration). The problems con�sidered here are arising when extended in a plane po�rous medium (for example thin membrane), which isconsisted from spherical particles, experiences de�formable loading in the same direction. In fact allmembranes during pressure�driven processes areworking under such loading. Moreover cake or gel�layer which is often formed on the membrane surfacefrom retarded particles also undergoes deformationand changes their permeability during filtration pro�cess. Therefore, for the first time, the cell models, sug�gested in the present paper, take into account how isthe membrane system permeability changed under dy�namic loading.

ACKNOWLEDGEMENT

Anatoly Filippov’s research was supported by theRussian Foundation for Basic Research, projectno. 11�08�01043.

2 2 2 2

2 2 2

12 { [ 6 (6 )]cosh( )

[ 3 (2 ) 2(3 )]sinh( )} { [ 54

oF Ua s s s s

s s s s

= πμ λ − + λ + +

+ − λ + + + − +

� �

� �

2 2 2 2(54 ( 9 2 (6 )))]cosh( )s s s+ λ + − + λ + −

2 2 2 2

2 4

3[ 6(3 ) (18 (3 4 )

2( 1 ) )]sinh( )},

s s

s s

− − + + λ + + λ +

+ − + λ

1

29 11 ,

2(1 )2 Qs

−

⎡ ⎤Ω = + +⎢ ⎥− λ − λλ⎣ ⎦

( )1

2

tanh( )31 1 .s

Qss

−

= + − −

1,λ =

a�

1

2 2

2

36 1 ,2

sinh( ) ( cosh( ) sinh( )),

oF aUs

s s s s

−

⎡ ⎤= πμ + λ +⎢ ⎥⎣ ⎦λ = −

� �

� �

Ω

( )11

2

tanh( ) 31 .2

ss s

−

−⎡ ⎤Ω = − +⎢ ⎥

⎣ ⎦

s → ∞ 0,ε =

6 5

11 8 3

2 3 3 2,

6 9L

− γ + γ − γ +

=

γ + γ

6 3

11 3

2( 5 9 5,

45L

− γ − γ + γ −

=

γ

3 3 2

11 8 3

( 1) (8 15 21 16,

18 72L

− γ − γ + γ + γ +

=

γ + γ

3 2

11 3 4 3 2

(1 ) (4 7 4),

18 ( 1)L

− γ γ + γ +

=

γ γ + γ + γ + γ +

,s → ∞ b → ∞� 0,γ → 0,ε =

F

6 .oF aU= πμ� �

� �

11L

,so λ

γ

ε

11L

622



REFERENCES

1. Brinkman, H.C., J. Appl. Sci. Res. A, 1947, vol. 1, p. 27.

2. Dassios, G., Hadjinicolaou, M., Coutelieris, F.A., andPayatakes, A.C., Int. J. Engng. Sci., 1995, vol. 33,p. 1465.

3. Davis, R.H. and Stone, H.A., Chem. Engng. Sci., 1993,vol. 48, p. 3993.

4. Datta, S. and Deo, S., Proc. Ind. Acad. Sci. (Math.Sci.), 2002, vol. 112, p. 463.

5. Deo, S. and Yadav, P.K., Bull. Cal. Math. Soc., 2008,vol. 100, p. 617.

6. Deo, S., J. Porous Media, 2009, vol. 12, p. 347.

7. Deo, S. and Gupta, B.R., Acta Mechanica, 2009,vol. 203, p. 241.

8. Yadav, P.K. and Deo, S., Meccanica, 2011, DOI10.1007/s11012�011�9533�y.

9. Epstein, N. and Masliyah, J.H., Chem. Engng. J. 1972,vol. 3, p. 169.

10. Filippov, A.N., Vasin, S.I., and Starov, V.M., ColloidsSurf. A, 2006, vol. 282–283, p. 272.

11. Boutros, Y.Z., Abd�el�Malek, M.B., Badran, N.A.,and Hassan, H.S., Meccanica, 2006, vol. 41, p. 681.

12. Mukhopadhyay, S. and Layek, G.C., Meccanica, 2009,vol. 44, p. 587.

13. Happel, J., A. I. Ch. E., 1958, vol. 4, p. 197.

14. Happel, J., A. I. Ch. E., 1959, vol. 5(2), p. 174.

15. Kuwabara, S., J. Phys. Soc. Japan, 1959, vol. 14, p. 527.

16. Kvashnin, A.G., Fluid Dynamics, 1979, vol. 14, p. 598.

17. Mehta, G.D. and Morse, T.F., J. Chem. Phys., 1975,vol. 63, p. 1878.

18. Cunningham, E., Proc. R. Soc. Lond. A, 1910, vol. 83,p. 357.

19. Palaniappan, D., Z. Angew. Math. Physik, 1994, vol. 45,p. 832.

20. Ramkissoon, H., Acta Mechanica, 1997, vol. 123,p. 227.

21. Qin, Yu. and Kaloni, P.N., J. Engng. Math. 1988,vol. 22, p. 177.

22. Srinivasacharya, D., Z. Angew. Math. Mech., 2003,vol. 83, p. 1.

23. Uchida, S., Int. Sci. Technol. Univ. Tokyo, 1954, vol. 3,p. 97 (Ind. Engng. Chem., vol. 46, p. 1194).

24. Pal, D. and Mondal, H., Meccanica, 2009, vol. 44,p. 133.

25. Vasin, S.I. and Filippov, A.N., Colloid J., 2004, vol. 66,p. 266.

26. Vasin, S.I., Filippov, A.N., and Starov, V.M., Adv. Col�loid Interface Sci., 2008, vol. 139, p. 83.

27. Vasin, S.I. and Filippov, A.N., Colloid J., 2009, vol. 71,p. 31.

28. Zlatanovski, T., Quart. J. Mech. Appl. Math., 1999,vol. 52, p. 111.

29. Happel, J., and Brenner, H., Low Reynolds Number Hy�drodynamics. Dordrecht: Kluwer, 1991.

30. Langlois, W.E., Slow Viscous Flow. New York: Macmill�an, 1964.

31. Abramowitz, M. and Stegun, I.A., Handbook of Mathe�matical Function. New York: Dover Publications, 1970.

On hydrodynamic permeability of a membrane built up by porous deformed spheroidal particles

Documents

Transcript of On hydrodynamic permeability of a membrane built up by porous deformed spheroidal particles