ISSN 2517-7516, Membranes and Membrane Technologies, 2019, Vol. 1, No. 6, pp. 394–405. © Pleiades Publishing, Ltd., 2019.

Hydrodynamic Permeability of a Membrane Built up by Non-Homogenous Porous Cylindrical Particles

Satya Deoa, Pankaj Kumar Mauryaa, and A. N. Filippovb, *aDepartment of Mathematics, University of Allahabad, Allahabad, 211002 India

bDepartment of Higher Mathematics, National University of Oil and Gas “Gubkin University,” Moscow, 119991 Russia*e-mail: filippov.a@gubkin.ru

Received August 18, 2019; revised August 28, 2019; accepted August 28, 2019

Abstract—This paper concerns the evaluation of the hydrodynamic permeability of a membrane built up bynon-homogeneous porous cylindrical particles using four known boundary conditions, such as Happel’s,Kuwabara’s, Kvashnin’s, Mehta-Morse/Cunningham’s. For different values of f low parameters, the varia-tions of the hydrodynamic permeability are presented graphically and discussed. Some earlier results reportedfor the drag force and the hydrodynamic permeability, have been verified.

Keywords: Brinkman equation, Stokes equation, drag force, hydrodynamic permeabilityDOI: 10.1134/S2517751619060088

INTRODUCTIONFlow through a porous medium has been a most

interesting topic for researchers due to its numerousapplications in various fields of sciences and engineer-ing, particularly in chemical engineering, petroleumreservoirs, membrane technology, etc. Nield andBejan [1] gave various models for describing f luid f lowthrough porous medium in their celebrated book‘Convections in Porous Media’. In 1856, Henri Darcyproposed an empirical law, which states that the rate offluid flow through a porous medium is proportional tothe pressure drop through a bed of fine particles. Mathe-matically, the Darcy law can be expressed in the form:

(1)

where, is the viscosity of the f luid, is the velocityof the f luid, is the pressure and is the permeabil-ity of the porous medium.

In 1947, Brinkman [2] proposed a modification ofthe Darcy law by adding the effective viscosity term inthe equation. Therefore, the Brinkman equation forthe f low in porous medium is expressed in the follow-ing mathematical form:

(2)

where, is the effective viscosity of the f luid in theporous medium.

Happel [3] proposed a cell model in which cylin-drical/spherical particles are enveloped by the same

geometric configurations. Happel model assumes theuniform radial velocity condition and zero tangentialstress at the outer cell surface, whereas, Kuwabara [4]proposed nil vorticity on the outer cell surface, insteadof the vanishing tangential stress. Kvashnin [5] modelassumes the symmetry of the f low and used the extre-mum condition for tangential velocity on the outer cellsurface. Mehta-Morse [6]/Cunningham [7] modelsuggests, uniform velocity condition on the outer cellsurface. Details about these cell models is discussed inthe celebrated book titled ‘Low Reynolds NumberHydrodynamics’ by Happel and Brenner [8].

Deo [9] studied the Stokes f low past a swarm ofporous circular cylinders with Happel and Kuwabaraboundary conditions and obtained the drag forceexperienced by a porous cylinder in a cell. Vasin andFilippov [10] studied the hydrodynamic permeabilityfor impenetrable cylindrical particles covered with aporous layer and evaluated the theoretical values ofKozeny constants. The drag force and the hydrody-namic permeability for the slow viscous f low throughan aggregate of concentric clusters of porous cylindri-cal particles by applying Happel boundary conditionwere evaluated by Deo et al. [11]. The hydrodynamicpermeability of aggregates of porous cylindrical parti-cles with impermeable core for various cell models andimpact of the jump shear stress at the f luid-membraneinterface were investigated by Deo et al. [12]. Theanalysis of the hydrodynamic permeability for the vis-cous f luid f low in the porous medium formed bycylindrical fibers was discussed by Vasin et al. [13] andthe solution of Brinkman equation for radially varying

(1) (1)μ ,p− ∇=v kμ (1)v

(1)p k

∇ − = ∇ 2 (1) (1) (1)μμ ,e p

kv v

μe

394

HYDRODYNAMIC PERMEABILITY OF A MEMBRANE BUILT 395

permeability in the cylindrical polar coordinates wasobtained by Mishra and Panday [14].

Yadav [15] investigated the drag coefficient andhydrodynamic permeability for the case of the slowmotion of a porous cylindrical shell in a concentriccylindrical cavity using cell models and discussedgraphically. The variation of hydrodynamic permea-bility of biporous medium with viscosity ratio, Brink-man constant and solid fraction, in which porouscylindrical particles embedded in another porousmedium by using cell model technique for four knownboundary conditions studied by Yadav et al. [16].Sherief et al. [17] were investigated the drag force forthe modified Happel boundary conditions and veloc-ity profile for two-dimensional quasi-steady micropo-lar f luid f low between two coaxial cylinders. Fluidflow in the fractal layer is investigated by Vasin andKharitonova [18] using the Brinkman equation, onconsidering that the viscosity of the effective mediumvaries from the viscosity of the pure liquid. Velocityand pressure distributions have been calculated, andthe viscous drag force applied to the capsule has beenobtained. Deo et al. [19] obtained the general streamfunction solution of the Brinkman equation in thecylindrical polar coordinates. The f low in compositeporous cylindrical channel of variable permeability foruniform flow in porous medium was studied by Vermaand Singh [20]. Recently, effect of various parametersfor the f low of a steady incompressible viscous f luidflow through a non-homogeneous porous medium forvariable permeability by using the Darcy equation dis-cussed by Yadav [21]. Some earlier results reported forthe drag force and the hydrodynamic permeability,have been verified. It is necessary to mention that asimilar problem for partly porous cylinders and jumpof tangential stresses on the porous-liquid interfacewas considered by Vasin [22]. He accounted for radialvarying Brinkman constant which is equal to the ratioof the liquid viscosity and specific permeability of aporous medium.

The aim of this work is to obtain explicit algebraicformulas for calculating the hydrodynamic permeabil-ity, when the structure of the initial membrane can besignificantly changed by dissolving the surfaces of theconstituent grains and fibers (loosening) or adsorptionof polymers (contamination). Both described pro-cesses lead to the formation of porous structures withradially variable permeability on the surface of thegrains or fibers that make up the membrane. Whenloosening the fiber surfaces, the permeability of themembrane increases by increasing its porosity; and, asa rule, selectivity decreases. In contrast, when themembrane is contaminated, porosity decreases and, asa rule, selectivity increases.

In this work, the f low outside the porous mediumis governed by Stokes equation, whereas, the f lowthrough non-homogeneous porous medium isdescribed by the Brinkman equation, in which the

MEMBRANES AND MEMBRANE TECHNOLOGIES V

permeability varies radially by the power law. Thehydrodynamic permeability of a membrane built up bynon-homogeneous porous cylindrical particles usingfour known boundary conditions of the cell models,such as Happel’s, Kuwabara’s, Kvashnin’s, Mehta-Morse/Cunningham’s is investigated. There are noconvincing physical arguments in favor of some orother model in the published literature. Moreover, forthe case of f low in the f lat slit (the limiting case of par-ticles with infinite radius), in the center of slit that cor-responds to the cell surface, boundary conditions cor-responding to all cell models are fulfilled. The same isobserved in the f low of viscous f luid along the axes ofthe system of parallel cylinders [10]. So, in this work,we consider all four models. Any of these models canbe used to calculate the hydrodynamic permeability offiber membranes with an ordered structure. Moreover,the first three models give slightly different results. Fordifferent values of f low parameters, the variations ofthe hydrodynamic permeability for non-homoge-neous porous medium are presented graphically anddiscussed.

MATHEMATICAL FORMULATION AND SOLUTION OF THE PROBLEM

Let us consider the steady, incompressible viscousfluid f lowing with uniform velocity through anassemblage of porous cylindrical particles of radius ,perpendicular to the axes of parallel cylinders (z-axis).Therefore, this assemblage of porous cylindrical parti-cles with radially varying permeability is hydrodynam-ically equivalent to a membrane. Further, it is assumedthat each non-homogeneous porous cylindrical parti-cle is enveloped by a concentric hypothetical cell ofradius (Fig. 1). The radius of the hypothetical cell is so chosen that the particle volume fraction of theswarm is equal to the particle volume fraction of thecell, i.e.

(3)

Let be the f luid veloci-

ties and be the pressures inside the non-homogeneous porous and outside regions, respec-tively.

The f low inside the non-homogenous porousregion I is governed by the Brinkman’sequation with continuity condition

(4)

(5)

where, is the specific permeability coefficient ofnon-homogenous porous region which is taken to be

Ua

b bλ

2

2πλ .π

ab

=

( ) ( ) ( )( , ,0), ( 1,2)i i ir iθ= =v v v

( ), ( 1,2)ip i =

(0 )r a≤ ≤

∇ − = ∇ 2 (1) (1) (1)μμ ,

( )e pk r

v v

(1) 0,∇ ⋅ =v

( )k r

ol. 1 No. 6 2019

396 SATYA DEO et al.

Fig. 1. Model of the Problem.

Hypothetical cell

U~

y-axis

z-axis

x-axis

Porous cylinder

II

I

θ

vθ ~vr

a~

b~

varying radially. In general, the relation betweenand depends on the nature of porous media. Here,

for simplicity, we assume that .The f low outside the non-homogeneous porous

region II is governed by the Stokes equa-tion together with continuity condition, respectively as:

(6)

(7)

Let us introduce the following dimensionless vari-ables and parameters:

(8)

where, is the specific permeability on the outerboundary of the porous cylinder. Accordingly, theparameter characterizes the filtration resistancewhen crossing the porous cylinder boundary.

Therefore, the above governing equations in thenon-dimensional form are become as follows.

For porous medium region ,

(9)

(10)

μe μμ μe =

( )a r b≤ ≤

2 (2) (2)μ ,p∇ = ∇v (2) 0.∇ ⋅ =v

= = = =

∇ = ∇ = = =

( )( )( ) ( )

22

0

, , ( 1,2), ,μ( ) 1, ( ) , γ , ,

iii i a rp i r

U U ak r aa k r m

b ka

pvv

0k

m

(0 1)r≤ ≤

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

pk r

∇ − = ∇v v

(1) 0.∇ ⋅ =v

MEMBRANES AND M

Assume that the permeability of the non-homoge-neous porous medium varies radially as ,because the outer layers of the cylinder dissolve usu-ally better than the inner ones and form more perme-able porous structures. Then equations (9) and (10)become:

(11)

(12)

For outside porous region

(13)

(14)

Equations (11)–(12) in the cylindrical polar coor-dinates can be written as:

(15)

(16)

with equation of continuity

(17)

Similarly, the above equations (13)–(14) willbecome as

20( )k r k r=

22 (1) (1) (1)

2 ,m pr

∇ − = ∇v v

(1) 0.∇ ⋅ =v

(1 1 γ):r≤ ≤2 (2) (2),p∇ = ∇v

(2) 0.∇ ⋅ =v

( ,θ, )r z(1)(1) 2

2 (1) (1)θ2 2

(1 )2 ,θr r

p mr r r

∂∂ += ∇ − − ∂ ∂

vv v

(1)(1) 22 (1) (1)

θ θ2 2(1 )1 2 ,

θ θrp m

r r r

∂∂ += ∇ + − ∂ ∂

vv v

(1)(1)(1) θ 0.

θr

rrr

∂∂ + + =∂ ∂

vvv

EMBRANE TECHNOLOGIES Vol. 1 No. 6 2019

HYDRODYNAMIC PERMEABILITY OF A MEMBRANE BUILT 397

(18)

(19)

with the equation of continuity

(20)

On solving equations (15)–(20) using method ofseparation of variables by assuming

(21)

and

(22)

we obtain analogous formulas as in [22],

(23)

(24)

(25)

where,

(26)

For regular solution, we can choose and bothare zero, then the equations (23)–(25) become as:

(27)

(28)

(29)

(2)(2)2 (2) (2)θ

2 2

2 1,

θr r

pr r r

∂∂ = ∇ − − ∂ ∂

vv v

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

θ θ2 2

1 2 1,

θ θ

rpr r r

∂∂ = ∇ + − ∂ ∂

vv v

(2)(2)(2) θ 0.

θ

rrr

r∂∂ + + =

∂ ∂vv

v

= ==

(1) (1)

1 θ 1

(1)

1

θ θ

θ,

( )cos , ( )sin ,

( )cos

r f r g r

p h r

v v

θ= ==

(2) (2)

2 2

(2)

2

θ θ

θ

( )cos , ( )sin ,

( )cos

r f r g r

p h r

v v

1 2 3 4(1) α α α α

1 2 3 4 θ,cosr b r b r b r b r = + + + v

1 2

3 4

(1) α α

θ 1 1 2 2

α α

3 3 4 4 θ,

(α 1) (α 1)

(α 1) (α 1) sin

b r b r

b r b r

= − + + ++ + + +

v

−

−

−

−

= + − + −+ + − + −+ + − + −

+ + − + −

1

2

3

4

(1) 3 2 2 2 α 1

1 1 1 1

3 2 2 2 α 1

2 2 2 2

3 2 2 2 α 1

3 3 3 3

3 2 2 2 α 1

4 4 4 4 θ,

{α α (2 )α }

{α α (2 )α }

{α α (2 )α }

{α α (2 )α } ]cos

p b m m r

b m m r

b m m r

b m m r

+ − +=

− + − +=

+ + +=

− + + +=

2 2

1

2 2

2

2 2

3

2 2

4

4 (16 )α ,

2

4 (16 )α ,

2

4 (16 )α ,

2

4 (16 )α .

2

m m

m m

m m

m m

2b 4b

1 3(1) α α

1 3 θ,cosr b r b r = + v

1 3(1) α α

θ 1 1 3 3 θ,(α 1) (α 1) sinb r b r = − + + + v

−

−

= + − + −+ + − + −

1

3

(1) 3 2 2 2 α 1

1 1 1 1

3 2 2 2 α 1

3 3 3 3 θ,

[ {α α (2 )α }

{α α (2 )α } ]cos

p b m m r

b m m r

MEMBRANES AND MEMBRANE TECHNOLOGIES V

and [8]:

(30)

(31)

(32)

where and are arbitrary constants.These constants can be determined by applying thefollowing suitable boundary conditions:

At :

(33)

(34)

(35)

(36)

Here, is a viscous stress tensor in the

appropriate region.

At :

(37)

Happel boundary condition:

(38)

Kuwabara boundary condition:

(39)

Kvashnin boundary condition:

(40)

Cunningham/Mehta–Morse boundary condition:

(41)

Applying the above boundary conditions (33)–(37)and one condition among from (38)–(41), values of allthe arbitrary constants are determined. The values ofthese constants for Happel, Kuwabara, Kvashnin andCunningham/Mehta-Morse boundary conditions aregiven in the Appendix I.

CALCULATION OF DRAG FORCE

The drag force experienced by the porous cylinderis evaluated by using the formula:

(42)

where dimensionless force looks like,

(2) 212 3 42

θ,ln cosrc c r c r cr = + + +

v

(2) 21θ 2 3 42

θ,(1 ln ) 3 sinc c r c r cr = − + − −

v

(2) 23 θ,

28 cos

cp c rr

= − +

1 3 1 2 3, , , ,b b c c c 4c

1r =(1) (2)

,r r=v v

(1) (2)

θ θ ,=v v

(1) (2)

θ θ ,r rT T=(1) (2)

.rr rrT T=

( )( ), 1,2

kijT k =

1 γr =(2)

θ.cosr =v

(2)

θ 0.rT =

(2)0.∇ × =v

(2)

θ 0.r

∂ =∂v

(2)

θ θsin .= −v

π

== − θ = 2

(2) (2)

θ

0

[( cos θ sin θ) ] μ ,rr r r aF T T r d F U

ol. 1 No. 6 2019

398 SATYA DEO et al.

(43)

Here, the stress tensor components are,

(44)

and

(45)

Substituting values of and from equa-tions (44) and (45) in equation (43) and integrating,we obtain

(46)

Hence, from the equation (42), we have

(47)

Case I. For Happel model:

When the permeability coefficient tends to zero,

, then the porous cylinder becomes

like a solid cylinder and the equation (47) reduces to

(48)

This is well-known result reported earlier by Hap-pel [3] for f low past a solid cylinder in a cell.

Case II. For Kuwabara model:

2

(2) (2)

θ 1

0

cos θ sin θ θ.( )rr r rF T T dπ

== −

(2) 1 233

4 cos θ,rrc cT c r

rr = − − +

(2) 1θ 33

4 sin θ.rcT c rr = − +

(2)

rrT (2)

rT θ

24π .F c=

24πμ .F Uc=

2 4 4

1

2 4 4 2 4

3

1

2 4 4 2

3

3 3 1

2

3

1[2( (2 2γ (1 γ )α )

α α (2 2γ (1 γ )α ) α ( (1 γ )

(4 )(1 γ ) 2( 1 γ )α ))],α

c mD

m

m

= − − + +

+ − + + + ++ + + − − +

= − − + + + − +− + + + + − +

+ − + − − + + −+ + + − + − ++ + + − − + − +

+ + + − − + α

2 4 2

1

2 4 2 4 4

2 4 4 4

1 3

4 2 2 2 4

3

2 4 2 4

2

3

3

1

3

3

4 4

4 ( 1 γ )(1 ln γ) α ( 8

(8 )γ 2 (1 γ ) ln γ 4( 1 γ )α )

α (4( 1 γ ) 4( 1 γ )α ln γ (1 γ

2(1 γ ) ln γ)α ) α ( 8 (8 )γ

2 (1 γ ) ln γ α ( ( 4 )( 1 γ )

2(4 )(1 γ ) ln γ 4( 1 γ ) ln γ))

D m m

m m

m m

m m

m .

( )0 0,k m→ → ∞

4

4 4

(1 )

1

8πμ γ.

γ γ ln1 ) γ2(

UF =+−+

− +

2

3

2

2 1

2

2

3 3

1 3 3

2

1[4( (2 α ) α α (2 α )

α ( α (4 2α )))],

c mD

m m

= − + + +

+ + + +

MEMBRANES AND M

When , then equation (47)

reduces to

(49)

This is well-known result reported earlier by Kuwa-bara [4] for f low past a solid cylinder in a cell.

Case III. For Kvashnin model:

When , then equation (47)

reduces to

(50)

This is a known result reported earlier by Kvashnin[5] for f low past a solid cylinder in a cell.

Case IV. For Mehta-Morse/Cunningham model:

3

3 3

3

2 2 2 4 2

2

2 2 4 2

2 2 2 4

1

2 4

2 2 4 2

4 2

2

1

2

4

3

3

4

[ 32γ 2 ( 5 4γ γ ) 8 ln γ

α ( 16 (3 4γ γ ) 4 ln γ

8( 1 γ )α ) α (8( 1 γ ) α (2 2γ

8 ln γ (3 4γ γ 4 ln γ)α ))

α ( 16 (3 4γ γ ) 4 ln γ

α ( 4 4γ (3 4γ γ )

4(4 ) ln γ (2 2γ 8 ln γ)α ))].

D m m

m m

m m

m

m

= − − − + + ++ − + − + +

+ − + + − + + −+ + − + +

+ − + − + ++ − + + − +

+ + + − +

( )0 0,k m→ → ∞

2 4

4πμ.

γ3/4 (1/4)γ ln γ

UF−

=− −+

2 4 4

2 3

3

2 4 4 2 4

1 3 3 1

2 4 4 2

3 3

1[ (6 2γ (3 γ )α )

α α (6 2γ (3 γ )α ) α ( (3 γ )

(4 )(3 γ )α 2( 3 γ )α )],

c mD

m

m

= − − + +

+ − + + + ++ + + − − +

= − + − − − ++ − + + − + + +

+ − + + + − + + − ++ + + − − + − − +

+ + + − − + + − ++ − +

2 2 2 2 4 2 4

3

4 2 2 2 4

3

2 4 4 2 2

3 1

2 4 2 2 2

3

2 4 4 4

3

2

1

(7 4(4 )γ 3 γ 2 ( 3 γ ) ln γ

α ( 2(6 2γ ( 1 γ )) (3 γ ) ln γ

2( 3 2γ γ )α ) α ( 2(6 2γ ( 1 γ ))

(3 γ ) ln γ α ( 2( 2 2γ )( 1 γ )

(4 )(3 γ ) ln γ ( 1 γ 2( 3 γ ) ln γ)α ))

α (2( 3 2γ

[D m m m m

m m

m

m m

m

+ + − − − ++ − + +

2 4 4 4

3

2 4

3

γ ) α (1 γ 2( 3 γ ) ln γ

(2 2γ (3 γ ) ln γ)α ))))].

( )0 0,k m→ → ∞

4

2 4

(3 )

2

4πμ γ.

γ2 (3 γ l) n γ

UF +− + +

=−

= + + −

+ + + − + − − +− + − + + +

2 4 4

2 3 3

4

2 4 4 2 4

1 3 3 3 1

2 4 4 2

3 3

1(2 2γ α γ α )

α α (2 2γ α γ α ) α ( ( 1 γ )

(4 )( 1 γ α 2(1 γ )α

[

) )],

c mD

m

m

EMBRANE TECHNOLOGIES Vol. 1 No. 6 2019

HYDRODYNAMIC PERMEABILITY OF A MEMBRANE BUILT 399

When , then equation (47)

reduces to

(51)

This is well-known result derived in [8] for f lowpast a solid cylinder in a cell.

HYDRODYNAMIC PERMEABILITY OF THE MEMBRANE

Now, we find out the hydrodynamic permeabilityof the membrane by using [10]:

(52)

where is the cell volume.

Using expression (47) in equation (52), we obtain

(53)

where, is the dimensionless hydrodynamic

permeability of the membrane.

Now, we find out the value of the dimensionlesshydrodynamic permeability of the membrane by sub-

stituting the value of the constant for all four cases.

Case I. Happel model:

(54)

= − − + − ++ − + − + − + + +

+ − + + − + + − +− + + − + − + +

+ − − − + + − ++ − − + + +

2 2 2 4 2 4

4

2 2 2 2 2 2

3

2 2 2 2 4)3 1 3

4 2 2 2

3

4 2 2 2 2 4)1

2 2 2 2

3

16γ (3 4γ γ ) 2 (1 γ ) ln γ

( 1 γ )α ( 4 (4 )γ (1 γ ) ln γ

2( 1 γ )α ) α (2( 1 γ α ( 1 γ

2(1 γ ) ln γ ( 1 γ )(1 γ ln γ γ ln γ)α ))

α (4 4γ ( 1 γ ( 1 γ ) ln γ

α ( ( 1 ) 4 )

[

γ (

D m m

m m m

m m

m m − + γ+ − + − +

4

4 4

3

( 1 γ ) ln

( 1 γ 2(1 γ ) ln γ)α ))].

( )0 0,k m→ → ∞

2

2 2

(1 )

1

4πμ γ.

γ (1 γ ) ln γ

UF =− +

+− +

= 11 ,

/

ULF V

2πV b=

=

2

11 11 ,μ

aL L

11 2

2

1

4γL

c=

2c

− − + + + − +− + + + + − +

+ − + − − ++ − + +

+ − + − + + ++ − − + − + + +

=

+− − +

2 4 2

2 4 2 4 4

2 4 4

1

4 4 2

3

2 2

11 3

3

3

1

4 2 4

2 4 2 4

4

3

3

[ 4 ( 1 γ )(1 ln γ) α ( 8

(8 )γ 2 (1 γ ) ln γ 4( 1 γ )α )

α (4( 1 γ ) 4( 1 γ ) ln γα

(1 γ 2(1 γ ) ln γ)α )

α ( 8 (8 )γ 2 (1 γ ) ln γ

α ( ( 4 )( 1 γ ) 2(4 )(1 γ ) ln γ

4( 1 γ ) ln γα ))]/

m m

m m

m

m

L

m m

m

− −+ + + − + +

+ + + + +− − +

3 3 3

2 2

1 3

3

4

4 2 4 4

1

2 4 2 4

4 2

[ 4γ 2( (2 2γ

(1 γ )α ) α α (2 2γ (1 γ )α )

α ( (1 γ ) (4 )(1 γ )α

2( 1 γ )α

(

)))].

m

m m

MEMBRANES AND MEMBRANE TECHNOLOGIES V

Case II. Kuwabara model:

(55)

Case III. Kvashnin model:

(56)

Case IV. Mehta-Morse/Cunningham model:

(57)

When i.e. porous cylinder becomes likea solid cylinder, then the hydrodynamic permeabi-lity (54)–(57) come out as:

− − − + + ++ − + − + + + − +

+ − + + − ++ − + +

+ − + − + ++ − + + − + + +

+ − + −

= 2 2 2 4 2

2 2 4 2 2

2

11

3 3

3

3

2 4

1

2 4

2 2 4 2

4 2 2 4

4

1

3

2

3

[ 32γ 2 ( 5 4γ γ ) 8 ln γ

α ( 16 (3 4γ γ ) 4 ln γ 8( 1 γ )α )

α (8( 1 γ ) α (2 2γ 8 ln γ

(3 4γ γ 4 ln γ)α ))

α ( 16 (3 4γ γ ) 4 ln γ

α ( 4 4γ (3 4γ γ ) 4(4 ) ln γ

(2 2γ 8 ln γ)α ))] / [

m m

m m

L

m m

m m

++ + + + + +

2 2

3

2 2 2

1 3 3 1 3 3

4γ 4( (2 α )

α α (2 α ) α ( α (4 2α )) )

(

) ].

m

m m

− + − − − ++ − + + − + + +

+ − + + + − + + − ++ + +

=

− − + −× − + + + +− − + + − +

+ − +

2 2 2 2 4 2 4

4 2 2 2 4

3

2 4 4 2 2

3 1

2 4 2 2

3

2 2 4

4 4

3

2

1

11 7 4(4 )γ 3 γ 2 ( 3 γ ) ln γ

α ( 2(6 2γ ( 1 γ )) (3 γ ) ln γ

2( 3 2γ γ )α ) α ( 2(6 2γ ( 1 γ ))

(3 γ ) ln γ α ( 2( 2 2γ )

( 1 γ ) (4 )(3 γ ) ln γ

( 1 γ 2( 3 γ ) ln γ)α ))

α (2( 3

[

2

m m m m

m m

m

m

m

L

m

+ + − − − ++ − + + − −

+ + + − + ++ + + + +

− − +

2 4 4 4

3

2 4 2 2 4

3

4 2 4 4

3 1 3 3

2 4 2 4

1 3

4 2

3

γ γ ) α (1 γ 2( 3 γ ) ln γ

(2 2γ (3 γ ) ln γ)α ))]/[ 4γ (6 2γ

(3 γ )α ) α α (6 2γ (3 γ )α )

α ( (3 γ ) (4 )(3 γ )α

2( 3 γ )α )].

(

)

m

m m

− − + − ++ − + − + − + + +

+ − + + − ++ − + − +

+ − + − + ++ − − − + + − +

+ − − + +

=

+

2 2 2 4 2 4

2 2 2 2 2 2

3

2 2 2 2

3 1

4 4

3

2 2 2

3

4 2 2 2 2 4

1

1

2

3

1

2 2 2

16γ (3 4γ γ ) 2 (1 γ ) ln γ

( 1 γ )α ( 4 (4 )γ (1 γ ) ln γ

2( 1 γ )α ) α (2( 1 γ )

α ( 1 γ 2(1 γ ) ln γ

( 1 γ )(1 γ ln γ γ ln γ)α ))

α (4 4γ ( 1 γ ) ( 1 γ ) ln γ

α ( ( 1 γ ) (4

[L m m

m m m

m m

m m − ++ − + − +

+ + −+ + + − + − − +

− + − + +

×

+

4

4 4 2

3

2 4 4

3 3

2 4 4 2 4

1 3 3 3 1

2 4 4 2

3 3

)( 1 γ ) ln γ

( 1 γ 2(1 γ ) ln γ)α ))]/[4γ

(2 2γ α γ α )

α α (2 2γ α γ α ) α ( ( 1 γ )

(4 )( 1 γ )α 2(1 γ )

(

)α ) ].

m

m

m

→ ∞m

ol. 1 No. 6 2019

400 SATYA DEO et al.

Fig. 2. Variation of natural logarithm of the dimensionlesshydrodynamic permeability with parameter for

in case of different cell models: Happel (1), Kvash-nin (2), Kuwabara (3), Cunningham (4).

8

6

4

2

0.80.60.40.2

ln(L11)

�

1–4

11L γ1m =

Fig. 3. Variation of the dimensionless hydrodynamic per-meability with parameter for in case of dif-ferent cell models: Happel (1), Kvashnin (2), Kuwabara (3),Cunningham (4).

1.0

0.8

0.6

0.4

0.2

862

L11

m104

1234

11L m =γ 0.5

Case I: Happel model:

Case II: Kuwabara model:

Case III: Kvashnin model:

Case IV: Mehta-Morse/Cunningham model:

Last four formulas are well-known results reportedearlier in [4, 5, 8].

DISCUSSION AND CONCLUSION

The natural logarithm of the dimensionless hydro-

dynamic permeability is depends upon the param-

eters and . The hydrodynamic permeabilitydecreases with increasing particle volume fraction

for inhomogeneous porous medium for all thefour cell models (Happel, Kuwabara, Kvashnin andCunningham) as shown in Fig. 2. This result is likethat in case of the homogeneous porous medium forall the four cell models [12]. For higher values of the Mehta-Morse model slightly deviates with othermodels which are almost indistinguishable. Thus,

is more inf luenced by particle volume fraction for

Happel model and least inf luenced for Mehta-Morse/Cunningham model. Here, the hydrody-namic permeability tends to infinite when particlevolume fraction tends to zero, i.e. no resistance in thef luid medium.

Figure 3 shows that the hydrodynamic permeabil-

ity decreases with increasing the value of m. In

other words, the hydrodynamic permeability of the

membrane is higher for the higher value of the per-

meability coefficient k0 for all cell models in the con-

sidered case of inhomogeneous porous medium. Thisresult coincides with the homogeneous porousmedium [10, 11] as well as with the inhomogeneousporous medium [22] for all the four cell models. ForHappel’s model the hydrodynamic permeability ishighest, whereas for Mehta-Morse/Cunninghammodel, it is lowest. For Cunningham’s model, thedecaying is most rapid as compare to the other models

4

11

4

2 4

γ γ1 2(1 ) ln γ

8 (1

.

γ γ )L

− ++

=− +

2

11

4

2

γ γ l3 4 n γ.

1

4

6γL − −+ −=

+ ++

= − −2 4

1 41 2

γ γ γ2 2 (3 ) ln

(3.

4γ γ )L

2 2

1 2 21

1 (1 ) ln

4

γ γ γ

).

1γ γ(L − −

+= + +

11Lγ m

2λ γ=

γ,

11L

11L

11L

MEMBRANES AND M

due to more restricting boundary condition on the

outer cell surface.

It is necessary to mention that in contrast to work

[22] we derived here exact analytical formulas (54)–

(57) for direct calculation of the hydrodynamic per-

meability of a fibrous membrane treated as a swarm of

parallel porous cylinders. It should also be noted that

a similar problem for a radially variable effective vis-

cosity was investigated in [23] in the case

of a jump in the shear stresses on the interphase sur-

face of a porous particle-pure f luid when flowing par-

allel to the axis of the cylinders. That model is more

suitable for superhydrophobic surfaces. Recently,

Ryzhikh and Filippov [24] dealt with another close

problem for a swarm of partly porous spherical parti-

cles with radially variable viscosity ,

where . All the mentioned models showed quali-

tatively similar results for the membrane permeability.

( )= 2μ μe r a

= αμ μ 1e r

α > 1

EMBRANE TECHNOLOGIES Vol. 1 No. 6 2019

HYDRODYNAMIC PERMEABILITY OF A MEMBRANE BUILT 401

APPENDIX ICase I. For Happel model:

= α + α α + α + αΔ

4 2 2 2 2

1 3 1 3 1 3

1

1[γ ( (1 ))],c m m

4 2 4 2

1 3

2 4 2 4

3

2 4 4

1 3

4 4 2 4 2

3 1

2 4 2 4 2

3

4 2 4

3

[ 4(1 γ ) ( 1 ln γ) ( γ ( 8 )

(8 ) 2(1 γ ) γ 4( 1 γ )α )

α ( 4 4γ 4( 1 γ ) ln γ

(1 γ 2(1 γ ) ln γ) ) ( γ ( 8 )

(8 ) 2(1 γ ) ln γ α ((1 γ )( 4 )

2(1 γ )(4 ) ln γ 4( 1 γ )α n )

n

) ]

l

l γ

m m

m m

m

m m m

m

Δ = − − − + + α − − ++ + + + + − +

+ − + + − + α+ − + + α + α − − +

+ + + + + − − ++ + + + − + ,

2 4 4

2 3

1

2 4 4 4 2

1 3 3 1

4 2 4 2

3 3

1[(2( ( 2 2γ (1 γ )α )

α α ( 2 2γ (1 γ )α ) α ((1 γ )

(1 γ )(4 )α 2( 1 γ )α ))],

c mD

m

m

= − + + +

+ − + + + + ++ + + + − +

− − − + + − − ++ + + + + − +

+ − + + − ++ − + + + − − +

+ + + + + − − ++ + + + − +

= 4 2 4 2

1 3

2 4 2 4

3

2 4 4

1 3

4 4 2 4 2

3 1

2 4 2 4 2

3

4 2 4

3

4(1 γ ) ( 1 ln γ) α ( γ ( 8 )

(8 ) 2(1 γ ) γ 4( 1 γ )α )

α ( 4 4γ 4( 1 γ ) ln γα

(1 γ 2(1 γ ) ln γ)α ) α ( γ ( 8 )

(8 ) 2( γ ) γ α (( γ )( 4 )

2( γ )(4 ) ln γ 4(

[

ln

1 ln 1

1 1 γ ) ln γα )))

D m m

m m

m

m m m

m ].

2 2 2 2

3 3 1 3 1 3

3

1[ ] ( (1 )) ,c m m= − α + α α + α + α

Δ4 2 4 2

3 3

2 4 2 4

3

2 4 4

1 3

4 4 2 4 2

3 1

2 2 4 4 2

3

4 2 4

3

[ 4(1 γ ) ( 1 ln γ) α ( γ ( 8 )

(8 ) 2( γ ) ln γ 4( γ )α )

α ( 4 4γ 4α ( γ ) ln γ

( γ 2

1 1

1

1

1

(1 γ ) ln γ)α ) α ( γ ( 8 )

(8 ) 2 ( γ ) ln γ α (( γ )( 4 )

2( γ )(4 ) ln γ 4α ( γ ) ln γ))]

1

1 1

m m

m m

m

m m m

m

Δ = − − − + + − − ++ + + + + − +

+ − + + − ++ − + + + − − +

+ + + + + − − ++ + + + − + ,

4 2 4 2 2

4 3

4

4 4 2 2

3 1

4 2 4 2 2

3 1 3

1[(4( 1 γ ) α (γ ( 8 ) (8 )

4α (1 γ )) α (γ ( 8 ) (8 )

α (1 γ )( 4 )) (1 γ )α ( 4 α ))],

c m m m

m m

m

= − − + + − + − +Δ

+ − + − + − +− − − + − − − +

4 2

4

4 2 2 4 2

3

4 2 4 4

3 1 3

4 4 2 4 2

3 1

2 4 2 4 2

3

4 2 4

3

[ 4(1 γ ) ( 1 ln γ)

α ( γ ( 8 ) (8 ) 2(1 γ ) ln γ

4( 1 γ )α ) α ( 4 4γ 4α ( 1 γ ) ln

(1 γ 2(1 γ ) ln γ)α ) α ( γ ( 8 )

(8 ) 2( γ ) ln γ α ((1 γ )( 4 )

2(1 γ )(4 ) ln γ 4

1

α ( 1 γ ) ln γ))]

m

m m m

m

m m m

m

Δ = − − − ++ − − + + + + +

+ − + + − + + − + γ+ − + + + − − +

+ + + + + − − ++ + + + − + ,

MEMBRANES AND MEMBRANE TECHNOLOGIES V

Case II. For Kuwabara model:

4 2 4 2

1 3

5

4 2 4 3

3 3

14((1 γ ) (1 γ ) α[

2( γ )α ( γ ) ) ,1 α ]1

m mb − + −

+ + +

= −Δ

− +

− − − − ++ − − + + + + + γ

+ − + + − + + − ++ − + + + − − +

+ + +

Δ

+ + − − ++ + + + − +

= 2 4

4 2 2 4

5 1 3

3

3

2

4 2 4 4

4 4

1 3

3 1

2 4 2

2 4 2 4 2

3

4

3

2

(α α )[( 4 ( γ )( 1 ln γ)

α ( γ ( 8 ) (8 ) 2(1 γ ) ln

4( 1 γ )α ) α ( 4 4γ 4α ( γ ) ln γ

(1 γ 2( γ ) ln γ)α ) α ( γ ( 8 )

(8 ) 2( γ ) ln γ α (( γ )( 4 )

2(1 γ )(4 ) ln γ

1

1

1

1 1

4α ( γ1

m

m m m

m

m m m

m 4) ln γ))],

4 2 4 2

3 1

6

4 2 4 3

1 1

4(( 1 γ ) ( γ ) α

2( γ )α ( γ )α )

1

1 1 ,

[

]

1 m mb − − + + − +

− + −

=

+Δ

2 4

6 1 3

4 2 2 4 2

3

4 2 4 4

3 1 3

4 4 2 4 2

3 1

2 4 2 4 2

3

4 2 4

(α α )[ 4 (1 γ )( 1 ln γ)

α ( γ ( 8 ) (8 ) 2(1 γ ) ln γ

4( 1 γ )α ) α ( 4 4γ 4( 1 γ )α ln γ

(1 γ 2(1 γ ) ln γ)α ) α ( γ ( 8 )

(8 ) 2(1 γ ) ln γ α ((1 γ )( 4 )

2(1 γ )(4 ) ln γ 4( 1 γ )α

m

m m m

m

m m m

m

− − − − ++ − − + + + + +

+ − + + − + + − ++ − + + + − − +

+ + + + + − − ++ + + + − +

Δ =

3 ln γ))].

2 2 2

1 3

1

2 2 2 2

1 3 3 1

2 2 2

3 3

1[γ ( ( 2 ( 2γ )α )

α α ( 2 (1 2γ )α ) α (( 2γ )

α ( 2γ (4 ) 2α ))

1

)],

1

c m

m

m m

= − − + −Δ

+ − + − + −+ − + + −

2 4 2 2 2 4 2

1

4 2 4 2 4 2

3

2 2 2 2 2

3 1

4 4 2 4 4 2

3 3

4 2 4 2 4 2

1

2 2 4 2 2

3

4 2

[2( 5γ 4γ (4 ) 4γ ln γ)

α (16γ (1 4γ 3γ ) 4γ ln γ

8γ ( 1 γ )α ) α (8γ ( 1 γ )

2(1 γ 4γ ln γ)α (1 4γ 3γ 4γ ln γ)α )

α (16γ (1 4γ 3γ ) 4γ ln γ

α (4γ γ (4 3 ) (4 )

4γ (4

m m m m

m m

m m

m m m

m

Δ = − + + ++ − − + +

+ − + + − ++ − + − − + −

+ − − + ++ + − − +

+ + 4 4

3) ln γ 2(1 γ 4γ ln γ)α ))],+ − +

4 2 2

2 3 1 3 3

2

2 2

1 3 3

1[4γ ( (2 α ) α α (2 α )

α ( α (4 2α )))],

c mD

m m

= − − + − +

− + + +

ol. 1 No. 6 2019

402 SATYA DEO et al.

= − + + ++ − − + +

+ − + + − ++ − +

− − + −+ − − + +

+ + − − ++ +

2 4 2 2 4 2

2

4 2 4 2 4 2

3

2 2 2 2 2

3 1

4 4

3

2 4 4 2

1

4 2 4 2 4 2

1

2 2 4 2 2

3

4

[2( (1 5γ ) 4γ (4 ) 4γ ln γ)

α (16γ (1 4γ 3γ ) 4γ ln γ

8γ ( 1 γ )α ) α (8γ ( 1 γ )

2(1 γ 4γ ln γ)α

( 4γ 3γ 4γ ln γ)α )

α (16γ ( 4γ 3γ ) 4γ ln γ

α (4γ γ (4 3 ) (4 )

4

1

γ (4

1

D m m m

m m

m m

m m m

m + − +2 4 4

3) ln γ 2(1 γ 4γ ln γ)α ))],

2 2 2

3 3 1 3 3

3

2 2

1 3 3

1[γ ( (2 α ) α α (2 α )

α ( α (4 2α )))],

c m

m m

= + + +Δ

+ + + +

Δ = − − + + ++ − + − + −+ − + − − −

+ + − + −+ − + − + −

+ − + − + − +

2 4 2 2 2 4 2

3

4 2 4 2 4 2

3

2 2 2 2 2 4

3 1

4 2 4 4

3 3

4 2 4 2 4 2

1

4 2 4 2 4 2

3

[ 2( 5γ 4γ (4 ) 4γ ln γ)

α ( 16γ (1 4γ 3γ ) 4γ ln γ

8γ (1 γ )α ) α (8γ (1 γ ) 2(1 γ

4γ ln γ)α (1 4γ 3γ 4γ ln γ)α ))

α ( 16γ (1 4γ 3γ ) 4γ ln γ

α (4 4γ (1 4γ 3γ ) 4γ (4 ) l

m m m m

m m

m m

m m

− − +4 4

3

n γ

2(1 γ 4γ ln γ)α ))],

2 2 2 2

4

4

2 2 2 2

3 3

2 2 2 2 2 2

1 3

2 2 2

1 3

1[2γ (16 4γ 4

α ( γ ( 8 ) 4(1 γ )α )

α ( γ ( 8 ) ( γ ( 4 ))α )

α ( 1 γ )( 4 α ))],

c m m

m m

m m m m

= − − +Δ

+ − + − −+ − + + − +

− − + +−

−

−−

2 4 2 2 4 2

4

4 2 4 2 4 2

3

2 2 2 2 2

3 1

4 4 2 4 4 2

3 3

4 2 4 2 4 2

1

2 2 4 2 2

3

4

[2( (1 5γ ) 4γ (4 ) 4γ ln γ)

α (16γ (1 4γ 3γ ) 4γ ln γ

8γ ( 1 γ )α ) α (8γ ( 1 γ )

2(1 γ 4γ ln γ)α ( 4γ 3γ 4γ ln γ)α )

α (16γ ( 4γ 3γ ) 4γ ln γ

α (4γ γ (4 3 ) (4 )

4γ (4

1

1

m m m

m m

m m

m m m

m

Δ = − + + ++ − − + +

+ − + + − ++ − + − − + −

+ − − + ++ + − − +

+ + 2 4 4

3) ln γ 2( γ 4γ ln γ)α )],1 )+ − +

2 2 2 2 2 2

1 3

5

2 2

3 3

1[8γ ((1 γ ) ( γ (4 )

(2γ ( 1 γ )α )))],

b m m m= − + α − + +Δ

+ α + − +

MEMBRANES AND M

Case III. For Kvashnin model:

Δ = − − − + ++ + − + − +− + − + −

− − + + − + −+ − + − + −

+ − + − + −

2 4 2 2

5 1 3

4 2 4 2 4 2

3

4 2 2 2 2 2 2

3 1

4 4 2 4 4 2

3 3

4 2 4 2 4 2

1

4 2 4 2

3

(α α )[( 2( (1 5γ ) 4γ (4 )

4γ ln γ) α ( 16γ (1 4γ 3γ )

4γ ln γ 8γ (1 γ )α ) α (8γ (1 γ )

2(1 γ 4γ ln γ)α (1 4γ 3γ 4γ ln γ)α )

α ( 16γ (1 4γ 3γ ) 4γ ln γ

α (4 4γ (1 4γ 3γ ) 4

m m

m m

m

m m

m +− − +

4 2

4 4

3

γ (4 ) ln γ

2(1 γ 4γ ln γ)α ))],

m

2 2 2 2 2 2

3 1

6

2 2

1 1

1[8γ ((1 γ ) α ( γ (4 )

α (2γ ( 1 γ )α )))],

b m m m= − − + − + +Δ

+ + − +

Δ = − − − + ++ + − + − +− + − + −

− − + + − +− + − + − +− + − + − +−

2 4 2 2 21 36

4 2 4 2 4 23

4 2 2 2 2 2 23 1

4 4 2 43

4 2 4 2 4 23 1

4 2 4 2 4 23

(α α )[( 2( 5γ 4γ (4 )

4γ ln γ) α ( 16γ (1 4γ 3γ )

4γ ln γ 8γ (1 γ )α ) α (8γ (1 γ )

2(1 γ 4γ ln γ)α (1 4γ 3γ

4γ ln γ)α ) α ( 16γ (1 4γ 3γ )

4γ ln γ α (4 4γ (1 4γ 3γ )

4γ

m m mm mm

mm m

+ − − +4 2 4 43(4 ) ln γ 2(1 γ 4γ ln γ)α ))].m

2 2 2

1 3

1

2 2 2 2

1 3 3 1

2 2 2

3 3

1[γ ( ( 2 (1 3γ )α )

α α ( 2 (1 3γ )α ) α ((1 3γ )

α ( 3γ (4 ) 2α )))],

c m

m

m m

= − − + −Δ

+ − + − + −+ − + + −

2 4 2 2 2 4 2

1

4 2 2 2 4 2

3

2 4 4

3 1

2 2 2 4 2

4 2 2 2 4 2

3

4 4 2

3 1

[2(3 7γ 4γ (4 ) 2(1 3γ ) ln γ

α (4(1 3γ ) 2γ (1 γ ) (1 3γ ) ln γ

2(1 2γ 3γ )α ) α (4(1 3γ )

2γ (1 γ ) (1 3γ ) ln γ

α ( 4 4γ 2γ (1 γ ) (1 3γ )(4 ) ln γ

(1 γ 2(1 3γ ) ln γ)α )) α ( 2(1

m m m m

m m

m m

m m

Δ = − + + − −+ + + − + +

− + − + ++ − + +

+ − + + − + + ++ − − − + − 2 2

4 4 2 2

3

4

3

γ )(1 3γ )

α (1 γ 2(1 3γ ) ln γ (2γ (1 γ )

(1 3γ ) ln γ)α )))],

− ++ − − − + −

+ +

2 4 4 4 2

2 1

3

4 2 4 2

1 1 3

4 4 2

1 1 3

1[( ( 2 6γ (1 3γ )α ) ((1 3γ )

(1 3γ )(4 )α 2(1 3γ )α )α

α ( 2 6γ (1 3γ )α )α )],

c m mD

m

= − − − + + + − +

+ + + − −− − + + +

EMBRANE TECHNOLOGIES Vol. 1 No. 6 2019

HYDRODYNAMIC PERMEABILITY OF A MEMBRANE BUILT 403

4 2 2 2 4 2

4 2 2 2 4 2

3

2 4 4 2 2 2

3 1

4 2 4 2 2 2

3

4 2 4 4

2

3

3

1

3 7γ 4γ (4 ) 2(1 3γ ) γ

α (4(1 3γ ) 2(1 γ )γ (1 3γ ) γ

2(1 2γ 3γ )α ) α (4(1 3γ ) 2(1 γ )γ

(1 3γ ) γ α ( 4 4γ 2(1 γ )γ

(1 3γ )(4 ) ln γ (1 γ 2(1 3γ )

[( )

ln γ

ln

ln

)α ))

α ( 2(1 γ

ln

D m m m

m m

m

m m

m

− + + − −+ + + − + +

− + − + + + −+ + + − + + −

+ + + +

=

− − −+ − − 2 2 4 4

3

2 2 4

3

)(1 3γ ) α (1 γ 2(1 3γ ) ln γ

(2(1 γ )γ (1 3γ ) ln γ)α ))],

+ + − − −+ − + +

= − + + +Δ+ + + +

+ + + +

2 2 2 2 2

3 3 1 3

3

2 2 2

2 2

3 1

2 2

3 3

1[( (2γ (1 γ )α ) α α (2γ

(1 γ )α ) α ((1 γ )

α (4γ (1 γ ) 2γ α )))],

c m

m

m

4 4

4 4

4 4

4 4

4

2 2 2 2

3

2 2 2 2

3

2 2 2 2

3 1

2 2 2 2

3

42

2

1

4

3

[2( (3 7γ ) 4γ (4 ) 2(1 3γ ) ln γ

α (4(1 3γ ) 2γ (1 γ ) (1 3γ ) ln γ

2(1 2γ 3γ )α ) α (4(1 3γ ) 2γ (1 γ )

(1 3γ ) ln γ α ( 4 4γ 2γ (1 γ )

(1 3γ )(4 ) ln γ (1 γ 2(1 3γ ) ln γ)α ))

α ( 2(1

m m m

m m

m

m m

m

Δ = − + + − −+ + + − + +

− + − + + + −+ + + − + + −

+ + + + − − −+ − 2 2

3

2 2

3

4 4

4

γ )(1 3γ ) α (1 γ 2(1 3γ ) ln γ

(2γ (1 γ ) (1 3γ ) ln γ)α )))],

− + + − − −+ − + +

2 4 2 2

4

4

2 2 4 2 2

1

2 4 2 2 4 2

1

2 2 2 2 4 2

1 3

2 4 2 2

1 3

1[( 4 (1 3γ ) 8γ (4 )

α ( 2γ 3γ ( 8 ) (8 )

4(1 2γ 3γ )α ) (2γ 3γ ( 8 )

(8 ) (2γ ( 4 ) 3γ ( 4 ))α )α

(1 2γ 3γ )( 4 α )α )],

c m m

m m m

m m

m m m m

= − − − − +Δ

+ − + − + − ++ + − − − − +

+ + + + − + − − +− + − − +

2 4 2 2 4 2

4

4

3

3

3

2 2 2 4 2

2 4 4 2 2 2

1

4 2 4 2 2 2

4 2 4 4

2

3

1

[2( (3 7γ ) 4γ (4 ) 2(1 3γ ) ln γ

α (4(1 3γ ) 2γ (1 γ ) (1 3γ ) ln γ

2(1 2γ 3γ )α ) α (4(1 3γ ) 2γ (1 γ )

(1 3γ ) ln γ α ( 4 4γ 2γ (1 γ )

(1 3γ )(4 ) ln γ (1 γ 2(1 3γ ) ln γ)α ))

α ( 2(1

m m m

m m

m

m m

m

Δ = − + + − −+ + + − + +

− + − + + + −+ + + − + + −

+ + + + − − −+ − 3

3

2 2 4 4

2 2 4

γ )(1 3γ ) α (1 γ 2(1 3γ ) ln γ

(2γ (1 γ ) (1 3γ ) ln γ)α )))],

− + + − − −+ − + +

= − + − + −Δ

+ + + + − + −

2 4 2 2 4

1 3

5

2 2 4 2 4 2

3 3

1[2((1 2γ 3γ ) α ( (1 3γ )

2γ (4 ) 2(1 3γ )α (1 2γ 3γ )α ))],

b m m

m

MEMBRANES AND MEMBRANE TECHNOLOGIES V

Case IV. For Mehta-Morse/Cunningham model:

2 4 2 2

5 1 3

4 2 4 2 2 2

3

4 2 2 4 4

3 1

2 2 2 4 2 4

3

2 2 2 4 2

4 4

3

α α ( (3 7γ ) 4γ (4 )

2(1 3γ ) ln γ α (4(1 3γ ) 2γ (1 γ )

(1 3γ ) ln γ 2(1 2γ 3γ )α ) α (4(1 3γ )

2γ (1 γ ) (1 3γ ) ln γ α ( 4(1 γ )

2γ (1 γ ) (1 3γ )(4 ) ln γ

(1 γ 2(1 3γ ) ln γ)α

)[

)

(

)

m m

m m

m

m m

m m

Δ = − − + +− − + + + −

+ + − + − + ++ − + + + − +

+ − + + ++ − − − + 2 2 2

1

4 4 2 2

3

4

3

α ( 2(1 γ )(1 3γ )

α (1 γ 2(1 3γ ) ln γ (2γ (1 γ )

(1 3γ ) ln γ)α ))],

− − ++ − − − + −

+ +

2 4 2 2 4

3 1

6

2 2 4 2 4 2

1 1

1[2( (1 2γ 3γ ) α ( (1 3γ )

2γ (4 ) 2(1 3γ )α (1 2γ 3γ )α ))],

b m m

m

= − − + − + − +Δ

− + − + + + −

2 4 2 2

6 1 3

4 2 4 2 2 2

3

4 2 2 4 4

3 1

2 2 2 4 2

4 2 2 2 4 2

3

4 4 2

3 1

α α (3 7γ ) 4γ (4 )

2(1 3γ ) ln γ α (4(1 3γ ) 2γ (1 γ )

(1 3γ ) ln γ 2(1 2γ 3γ )α ) α (4(1 3γ )

2γ (1 γ ) (1 3γ ) ln γ

α ( 4 4γ 2γ (1 γ ) (1 3γ )(4 ) ln γ

(1 γ 2(1 3

( )[

γ ) ln γ)α )) α

m m

m m

m

m m

m m

Δ = − − + +− − + + + −

+ + − + − + ++ − + +

+ − + + − + + ++ − − − + 2

2 4 4 2 2

3

4

3

( 2(1 γ )

(1 3γ ) α (1 γ 2(1 3γ ) ln γ (2γ (1 γ )

(1 3γ ) ln γ)α ))].

− −× + + − − − + −

+ +

= − −Δ

− −−

+ +

+ + + ++ +

2 2 2

2 2 2 2

2 2

1 3

1

1 3 1

2

3

3 3

[ ( (2 ( 1 ) )

+ (2 ( 1 )

1γ γ α

α ) (( 1 )

+ ( (4 ) 2 )))]

α γ α α

,

γ

α γ α

c m

m

m m

Δ = −− − − −

−−

++ +

−− − −

− −

+

−

−

2 4 2 2 2

1

4 2 2 2 2

2 2 2 2

2 2 2 4

1

4 2 2 4

4 2 2 2 4 2

1

3

3

3

3

3

γ γ

γ ln γ γ α γ

γ ln γ γ α

α

[2 ( + 3 4 (4 + )

2(1 ) +(1 ) ( ( 4 + )

+ (4 + ) + (1 ) 2( 1 ) )

+ ( 2(1 ) + ( 1 +

2(1+ ) +

γ α γ

γ ln γ γ γ ln γ α

α γ

((1 ) + (1 ) ) ))

+ (4 4 + (1 ) + (1 )

+ (

γ γ ln γ

α

m m m

m m

m m

m m

+− −

− −

2 2 2

3

4 2

4 4

γ γ ln γ(1 ) +(1 )(4 + )

(1 +2(1 ) )γ γ ln γ α )))],

m m

+ + +

+ + + ++ +

= − −

−− +

−−

2 4 4

2 3

2 4 4 4 2

1 3 3 1

4

4 2 4 2

3 3

1γ γ α

α α γ γ α α γ

γ

[ (2(1 ) ( 1 ) )

+ (2(1 )+( 1 ) ) (( 1 )

(1 ) α γ(4 ) 2(1 ) )],α

c mD

m

m

ol. 1 No. 6 2019

404 SATYA DEO et al.

− −− −

= + + ++ + +

+ ++ + +

−−

− − − −− + −

− − −+ −

++ −

+

2 4 2 2 2 4 2

2 2 2 2

3

2 2 2

3

2 2 2 4 4

1 3

2 2 4

3

4 2 2 2 4

4

2 1

2 2 2

3

γ γ γ ln γ

γ α γ

γ ln γ γ α

α γ α γ γ ln γ

γ γ ln γ α

[ 3 4 (4 ) 2(1 )

+ (1 ) ( ( 4 ) (4 )

+ (1 ) + 2( 1 ) )

( 2 (1 ) + ( 1 2 (1 )

+ ((1 ) (1 ) ) ))

(4 4 (α γ γ γ ln γ

α

1 ) + (1 )

((1 )γ γ(1

D m m m m

m m

m

m m

m +× + +− −

4 2

4 4

3

)(4 )

(1 ln γ γ γ l2(1 ) )n γ α ))],

m

= − − −Δ

− − −− −

+

+ ++ + +

2 2 2

3 3

3

2 2 2 2 2

1 3 3 1

2 2 2 2

3 3

[( ( 2 (1 ) )

+ ( 2 (1 ) ) ((1 )

(4 ( 1 )

1γ γ α

α α γ γ α α γ

α γ γ α2 )) ,γ ) ]

c m

m

m

−− − − −

−− − − −

− −

Δ = + ++ ++ + + +

+ +

+ ++

+

− −− −

2 4 2 2 2

3

4 2 2 2 2

3

2 2 2 2

3

2 2 2 4 4

1 3

2 2 4

3

4 2 2 2

1

4 2 2 2 2

3

[2( 3 4 (4 )

2(1 ) + (1 ) ( ( 4 )

+ (4 ) (1 ) + 2( 1 ) )

+ ( 2(1 ) + ( 1 2(1 )

+

γ γ

γ ln γ γ α γ

γ ln γ γ α

((1 ) + (1 ) ) ))

(4 4 (1 )

+ (1

α γ α γ γ ln γ

γ γ ln γ α

α γ γ

γ ln) ((1γ α γ )

(1

m m m

m m

m m

m

m m

++− +

−−

4 2

4 4

3

γ ln γ

γ γ l

) (

n γ

4 )

(1 2(1 ) )α )))],

m

= − − −Δ

−

+ +

× + + + + ++ + +

+ +

− −− − −

− − −

2 2 2 2 2

4

4

2 2 2 2

3 3

4 2 2 2 4

1

2 2 2 2 2 2 2

3 1 3

[ 32 4(1 ) (1 )

( ( 8 ) (8 ) 4( 1 ) )

+ (8 8 (1 ) ( 4(1 )

+ (1 ) ) (1 ) ( 4 )

1γ γ γ

α γ γ α

α γ γ

],

γ

γ α γ α α

c m

m m

m

m

Δ = − −× − − −

−− −

+ + ++ + + +

+ + ++ +− −

− −−

+

+ ++

− −− −

2 4 2 2 2 4 24

2 2 2 23

2 2 23

2 2 2 4 41 3

2 2 43

4 2 2 2 4 21

2 2 23

γ γ γ

ln γ γ

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

(1 ) ( ( 4 ) (4 )

+ (1 ) 2( 1 ) )

+ ( 2(1 ) ( 1 2(1 )

+ ((1 ) + (1 ) ) ))

α γ

γ ln γ γ α

α γ α γ γ

+ (4 4 (1 ) (1 )

ln γ

γ γ ln γ α

α γ γ γ ln γ

α γ+ ((1 ) (1

m m m mm m

m

m mm +

+− − +

4 2

4 43

)(4 )

(1

γ ln γ

γ γ l2 (1 ) ) )n α ,γ ))]

m

= − − − −

−

+ +Δ

− −

2 2 2 2 2 2 2

1 3

5

4 2 2 2

3 3

[2(1 ) 2 ( 8 (1 )

+ 2(1 ) (1 )

1γ α γ γ

γ α γ ]α ) ,

b m m

MEMBRANES AND M

FUNDING

This work was carried out on an initiative basis without

attracting funding.

CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.

REFERENCES

1. D. A. Nield and A. Bejan, Convection in Porous Media,3rd Ed. (Springer, New York, 2006).

2. H. C. Brinkman, J. Appl. Sci. Res., Sect. A 1, 27 (1947).

3. J. Happel, AIChE J. 5, 174 (1959).

4. S. Kuwabara, J. Phys. Soc. Jpn. 14, 527 (1959),

5. A. G. Kvashnin, Fluid Dyn. 14, 598 (1979).

6. G. D. Mehta, and T. F. Morse, J. Chem. Phys. 63, 1878(1975).

7. E. Cunningham, Proc. R. Soc. London A 83, 357(1910).

8. J. Happel, and H. Brenner, Low Reynolds Number Hy-drodynamics (Martinus Nijhoff, The Hague, 1983).

9. S. Deo, Sadhana 29, 381 (2004).

10. S. I. Vasin and A. N. Filippov, Colloid. J. 71, 141(2009).

11. S. Deo, P. K. Yadav, and A. Tiwari, Appl. Math. Mod.34, 1329 (2010).

Δ = − −− γ − − −

−−

+ ++ + ++ + + +

+ + ++

+

− − −− −

− − + −+−

2 4 2 2 2

5 1 3

4 2 2 2 2

3

2 2 2 2

3

2 2 2 4 4

1 3

2 2 4

3

4 2 2 2 4 2

1

2 2 2

3

α α γ γ

γ ln γ α γ

γ ln γ γ α

α γ α γ γ ln γ

γ γ ln γ

( )[( 3 4 (4 )

2(1 ) (1 ) ( ( 4 )

+ (4 )+(1 ) 2 ( 1 ) )

+ ( 2(1 ) ( 1 2(1 )

+ ((1 ) (1 ) ) ))

+

α

α γ γ γ (4 4 (1 ) (1 ) ln γ

)γ+ 1α ((

m m m

m m

m m

m m

m ++

−− +−

4 2

4 4

3

γ ln γ

γ

(1 )(4 )

(1 2(1 )γ ln γ α) ))],

m

= − − − − −Δ

− −

+

+ +

2 2 2 2 2 2 2

3 1

6

4 2 2

1 1

[ 2(1 ) 2 (8 (1 )

+ ( 2 2 (1

1γ α γ γ

α γ γ ) ],α ))

b m m

Δ = − −− γ − − −

−− − −

− −− − −−

+ ++ + +

+ + + + ++ +

+ ++ +

+ +−

2 4 2 2 2

6 1 3

4 2 2 2 2

3

2 2 2 2

3

2 2 2 4 1 3

4 2 2

4 4 2 2 2

3 1

4 2 2 2 2

3

α α γ γ

γ ln γ α γ

γ ln γ γ α

α γ α γ

γ ln γ γ

γ ln γ

( )[ 3 4 (4 )

2(1 ) (1 ) ( ( 4 )

(4 ) + (1 ) 2( 1 ) )

( 2(1 ) + ( 1

2(1 ) ((1 )

+ (1 ) ) )) (4 4 (1 )

+

α α γ γ

γ ln γ α γ(1 ) ((1 )

m m m

m m

m m

m

m m

× + +− +−

−

4

2 4 4

3

γ

ln γ γ

(1 )

γ ln(4 γ) (1 2(1 ) )α ))].m

EMBRANE TECHNOLOGIES Vol. 1 No. 6 2019

HYDRODYNAMIC PERMEABILITY OF A MEMBRANE BUILT 405

12. S. Deo, A. Filippov, A. Tiwari, et al., Adv. Colloid In-terface Sci. 164, 21 (2011).

13. S. I. Vasin, E. E. Sherysheva, and A. N. Filippov, Col-loid J. 73, 167 (2011).

14. B. Mishra and S. N. Panday, Adv. Theor. Appl. Mech.5, 113 (2012).

15. P. K. Yadav, Meccanica. 48, 1607 (2013).

16. P. K. Yadav, A. Tiwari, S. Deo, et al., Colloid J. 75, 473(2013).

17. H. H. Sherief, M. S. Faltas, E. A. Ashmawy, andA. M. Abdel-Hameid, Eur. Phys. J. Plus 129, 217(2014).

18. S. I. Vasin and T. V. Kharitonova, Colloid J. 76, 662(2014).

19. S. Deo, I. A. Ansari, and B. G. Srivastava, Adv. Theor.Appl. Mech. 9, 21 (2016).

20. V. K. Verma and S. K. Singh, Int. J. Pure Appl. Math.118, 321 (2018).

21. P. K. Yadav, Eur. Phys. J. Plus, 133, 1 (2018).

22. S. I. Vasin, Colloid J. 72, 315 (2010).

23. A. Filippov and Yu. Koroleva, Appl. Math. Comp. 338,363 (2018).

24. P. O. Ryzhikh and A. N. Filippov, Colloid J. 80, 199(2018).

MEMBRANES AND MEMBRANE TECHNOLOGIES Vol. 1 No. 6 2019