Hydrodynamic Permeability of Biporous Membrane

ISSN 1061�933X, Colloid Journal, 2013, Vol. 75, No. 4, pp. 473–482. © Pleiades Publishing, Ltd., 2013.

473

1 INTRODUCTION

Flow through porous media has been a topic oflongstanding interest for researchers due to its numer�ous applications in bio�mechanics, physical sciencesand chemical engineering etc. For effective use of aporous medium in the above areas, the structure ofporous layer should be viewed from all angles e.g. it isnot necessary that the particles always have a smoothhomogeneous surface but also have a rough surface ora surface covered by porous shell. For the medium ofhigh porosity, the sum suggested by Brinkman [1] ismore suitable for describing the flow through theporous medium. He evaluated the viscous forceexerted by a flowing fluid on a dense swarm of particlesby modifying Darcy’s equation for porous medium,which is commonly known as Brinkman equation. Inmany technical and technological problems that ariseduring the study of the permeability of aqueous con�glomerates composed primarily of one size particles, itis important to determine the regularities of its varia�tions upon the addition of a certain amount of parti�cles with quite different characteristic sizes. Theseproblems include the determination of the permeabil�ity of sugar syrups on the growth of large crystals andthe formation of clusters [2], finding the permeabilityof forming ion�exchange membranes during the vari�ations in their structural composition, the determina�

1 The article is published in the original.

tion of the permeability of liquid concrete upon theaddition of large gravel, etc.

The problem of flow through a swarm of particlesbecome complex, if we consider the solution of theflow field over the entire swarm by taking exact posi�tions of particles. In order to avoid the above compli�cation, it is sufficient to obtain the analytical expres�sion by considering the effects of the neighboring par�ticles on the flow field around a single particle of theswarm, which can be used to develop relatively simpleand reliable models for heat and mass transfer. Thishas lead to the development of particle�in�cell models.

Uchida [3] proposed a cell model for a sedimentingswarm of particles, considering spherical particle sur�rounded by a fluid envelope with cubic outer bound�ary. Happel [4, 5] proposed cell models in which theparticle and outer envelope, both are spherical/cylin�drical. He solved the problem when the innersphere/cylinder is solid with respective boundary con�ditions on the cell surface. The Happel model assumesuniform velocity condition and no tangential stress atthe cell surface. The merit of this formulation is that,it leads to an axially symmetric flow that has a simpleanalytical solution in closed form, and thus can beused for heat and mass transfer calculations. Analyti�cal solutions of particle�in�cell models discussedabove are always practically useful to many industrialproblems, but the solutions of creeping flow for theabove models have not been found in case of complexgeometry. Kuwabara [6] proposed again a cell model

Hydrodynamic Permeability of Biporous Membrane1 Pramod Kumar Yadava, Ashish Tiwarib, Satya Deoc, Manoj Kumar Yadava, Anatoly Filippovd,

Sergey Vasind, and Elena Sheryshevae

a Department of Mathematics, National Institute of Technology Patna, Patna�800005 (Bihar), Indiab Department of Mathematics, Birla Institute of Technology and Science, Pilani�333031, (Rajasthan), India

c Department of Mathematics, University of Allahabad, Allahabad�211002 (U.P.), Indiad Department of Higher Mathematics, Gubkin Russian State University of Oil and Gas,

Leninskii pr. 65–1, Moscow, 119991 Russiae Frumkin Institute of Physical Chemistry and Electrochemistry RAS, Leninskii pr. 31, Moscow, 119991 Russia

e�mail: a.filippov@mtu�net.ruReceived September 12, 2012

Abstract—This paper concerns the hydrodynamic permeability of biporous medium built up by porous cylin�drical particles located in another porous medium by using cell model technique. It is continuation of the pre�vious work of authors where biporous membrane was built up by porous spherical particles embedded inaccompanying porous medium. Four known boundary conditions, namely, Happel’s, Kuwabar;s, Kvashnin;sand Cunningham/Mehta�Morse;s, are considered on the outer surface of the cell. The variation of hydrody�namic permeability of biporous medium (membrane) with viscosity ratio, Brinkman constants, and solidfraction are presented and discussed graphically. Comparison of the resulting hydrodynamic permeability isundertaken. Some previous results for dimensionless hydrodynamic permeability have been verified.

DOI: 10.1134/S1061933X13040182

474

COLLOID JOURNAL Vol. 75 No. 4 2013

PRAMOD KUMAR YADAV et al.

in which he used the nil vorticity condition on the cellsurface to investigate the flow through swarm of spher�ical/cylindrical particles. However, Kuwabara formu�lation requires a small exchange of mechanical energywith the environment. The mechanical power given bythe sphere to the fluid is not all consumed by viscousdissipation in the fluid layer. Apart from this, Kvashnin[7] and Mehta�Morse [8] gave their respective bound�ary conditions for the outer cell surface. Kvashnin [7]proposed the condition that the tangential componentof velocity reaches a minimum at the cell surface withrespect to radial distance, signifying the symmetry onthe cell. However, Mehta�Morse [8] used Cunning�ham’s [9] approach by assuming the tangential veloc�ity as a component of the fluid velocity, signifying thehomogeneity of the flow on the cell boundary. Theimportance of the Mehta�Morse [8] boundary condi�tion is that since we are interested in the flow behavioron a large scale, we shall average the flow variables onthe small scale over a cell volume to obtain large scalebehavior.

A Cartesian�tensor solution of the Brinkman equa�tion was investigated by Qin and Kaloni [10] and theyalso evaluated the drag force on a porous sphere in anunbounded medium. Flow through beds of porousparticles was studied by Davis and Stone [11] and theyevaluated the overall bed permeability of swarm byusing cell model. Vasin and Filippov [12], Filippov etal. [13] evaluated the hydrodynamic permeability ofmembrane of porous spherical particles using Mehta�Morse condition on the cell surface. Recently, Vasin etal. [14, 15] compared all four cell models to evaluatethe permeability of membrane of porous spherical par�ticles with a permeable shell and discussed the effect ofdifferent parameters on the hydrodynamic permeabil�ity of the membrane for all the four above mentionedboundary conditions. Deo and Yadav [16] studied theproblem of Stokes flow through a swarm of porous cir�cular cylinder�in�cell enclosing an impermeable corewith Kuwabara’s and Happel’s boundary conditions.The stream function for a slow viscous flow through anarray of porous cylindrical particles with Happel’sboundary condition was considered in [17]. The flowpatterns along with the drag force exerted to eachporous cylindrical particle in a cell were evaluated.

The hydrodynamic permeability of membranesbuilt up by spherical particles covered by porous shellswas discussed by Yadav et al. [18]. Deo et al. [19] stud�ied hydrodynamic permeability of membranes built upby porous cylindrical or spherical particles withimpermeable core using cell model technique. Theyused different versions of a cell method to calculate thehydrodynamic permeability of the membranes andutilized the boundary condition of tangential stressjump at the interface between porous shell and clearliquid. They studied also both transversal and normalflows of liquid with respect to the cylindrical fibers thatcompose the membrane. The hydrodynamic perme�ability of biporous medium (membrane) modeled by

the set of porous spherical particles located in theporous medium with other rheological properties iscalculated using the cell method by Vasin et al. [20].The motivations of mentioned papers and especiallylast article [20] lead us to discuss the present problemfor porous cylindrical particles located not in clear liq�uid but in the porous medium, which includes someearlier known results.

Therefore this paper concerns the hydrodynamicpermeability of biporous medium (membrane) builtup by porous cylindrical particles embedded in anoth�er porous medium by using cell model technique. It isassumed that the biporous medium composed twotypes of particles – conventionally small (as referred totheir diameter) cylindrical or spherical particles andlarge cylindrical particles, which are parallel each oth�er. So, in comparison to large particle, the concentra�tion of small particles is higher and characteristic sizeis lower. The medium formed by the particles of thefine fraction is assumed to be continuous as it makesthe large particles submerge into the porous mediumof small cylindrical or spherical particles. Such kind ofmedium can be constructed, for example, when ho�mogeneous porous pattern (ion�exchange membranematerial) is reinforced by the set of identical parallelporous fibers. Four known boundary conditions,namely, Happel’s, Kuwabara’s, Kvashnin’s and Cun�ningham/Mehta–Morse’s, are considered on the out�er surface of the cell. The variation of hydrodynamicpermeability of biporous medium (membrane) withviscosity ratio m, Brinkman constants si, i = 1, 2, andsolid volume fraction γ are presented and discussedgraphically. Some previous results for dimensionlesshydrodynamic permeability have been verified. Theproblem of flow in media with low and high concen�trations of large particles leads precisely to this physi�cal and mathematical formulation. For the medium ofhigh concentrations of large particles, small cylindri�cal particles merely fill the space between large cylin�drical or spherical particles, thus forming a highly po�rous medium. The only distinction between two afore�mentioned situations is in the difference of the relativeconcentrations of large and small particles in a con�glomerate.

MATHEMATICAL FORMULATION OF THE PROBLEM

In the mathematical model, we will represent a dis�perse system (membrane) by a periodic net of identical

porous cylindrical particles of radius the permeabil�

ity of that system is proportional to and effective

viscosity is The particles are located in anotherporous medium with permeability proportional to

,R�( )11 k� ,

( )1 .µ�


HYDRODYNAMIC PERMEABILITY OF BIPOROUS MEMBRANE 475

and effective viscosity (Fig. 1). Each particleis placed into a cylindrical cell with radius Let usconsider that porous cylindrical shells are stationaryand external flow has been established by a uniform

velocity = which is perpendicular to the cyl�inders axis (Fig. 1). The radius of hypothetical cell isso chosen that the ratio of particle to cell volume isequal to γ2 i.e., to the volume fraction of internal par�ticles in the real membrane:

(1)

where ε is the porosity of medium.The flows of liquid in inner and outer porous media

(region 1 and 2 respectively) are governed by theBrinkman [1] equation and the continuity condition:

(2)

(3)

where i is superscript, which mark flow in the both re�

gions, tilde sign denotes the dimensional values,

are velocity vectors, are pressures, and are theBrinkman constants which are inversely proportionalto the hydrodynamic permeability of porous media.

SOLUTION OF THE PROBLEM

By using the following dimensionless variables

where = is the Brinkman’s length, the gov�

erning Brinkman’s equations (2) and the continuitycondition (3) in non�dimensional form can be writtenas follows:

(4a)

(4b)

The stream functions in both regions, sat�isfying equations of continuity (4b) may be defined as

(5)

Using the stream functions and eliminating thepressures from equations (4a) and using definition (5),we arrive to the following fourth order partial differen�tial equations for stream functions:

(6)

( )21 k� ( )2µ�

.a�

U� (U� )U�

a�

221 ,

aR⎛ ⎞− ε = γ = ⎜ ⎟⎝ ⎠�

�

( ) ( ) ( ) ( ) ( ),i i i i ip k∇ = μ Δ −v v� � �

� � � �

( ) 0, 1,2,i i∇ ⋅ = =v�

�

( )iv�( )ip�

( )ik�

,Ra

γ =�

�

,ra

r =

�

�

,a∇ = ∇� �2,aΔ = Δ� �

( )( )

( )0

,i

i

i

pp

p=

�

�

( )( )

0 ,i

i Up

a

µ=

�

�

�

�

( ),i i

b

asR

=

�

�

( )

( )

1

2,m

µ=

µ

�

�

( )( )

v ,i

i

U=

v��

( )ibR�

( )

( )

i

ik

µ�

�

( ) ( ) ( )2 ,i i iip s∇ = Δ −v v

( ) 0, 1,2.i i∇ ⋅ = =v( )i

ψ ( , )r θ

( )( )

( )( ) 1 , .

ii

r

ii

r rθ

∂ψ ∂ψ= = −

∂θ ∂v v

( )2 2 2 ( ) 0,iis∇ ∇ − ψ =

where

(7)

The pressure may be obtained in both regions(Happel and Brenner [21]) by integrating the follow�ing relations respectively as:

(8)

(9)

The range of r and θ in the above equations (6), ina cylinder is given by:

(10)

A solution of the Brinkman equation (6) can bewritten as

(11)

Here, and are the modified Bessel’sfunctions of the order one of the first and second kinds(Abramowitz and Stegun [22]), respectively.

In order to close the boundary value�problem forequation (4), it is necessary to specify the boundaryconditions. At the interface of porous media, we setcontinuity conditions for the components of velocityvector and stress tensors as follows:

(12)

On the cell surface = four variants of boundaryconditions are known i.e. the Happel, Kuwabara,Kvashnin, and Cunningham (Mehta–Morse) models.

2 22

2 2 21 1 .r rr r

∂ ∂ ∂∇ = + +

∂∂ ∂θ

( )( )( )2 ( ) 2 ( )

2 22 ,

iiii ir

r i rp

sr r r

θ∂∂= ∇ − − −

∂ ∂θ

vv

v v

( ) ( )( )2 ( ) 2 ( )

2 21 2 .

i iii ir

ip

sr r r

θθ θ

∂∂= ∇ − + −

∂θ ∂θ

v v

v v

0 1, 0 2 .r≤ ≤ ≤ θ ≤ π

( )

( ) ( ) ( ) ( )1 2 3 1 4 1

( , )

( ) ( ) sin .

i

i i i ii i

r

c r c r c s r c s r

ψ θ =

⎡ ⎤= + + + θ⎣ ⎦I K

1( )is rI 1( )is rK

,rrσ� rθσ�

(1) (2)

(1) (2)

(1) (2)

,

,

.

rr rr

r rθ θ

=

σ = σ

σ = σ

v v� �

� �

� �

r� a�

U~

U~

a~

R~

1

2

Fig. 1. Schematic representation of cell with porous cylin�drical particle located in another porous medium.

476



Moreover, in all four models, the continuity of the ra�dial component of the velocity of liquid on the cell sur�face = is assumed to be as follows:

(13)

Let us consider additional conditions that are usedin the above models.

According to Happel model [5] the tangentialstress vanishes on the cell surface, i.e.

(14a)

According to Kuwabara model [6] the vorticityvanishes on the cell surface (flow potentiality), i.e.

(14b)

According to Kvashnin model [7] a symmetry con�dition is introduced on the cell surface, i.e.

(14c)

Mehta�Morse’s [8] assumes homogeneity of theflow on the cell surface, i.e.

(14d)

In cylindrical coordinate system, boundary condi�tions (12)–(14) in the dimensionless variables arewritten, respectively, as:

at r = γ

(15)

at r = 1

(16)

(17a)

(17b)

(17c)

(17d)

Substituting expression (11) into Eqs. (5), (8), (9)we arrive at

(18)

(19)

(20)

Here, and are the modified Bessel’s func�tions of the order l of the first and second kinds, re�spectively.

Solution (5) and hence (18)–(20) must be regularin the centre of particle that is achieved if we assume

that = = 0. Substituting the values of

p(i) from solutions (5), (18)–(20) into boundaryconditions (15)–(17), we have six algebraic equations in�

volving six arbitrary constants j = 1, 2, 3, 4.Thus the values of arbitrary constants mentionedabove depends on the selection of the model used, i.e.,on the boundary condition (17a), (17b), (17c) or(17d). Solving the resulting equations by taking eachmodel respectively, we find the values of all arbitrary

constants j = 1, 2, 3, 4. The solution of thealgebraic system is rather cumbersome and not report�ed here due to brevity.

RESULTS AND DISCUSSION

The main characteristic of the problem is force applied to the particle located in the cell from the sideof a liquid and it can be evaluated by integrating thenormal and tangential stresses over the porous cylin�

drical shell surface of radius in a cell as follows:

(21)

r� a�

(2) U cos .r = θv�

�

(2) 0.rθσ =�

( )(2)rot 0.=v�

(2)

0.rθ∂

=∂

v�

(2) U sin .θ = − θv�

�

(1) (2)

(1) (2)

,

,

r r

θ θ

=

=

v v

v v

2 (1) (1)(1)

2

2 (2) (2)(2)

2

2 2

2 2 ,

m pr r r

pr r r

⎡ ⎤∂ ψ ∂ψ− + − =⎢ ⎥∂ ∂θ ∂θ⎣ ⎦

∂ ψ ∂ψ= − + −∂ ∂θ ∂θ

2 (1) (1) 2 (1)

2 2 2

2 (2) (2) 2 (2)

2 2 2

1 1

1 1 .

mr rr r

r rr r

⎡ ⎤∂ ψ ∂ψ ∂ ψ+ − =⎢ ⎥∂∂θ ∂⎣ ⎦

∂ ψ ∂ψ ∂ ψ= + −∂∂θ ∂

(2) cos ,r = θv

2 (2) (2) 2 (2)

2 2 21 1 0,

r rr r

∂ ψ ∂ψ ∂ ψ+ − =

∂∂θ ∂

2 (2) (2) 2 (2)

2 2 21 1 0,r rr r

∂ ψ ∂ψ ∂ ψ+ + =

∂∂ ∂θ

(2)

0,rθ∂

=∂

v

(2) sin .θ = − θv

2 ( ) ( ) ( ) ( )1 2 1 3 1 4( )

2

( ) ( ) cos,

i i i ii ii

r

r c c r s r c r s r c

r

⎡ ⎤+ + + θ⎣ ⎦=v

I K

2 ( ) ( ) 2 ( ) 2 ( )1 2 0 2 3 0 2 4( )

2

2 2 ( ( ) ( )) ( ( ) ( )) sin,

2

i i i ii i i i i ii

r c c r s s r s r c r s s r s r c

rθ

⎡ ⎤− + + − + θ⎣ ⎦= −v

I I K K

( )( ) 21

( ) 2 cos .i

ii

i cp s rc

r

⎛ ⎞= − + θ⎜ ⎟

⎝ ⎠

( )l is rI ( )l is rK

(1)2c (1)

4c ( ),iψ

( ),irv

( ),iθv

(1)1 ,c (1)

3 ,c (2);jc

(1)1 ,c (1)

3 ,c (2);jc

,F�

R�

( )(2) (2) (2)

0

2

cos sin ,rr rr a

F ad F Uθ=

π

= σ θ − σ θ θ = μ∫� �

� �

� � � �



were

(22)

is the dimensionless force.

Inserting the values of and into the aboveequation (22) and integrating, we get

(23)

Hydrodynamic permeability of the system un�der consideration, representing one of the elements ofthe Onsager matrix [23], is defined as the ratio of the

uniform flow rate to the cell gradient pressure [21]:

(24)

where = is the volume of the cell of the unitlength.

Substituting the value of from equation (21) and

the value of from above in equation (24), we have

(25)

where L11 = is the dimensionless hydrodynamic

permeability of filtration system.Using the expression (23) in the above dimension�

less hydrodynamic permeability L11, we find the fol�lowing relation:

(26)

where constants j = 1, 2, 3, 4 are calculated fromthe solution of system (15)–(17). The hydrodynamicpermeability is a function of four dimensionless pa�rameters, i.e. L11

Let us now consider the limiting cases of solvingthe problem under consideration.

If γ → 0 (i.e. added porous inclusions are absent)then the hydrodynamic permeability will become asfollows:

(27)

i.e. in place of above model we deal with uniform po�rous medium with hydrodynamic permeability L11

=

( )(2) (2)

0

2

1cos sinrr r r

F dθ

π

=

= σ θ − σ θ θ∫

(2)rrσ

(2)rθσ

(2) (2) (2) (2) 21 2 1 2 3 1 2 4 2( ) ( ) .F c c s c s c s⎡ ⎤= π − + +⎣ ⎦I K

11L�

U� F V� �

11 ,ULF V

=

�

�

� �

V�2aπ �

F�

V�

2 2

11 11(2) (2),a aL L

Fπ

= =

μ μ

� �

�

� �

Fπ

11 (2) (2) (2) (2) 21 2 1 2 3 1 2 4 2

1 ,( ) ( )

Lc c s c s c s

=⎡ ⎤− + +⎣ ⎦I K

(2);jc

1 2( , , , ).s s mγ

11 22

1 ,Ls

=

221 .s

If γ → 1 (i.e. external porous medium is absent)then the hydrodynamic permeability will become asfollows:

(28)

i.e. at γ → 1, we also obtain a uniform medium, albeit

with another hydrodynamic permeability L11 =

When s2 → 0, external porous medium becomesviscous fluid. In this case, the value of the hydrody�namic permeability L11 can be evaluated by the formu�las derived in [19].

When → ∞ or → ∞, then the porous cylindri�cal particle becomes rigid. In this case, the value of thehydrodynamic permeability L11 comes out as follows:

for the Happel model,

(29)

for Kuwabara model,

(30)

for Kvashnin model,

11 21

1 ,Lms

=

211 .ms

(1)k� (1)µ�

( )((

( ) ( ) )

211 2 0 2 0 2 2

1 2 1 2 2 0 2 1 2

20 2 2 1 2

0 2 1 2 2 1 2 0 2

( )

( ) ( )(2

2 ( )

(2 ( ) ) ( )

L s s s s

s s s s s

s s s

s s s s s

= − + − − + γ γ −

− γ γ + γ γ +

+ − + γ γ + γ γ ×

× + + γ ×

8 2 1 I K ( )

2 I K ( ) 2I K ( )

1 K I

K ( ) K I K ( )

( )( )( )( ))) ( ( )(

2 22 0 2 1 2

2 2 2 22 2

20 2 0 2 2 0 2 2 1 2

( ) ( )

8 2

( ) ( ) 2 ( ) (2

s s s

s s

s s s s s s

× − − + γ + γ ×

× − − + γ − + + γ ×

× γ − γ γ +

4 1 I K

4 1 1

I K I K ( )

( ) ( ) ) ( )( )( )( )( )

( ))))

2 20 2 2 1 2 1 2 2

2 2 21 2 0 2 2 2

2 2 20 2 1 2 2 2

21 2 0 2 2 1 2 2

( ) 8

( ) 4 8

( ) 4 8

( )(2

s s s s s

s s s s

s s s s

s s s s s

+ + γ γ γ γ + +

+ γ + γ + + γ +

+ γ + γ + + γ −

− γ γ + +

1 K 2 I K ( )

I K ( ) 1

I K ( ) 1

2 I K ( ) K ( ) 8

( )(( ) (

11 1 2 1 2

21 2 1 2 1 2 0 2

20 2 1 2 2 2 1 2 1 2

21 2 1 2 1 2 0 2

0 2 1 2 2

( ( ( )

( ) ) ( ( )

( ) ( 2 ( )

2 ( ) ( )

( ) )

L s s

s s s s

s s s s s s

s s s s

s s s

= γ γ −

− γ − − + γ γ +

+ γ − γ γ +

+ γ γ + + γ γ +

+ γ

2 I K ( )

I K ( ) 1 I K ( )

I K ( )) I K ( )

I K ( ) 1 I K ( )

I K ( )) )

(

(

311 2 2 2 22

2

1 2 2 2 2 2 2 2 2

1 2 2 2 1 2 1 2 2

1 ( )

( ) ( )(

( 2 ( )

L s ss

s s s s s

s s s s s s

= + γ − γ +

+ γ + γ +

+ γ − − γ +

(2 I K ( )

I K ( ) I K ( )

K ( ) ))) 4 I K ( )

478



(31)

and for Cunningham/Mehta�Morse’s model

(32)

The dependence of hydrodynamic permeability L11

of biporous medium with the parameter γ, which char�acterized the fraction of porous media in the mem�brane for all four known boundary conditions on thecell surface are shown in the Fig. 2. From the Fig. 2 weobserved that if dimensionless Brinkman constant ofthe inner porous medium is greater than that of exter�nal porous medium then the hydrodynamic perme�ability L11 decreases with increase of γ for all modelsand if dimensionless Brinkman constant of the inner

( )(

32 2 2 2 2 1 2

2 2 1 2 2 2 2 1 2

( )( 2 ( )

( ( ) 2

s s s s

s s s s s

+ γ + γ γ + γ ×

× + + γ −

I K ( ) I

K ( ) K ( ) ) I K ( )

( ) )( )( )

( ) )))(

32 2 2 2 2 1 2

2 2 22

2 2 21 2 2 2 2

( )

( )

s s s s

s

s s s

− γ + γ γ + γ γ ×

× γ + + γ +

+ γ γ γ + γ

K ( ) I K ( )

2 1

I K ( ) 2 1+

( )

11 1 2 0 2

1 2 0 2 1 2 0 2

20 2 1 2 2 0 2 0 2

( 4 2( ( )

( ) ( )

( ) ( )

L s s

s s s s

s s s s s

= − + γ γ +

+ γ + γ +

+ γ γ − − + γ γ −

I K ( )

I K ( ) K I ( )

I K ( )) 1 (I K ( )

) (2 20 2 0 2 2 2 1 2 1 2

1 2 1 2 2 1 2 0 2

( ) ( 4 8 ( )

8 ( ) ( 2 ( )

s s s s s s

s s s s s

− γ − − γ γ +

+ γ γ + − γ γ +

I K ( )) I K ( )

I K ( ) I K ( )

( ) ( )

( )

21 2 0 2 0 2 1 2

20 2 1 2

0 2 0 2 0 2 0 2 2

( ) ( )

( )

( ) ( ) ))).

s s s s

s s

s s s s s

+ + γ γ + γ −

− γ γ + + γ ×

× γ − γ

2 1 2 I K ( ) I K ( )

2 I K ( ) 1

(I K ( ) I K ( ))

porous medium is less than that of external porous me�dium then the hydrodynamic permeability L11 in�creases with increase of the parameter γ for all models.Initially, it increases slightly but for γ ≥ 0.5 it exponen�tially increases with increase of parameter γ. It is inter�esting to note that the values of permeability L11 for cy�lindrical particles as well as for spherical particles (dis�

cussed by Vasin et al. [20]) can vary from to

which are achieved at γ values equal to 0 and

1. As the permeability varies between to

it is observed that for a biporous medium of cylindricalparticles, the hydrodynamic permeability L11 of medi�

um become independent of parameter γ when = i.e. in this case the permeability of medium does notchange upon addition of porous particles. The result ison the lines of biporous medium of spherical particles[20].

Figure 3 shows the dependence of hydrodynamicpermeability L11 of biporous medium with the Brink�man constant s2 for all four known boundary condi�tions on the cell surface. It is seen that the hydrody�namic permeability L11 decreases with increase ofBrinkman constant s2. A rapid decrease followed bysteady decrease with s2 is observed for L11. It is also ob�served that the values of L11 for all four models are al�most same for s2 < 4. The deviation of permeability to�wards a decrease is observed for Cunningham/Meh�ta–Morse’s cell model compared to other threemodels. At s2 → ∞, the external porous medium be�comes absolutely impermeable and hence L11 tends tozero. At s2 → 0, the external porous medium becomes

221 s

( )211 ,ms

221 s ( )2

11 ,ms

( )1k� ( )2k�

1.5

1.0

0.5

1.0 0.80.60.40.2

1 '–4 '

1–4

γ

L11

Fig. 2. Variation of the dimensionless hydrodynamic per�meability L11 of biporous medium with the parameter γ for(1, 1') Happel, (2, 2') Kvashnin, (3, 3') Kuwabara, and (4,4 ') Cunningham models at (1–4) m = 0.5, s1 = 2, s2 = 1and (1'– 4') m = 0.5, s1 = 1, s2 = 2.

0.25

0.20

0.05

353025155

1–3

L11

0.30

0.15

0.10

2010

4

s2

Fig. 3. Variation of the dimensionless hydrodynamic per�meability L11 of biporous medium with the dimensionlessBrinkman constant s2 at m = 1, s1 = 2 and γ = 0.9 for (1)Happel, (2) Kvashnin, (3) Kuwabara, and (4) Cunning�ham models.



Newtonian viscous liquid and in this case the value ofthe hydrodynamic permeability L11 can be evaluatedby the formulas derived in [19].

Figure 4 shows the effect of Brinkman constant s1

on the hydrodynamic permeability L11 of biporousmedium for all four known boundary conditions onthe cell surface. We observe that the hydrodynamicpermeability L11 decreases with increase of Brinkmanconstant s1. It is also observed that the values of L11 forall four models are almost same for s1 > 1.5. For s1 < 1.5,the decrease in the values of L11 is observed for a se�quence of models (Happel, Kuwabara, Kvashnin, andCunningham/Mehta–Morse’s). At s1 → ∞, the po�

rous particle becomes impermeable and in this casethe values of L11 for all four models can be evaluated byformulas (29)–(32).

Figure 5 represents the dependence of the dimen�sionless hydrodynamic permeability of a membraneon viscosity ratio m of porous media. At m → ∞, theinternal porous particle becomes impermeable andagain in this case we can evaluate the hydrodynamicpermeability of a membrane L11 from the formulas(29)–(32).

It is interesting to note that the nature in variationof hydrodynamic permeability L11 with all four param�eters s1, s2, m, and γ for porous cylindrical particle are

1.0

0.8

0.2

864

1–4

L11

1.4

0.6

0.4

2

1.2

s1

Fig. 4. Variation of the dimensionless hydrodynamic per�meability L11 of biporous medium with the dimensionlessBrinkman constant s1 at m = 0.5, s2 = 2 and γ = 0.9 for (1)Happel, (2) Kvashnin, (3) Kuwabara, and (4) Cunning�ham models.

0.6

0.4

0.2

8 642

1–4

m

L11

0

0.3

0.5

0.7

0.8

Fig. 5. Variation of the dimensionless hydrodynamic per�meability L11 of biporous medium with the ratio of effec�tive viscosities m at s1 = 1, s2 = 3 and γ = 0.9 for (1) Happel,(2) Kvashnin, (3) Kuwabara, and (4) Cunningham models.

0.05

0.04

0.01

14121062

1

L11

0.06

0.03

0.02

84

4

3

2

s2

Fig. 6. Variation of the dimensionless hydrodynamic permeability L11 when cylindrical particles becomes rigid with the dimen�sionless Brinkman constant s2 at γ = 0.5 for (1) Happel, (2) Kvashnin, (3) Kuwabara, and (4) Cunningham models.

480



almost the same as in a case of porous spherical parti�cle [20].

Figure 6 exhibits the dependences of the dimen�sionless hydrodynamic permeability of a membraneon parameter s2 when the inner cylindrical fibers areimpermeable (rigid). The permeability was calculatedby formulas (29)–(32). It is seen the behavior charac�ter of graphics is similar to that of Fig. 3, but the differ�ence between models is more pronounced for s2 ≤ 8.

Hydrodynamic permeability and stream lines andare two important fluid�flow properties that are prac�tically useful to describe flow through porous media.Figures 7–10 show the flow patterns by stream lineswhich are important for finding the flow rate through theinternal part of the porous particles. From the Figs. 7–10, it is observed that the flow patterns for all fourknown models are almost parallel and density ofstream lines increases when the Brinkman constant s2 of

1.0

0.5

–0.5

–1.0

1.00.5–1.0 –0.5

(а)

1.0

0.5

–0.5

–1.0

1.00.5–1.0 –0.5

(b)

Fig. 7. Stream lines for Happel’s cell model at γ = 0.7, m = 3 and (a) s1 = 16, s2 = 2, (b) s1 = 2, s2 = 16.

1.0

0.5

–0.5

–1.0

1.00.5–1.0 –0.5

(а)

1.0

0.5

–0.5

–1.0

1.00.5–1.0 –0.5

(b)

Fig. 8. Stream lines for Kuwabara’s cell model at γ = 0.7, m = 3 and (a) s1 = 16, s2 = 2, (b) s1 = 2, s2 = 16.



the external porous medium is greater than that of theBrinkman constant s1 of the internal porous medium.

ACKNOLEDGMENTS

Anatoly Filippov’s and Sergey Vasin’s researcheswere supported by the Russian Foundation for BasicResearch, project nos. 11–08–01043 and 11–08–00807, correspondingly.

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1.0

0.5

–0.5

–1.0

1.00.5–1.0 –0.5

(а)1.0

0.5

–0.5

–1.0

1.00.5–1.0 –0.5

(b)

Fig. 9. Stream lines for Kvashnin’s cell model at γ = 0.7, m = 3 and (a) s1 = 16, s2 = 2, (b) s1 = 2, s2 = 16.

1.0

0.5

–0.5

–1.0

1.00.5–1.0 –0.5

(а)1.0

0.5

–0.5

–1.0

1.00.5–1.0 –0.5

(b)

Fig. 10. Stream lines for Cunningham/Mehta–Morse’s cell model at γ = 0.7, m = 3 and (a) s1 = 16, s2 = 2, (b) s1 = 2, s2 = 16.

482



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SPELL: 1. Naukova

Hydrodynamic Permeability of Biporous Membrane

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