Applied Mathematical Modelling 34 (2010) 1329–1343

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Slow viscous flow through a membrane built up from porouscylindrical particles with an impermeable core

Satya Deo a, Pramod Kumar Yadav b,*, Ashish Tiwari a,*

a Department of Mathematics, University of Allahabad, Allahabad 211002, Indiab Mathematics Group, Birla Institute of Technology and Science, Pilani 333031, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 July 2008Received in revised form 9 August 2009Accepted 19 August 2009Available online 23 August 2009

Keywords:Cylindrical porous shellParticle-in-cell modelStokes flowBrinkman equationDrag force

0307-904X/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.apm.2009.08.014

* Corresponding authors.E-mail addresses: satyadeo@allduniv.ac.in (S. De

This paper concerns the slow viscous flow through an aggregate of concentric clusters ofporous cylindrical particles with Happel boundary condition. An aggregate of clusters ofporous cylindrical particles is considered as a hydro-dynamically equivalent to solid cylin-drical core with concentric porous cylindrical shell. The Brinkman equation inside the por-ous cylindrical shell and the Stokes equation outside the porous cylindrical shell in theirstream function formulations are used. The drag force acting on each porous cylindricalparticle in a cell is evaluated. In certain limiting cases, drag force converges to pre-existinganalytical results, such as, the drag on a porous circular cylinder and the drag on a solidcylinder in a Happel unit cell. Representative results are then discussed and presented ingraphical forms. The hydrodynamic permeability of the membrane built up from porousparticles is evaluated. The variation of hydrodynamic permeability with different parame-ters is graphically presented. Some new results are reported for flow pattern in the porousregion. Being in resemblance with the model of colloid particles with a coating of porouslayers due to adsorption phenomenon, results obtained through this model can be usefulto study the membrane filtration process.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

The classical problems of motion of objects through fluids continue to be of interest because of their applications in phys-ical sciences and chemical engineering. A variety of physical situations arises in which the size of moving objects varies frommicro (10�6 m) to nano (10�9 m) scales.

The computational predictions of the relevant hydro-dynamical parameters of the flow of a viscous incompressible fluidpast a swarm of porous particles at nano scale are of considerable practical and theoretical interest to many physical, engi-neering and medical problems [1]. It occurs in the movement of water and other fluids in the sandy or the earthen soil, theflow of water through the porous bank of rivers, intrusion of sea water to coastal areas, etc. In reality, the particle may bearbitrarily shaped and of a fairly odd structure, such as solid core covered with one or more porous layers, a void one layeror multi layered shell of porous material, or only a single porous particle with uniform or varying permeability, exhibiting auniform slow translation motion relative to the quiescent unbounded fluid. Despite the inherent as well as practical useful-ness of the above mentioned creeping flow problems, the analytical solutions to these problems have not been found exceptfor the geometrically simpler cases like cylinders, spheres, etc. Earlier, these cases were investigated by various authors indifferent ways as briefly described here. Brinkman [2] evaluated the viscous force exerted by a flowing fluid on a dense

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o), pramod547@gmail.com (P.K. Yadav), ast79.ashish@gmail.com (A. Tiwari).

Nomenclature

a radius of inner solid cylindrical coreb radius of porous cylindrical particlec radius of cylindrical cellr, h, z cylindrical polar co-ordinatesF drag forcep fluid pressure at any pointvr, vh velocity components of fluid at any pointTrr normal stressTrh tangential stressU uniform velocity of fluidCD drag coefficientI±n(rr) modified Bessel functions of the first kind and of order nRe Reynolds numberk permeabilityr dimensionless permeability parameterl1 viscosity of clear fluidl2 effective viscosity of porous mediumw stream functionc particle volume fraction‘ ¼ a

b and m ¼ cb dimensionless quantities

v fluid velocity at any pointv(1) quantity v for the region outside the porous cylinderv(2) quantity v for the region inside the porous cylinderr2 Laplacian operatoreL11 dimensional hydrodynamic permeabilityL11 non-dimensional hydrodynamic permeability

1330 S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343

swarm of particles by introducing modified Darcy’s equation for porous medium, which is commonly known as Brinkmanequation.

In the analysis of flow through swarm of particles, we get cumbersome calculations, if we consider the solution of the flowfield over the entire swarm by taking exact positions of the particles. In order to avoid the above complication, it is sufficientto obtain the analytical expression by considering the effects of the neighboring particles on the flow field around a singleparticle of the swarm, which can be used to develop relatively simple and reliable models for heat and mass transfer. This haslead to the development of particle-in-cell models.

Filippov et al. [3] used the cell method to model the permeability of a membrane of porous particles with a permeableshell. They studied the influence of porous shell on permeability by applying Mehta–Morse boundary conditions on the cellboundary.

Uchida [4] proposed a cell model for a sedimenting swarm of particles, considering spherical particle surrounded by afluid envelope with cubic outer boundary. This was accurately solved by Brenner [5].

Happel [6,7] and Kuwabara [8] proposed cell models in which both particle and outer envelope are spherical/cylindrical.The merit of this formulation is that, it leads to an axially symmetric flow that has a simple analytical solution in closed formand thus can be used for heat and mass transfer calculations. However, it has a disadvantage that the outer envelope is notspace filling, a difficulty which must be dealt with, when one tries to pass from the single unit cell to the assemblage of par-ticles. The Happel model assumes that the inner sphere – while at the center – moves with a constant velocity along the axis.Along with this, it assumes no slip flow on the inner sphere, nil radial velocity and nil shear stress on the outer envelope. TheKuwabara model assumes that the inner sphere is stationary and that fluid passes through the unit cell. The following bound-ary conditions are imposed: nil radial and tangential velocity on the inner sphere/cylinder, velocity with axial componentequal to a constant approach velocity on the outer envelope and nil vorticity on the outer envelope. Although, both the formu-lations give the same velocity fields and approximately equal drag forces yet, Happel formulation has a significant advantage inthat it does not require an exchange of mechanical energy between the cell and the environment. However, the Kuwabara for-mulation requires a small exchange of mechanical energy with the environment. The mechanical power given by the sphere tothe fluid is not all consumed by viscous dissipation in the fluid layer. Rather, a small part is given to the environment.

Dassios et al. [9] did a path breaking work by finding the solution of Stokes equation in spheroidal co-ordinates. Later,Dassios et al. [10] used the above solution to apply the cell model technique on Stokes flow in spheroidal particle-in-cellmodels, which has wide ranging applications in problems concerning the flow through swarms of particles. Dassios and Vaf-eas [11] studied the 3-D Happel model for Stokes flow through a swarm of spherical particles by solving the boundary valueproblem using Papkovich–Neuber differential representation of the solutions for the Stokes flow. They obtained analyticalexpressions for the velocity, the pressure and the stress tensor.

S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343 1331

Analytical solutions of particle-in-cell models discussed above are always practically useful to many industrial problems,but the solutions of creeping flow for the above models have not been found in case of complex geometry. However, for geo-metrically simpler cases like cylinder, the analytical solutions were investigated by various authors along with Happel andKuwabara. Here is a brief description of their work:

Stechkina [12] evaluated the drag force experienced by porous cylinders in a viscous fluid at low Reynolds number. Popand Cheng [13] reported an analytical study of the steady incompressible flow past a circular cylinder embedded in a con-stant porosity medium based on the Brinkman model and obtained an exact solution for the governing equations. They hadproved that the separation of the flow does not take place at the surface of the cylinder. They had also explained the velocityovershoot behaviour (i.e. tangential velocity increases from zero at the wall to a maximum at small distance from the wall,and then decreases to its asymptotic value far away from the wall). Singh and Gupta [14] had discussed the problem of uni-form flow past a permeable inhomogeneous circular cylinder by assuming that the flow in the porous cylinder is governed byDarcy’s law. Gupta [15] had solved the problem of flow past a porous cylinder by matched asymptotic expansion as done byKaplun [16] for an impervious circular cylinder. Stokes flow past a swarm of porous circular cylinders with Happel andKuwabara boundary conditions was discussed by Deo [17]. The flow around a cylinder, where the liquid is viscoelastic ratherthan viscous can also be solved by using numerical scheme based on smoothed particle hydrodynamics [18]. Recently, a newmodel for calculating specific resistance of aggregated colloidal cake layers in membrane filtration process was discussed byKim and Yuan [19]. The flow around nanospheres and nanocylinders was investigated by Matthews and Hill [20] and theyemployed a boundary condition that attempts to account for boundary slip due to the tangential shear at the boundary byusing a slip length parameter ‘. For large values of the slip length parameter (‘?1), their results matched with certain lowReynolds number experimental results. They had also shown that in comparison to no-slip boundary condition, their resultsusing slip boundary condition show better agreement with the experimental results. Palaniappan et al. [21] studied the twodimensional Stokes flow with permeable cylinders. Datta and Shukla [22] calculated the drag on a cylinder, using slip bound-ary condition and also deduced that the slippage on the cylinder reduces the drag.

In the present work, the problem of the slow viscous flow through an aggregate of concentric clusters of porous cylindri-cal particles with Happel boundary condition is considered. The purpose of considering a porous layer on the surface of animpermeable cylindrical core is that due to dissolution and adsorption of polymers, formation of porous layers on the surfaceof rigid particles takes place. This in turn affects total permeability of the membrane. Formation of porous layers on surfacesof rigid particles results in a modification of frictional force on interface between fluid and solid surface, like the surfaces ofcolloid particles. The Brinkman equation for the flow inside the porous cylindrical shell and the Stokes equation outside theporous cylindrical shell in their stream function formulations are used. As boundary conditions, continuity of velocity, sur-face stresses at the porous cylindrical shell and the vanishing of velocity components on the solid cylindrical core are em-ployed. On the hypothetical surface, uniform velocity and Happel boundary conditions are used. The drag force experiencedby each porous circular cylindrical particle in a cell is evaluated. The earlier results reported for the drag by Happel [6] andDeo [17] for flow past a solid cylinder and for porous cylinder, respectively, in the Happel unit cell, have been then deduced.Representative results are presented in graphical form by using Mathematica software and they are compared in both cases.Analytical expressions for velocity components, pressures, vorticity and stress components in both regions, have been alsoreported. In addition to this, the hydrodynamic permeability of the membrane has been calculated and some limiting caseshave been discussed. Dependence of hydrodynamic permeability on some parameters has been graphically discussed.Finally, we have analyzed flow patterns in the porous medium by stream lines. Some interesting changes in flow patternsare reported with particle volume fraction and permeability parameter. A limiting case of perfectly porous cylindricalparticle-in-cell is also discussed.

2. Statement and mathematical formulation of the problem

2.1. Flow through a swarm of clusters of porous cylindrical particles

A primary assumption employed in this study is that a swarm of porous co-axial (along z-axis) cylindrical particles sur-rounding a solid cylindrical core having the same axis is hydro-dynamically equivalent to a co-axial porous cylindrical shellsurrounding the solid core. Let the radius of the solid cylindrical core be a and the radius of the concentric porous cylindricalshell enclosing the solid cylindrical core be b(b > a). Furthermore, we assume that this porous shell is enveloped by a con-centric cylinder of radius c(c > b) named as cell surface and the radius of each porous cylindrical particle is ap(Fig. 1). There-fore, the thickness (b � a) of the porous cylindrical shell characterized by its constituent primary particles of radius ap andthereby accompanying permeability k. Relative to this composite cylinder (i.e. a core with porous shell) in the hypotheticalcell, the creeping flow of a Newtonian fluid with absolute fluid viscosity is considered to be steady and axi-symmetric. Weassume that the fluid is approaching towards the cell surface as well as partially passing through the composite cylinderperpendicular to the axis of cylinder (z-axis) with velocity U from left to right. We shall denote i = 1 in an entity for outsideand i = 2 for inside regions of the porous cylindrical shell, respectively. The radius c of the hypothetical cell is so chosenthat the particle volume fraction c of the swarm is equal to the particle volume fraction of the cell (More precisely, thevolume fraction of the partially porous particles to the volume of a cell is equal to the volume fraction of particles in themembrane.), i.e.

z- axis

U Porous cylindrical particle of radius pa

U

Porous cylindrical shell

Hypothetical surface

Solid cylindrical core

θ a

b

c

rvθv

a

bc

θ

Fig. 1. The Physical model and the co-ordinate system.

1332 S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343

c ¼ pb2

pc2 : ð1Þ

2.2. Governing equations

The governing equation of an incompressible Newtonian creeping flow for free space, i.e. outside the porous cylindricalshell is governed by Stokes equation [23]

l1r2vð1Þ ¼ rpð1Þ: ð2Þ

Also, we assume that the flow inside the porous cylindrical shell is governed by the Brinkman equation [24]

r2vð2Þ � r2

b2

� �vð2Þ ¼ 1

l2rpð2Þ: ð3Þ

Here, r2 ¼ bb2

k , with b ¼ l1l2;l1 is the viscosity of the fluid, l2 denotes the effective viscosity of porous medium, k being the

permeability of porous medium. Since, r is dimensionless variable related inversely with the permeability, therefore wecalled it as dimensionless permeability parameter. The viscosity coefficients l1 and l2 are assumed to be constant and equal.Here, v(i), p(i), i = 1, 2 are the velocity vector and pressure outside and inside the porous cylindrical shell, respectively.

For high permeability k, Eq. (3) reduces to Stokes equation (l2r2v(2) =rp(2)), whereas, for low permeability this equationresembles with Darcy empirical equation (�(l1/k)v(2) =rp(2)).

In addition, the equations of continuity for incompressible fluids must be satisfied in both regions:

r � vðiÞ ¼ 0; i ¼ 1;2: ð4Þ

These equations of continuity for axi-symmetric, incompressible viscous fluid in cylindrical polar co-ordinates (r,h,z) inboth regions can be written as

@

@rðrv ðiÞr Þ þ

@

@hðv ðiÞh Þ ¼ 0; ð5Þ

S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343 1333

where v ðiÞr and v ðiÞh ; i ¼ 1;2 are component of velocities in the direction of r and h, respectively. The stream functions w(i)(r,h)in both regions satisfying equations of continuity (5) may be defined as

v ðiÞr ¼1r@wðiÞ

@h; v ðiÞh ¼ �

@wðiÞ

@r: ð6Þ

Thus, on eliminating pressure in both Eqs. (2) and (3) and using (6), we get the following fourth order partial differentialequations, respectively,

r4wð1Þ ¼ r2ðr2wð1ÞÞ ¼ 0 ð7Þ

r4wð2Þ � r2

b2

� �r2wð2Þ ¼ r2 r2 � r2

b2

� �wð2Þ ¼ 0 ð8Þ

with the Laplacian operator

r2 ¼ @2

@r2 þ1r@

@rþ 1

r2

@2

@h2 : ð9Þ

Furthermore, the expressions for tangential and normal stresses T ðiÞrh ; TðiÞrr ; i ¼ 1;2, respectively, are given by

TðiÞrh ¼ li1r2

@2wðiÞ

@h2 þ1r@wðiÞ

@r� @

2wðiÞ

@r2

" #; ð10Þ

TðiÞrr ¼ �pðiÞ þ 2li

r@2wðiÞ

@r@h� 1

r@wðiÞ

@h

" #: ð11Þ

Also, the pressure may be obtained in both regions [23] by integrating the following relations, respectively, as

@pðiÞ

@r¼ li r2v ðiÞr �

v ðiÞr

r2 �2r2

@v ðiÞh

@h� d2i

rb

� �2v ðiÞr

" #; ð12Þ

1r@pðiÞ

@h¼ li r2v ðiÞh �

v ðiÞhr2 þ

2r2

@v ðiÞr

@h� d2i

rb

� �2v ðiÞh

" #; ð13Þ

where d2i is the Kronecker delta function.A suitable stream function solution of the Stokes Eq. (7) can be expressed as

wð1Þðr; hÞ ¼ Ub½A1r0 þ B1r03 þ C1=r0 þ D1r0 ln r0� sin h: ð14Þ

A particular solution of the Brinkman Eq. (8) may be written as

wð2Þðr; hÞ ¼ Ub½A2r0 þ B2=r0 þ C2I1ðrr0Þ þ D2K1ðrr0Þ� sin h: ð15Þ

Here, I1(rr0) and K1(rr0) are the modified Bessel’s functions of the order one of the first and second kinds [25], respectively,and the dimensionless variable r0 = r/b.

3. Boundary conditions

The boundary conditions those are physically realistic and mathematically consistent for this problem can be taken asfollows:

On the solid cylindrical core:

v ð2Þr ða; hÞ ¼ 0; v ð2Þh ða; hÞ ¼ 0: ð16Þ

On the porous surface:

v ð2Þr ðb; hÞ ¼ v ð1Þr ðb; hÞ; v ð2Þh ðb; hÞ ¼ v ð1Þh ðb; hÞ; ð17ÞTð2Þrr ðb; hÞ ¼ Tð1Þrr ðb; hÞ; T ð2Þrh ðb; hÞ ¼ Tð1Þrh ðb; hÞ: ð18Þ

On the hypothetical cell surface:

v ð1Þr ðc; hÞ ¼ U cos h: ð19Þ

The vanishing of shear stress on the cell surface, i.e. Happel condition implies that

Tð1Þrh ðc; hÞ ¼ 0: ð20Þ

No-slip and impenetrability boundary conditions are applied on the surface of the solid cylindrical core of radius a(Eq. (16)). The normal and tangential components of velocity and stress tensor are considered to be continuous across the

1334 S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343

permeable interface at r = b (Eqs. (17) and (18)), and fluid velocity U exerted by the composite cylinder is reflected in Eq. (19),which is also corresponding to such a solution that the cylinder moves in the same direction with velocity U where the fluidis stationary. The condition of the free shear stress as proposed by Happel on the surface of the hypothetical cell is imple-mented in Eq. (20) to mimic the adjacent presence of other aggregates of similar size.

3.1. Determination of arbitrary constants

As a result of application of the boundary conditions (16)–(20) and solving the resulting equations, we get the values ofarbitrary constants A1, B1, C1, D1, A2, B2, C2 and D2, which are given in the Appendix. Thus, all the coefficients have been deter-mined and hence, we get the explicit expressions for stream functions from Eqs. (14) and (15), in both regions.

4. Evaluation of the drag force

Integrating the normal and tangential stresses over the porous cylindrical shell of radius b in a cell yields the experienceddrag force per unit length F as given below:

F ¼Z 2p

0ðTð1Þrr cos h� Tð1Þrh sin hÞr¼bbdh: ð21Þ

On evaluating the values of T ð1Þrr and Tð1Þrh from Eqs. (10) and (11), we obtain

Tð1Þrr ¼ �4l1U

br0B1 þ

1r03

C1 �1r0

D1

� �cos h; ð22Þ

Tð1Þrh ¼ �4l1U

br0B1 þ

1r03

C1

� �sin h: ð23Þ

Inserting the values of (22) and (23) in Eq. (21) and integrating, we get

F ¼ 4pl1UD1; ð24Þ

where the value of constant D1 is given in the Appendix.Also, the drag coefficient CD can be defined as

CD ¼F

ð1=2ÞqU22b¼ 8pD1

Re; ð25Þ

where Re = 2bU/m1 is the Reynolds number and m1 = l1/q being the kinematic viscosity of a fluid.

5. Deductions of some known results

5.1. Drag on a porous circular cylinder in a cell

If a ? 0, i.e. ‘ = a/b ? 0, then cylindrical shell will reduces to a porous circular cylinder of radius b. In this case, the value ofdrag force experienced by the porous circular cylinder in a cell is given as

F ¼ 8pl1Ur2½rI1ðrÞð1þ c2Þ � 4c2I2ðrÞ�r2ðc2 � 1Þ½rI1ðrÞ � 2I2ðrÞ� þ ð8� r2 ln cÞ½rð1þ c2ÞI1ðrÞ � 4c2I2ðrÞ�

; ð26Þ

and the drag coefficient CD as

CD ¼16pr2½rI1ðrÞð1þ c2Þ � 4c2I2ðrÞ�

Re½r2ðc2 � 1Þ½rI1ðrÞ � 2I2ðrÞ� þ ð8� r2 ln cÞ½rð1þ c2ÞI1ðrÞ � 4c2I2ðrÞ��ð27Þ

where c ¼ pb2

pc2 ¼ 1=m2 being the particle volume fraction.Deo [17] had evaluated the drag force on the porous circular cylinder in a cell, which when expressed in our notations,

comes out as

F ¼ 8pl1Ur2½rI1ðrÞð1þ c2Þ � 4c2I2ðrÞ�ðc2 � 1Þ½r2frI1ðrÞ � 2I2ðrÞg � 8I2ðrÞ� þ ð8� r2 ln cÞ½rð1þ c2ÞI1ðrÞ � 4c2I2ðrÞ�

; ð28Þ

this is almost identical with expression (26). Here, it may be mentioned that when we rechecked the calculations of Deo [17],we found that the expression (26) is correct.

S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343 1335

5.2. Drag on a solid cylinder in Happel cell model (k ? 0)

When permeability k vanishes, i.e. permeability parameter r ?1, then the porous circular cylinder behaves like a solidcylinder of radius b. In this case, the value of the drag force experienced by a porous circular cylinder in a cell comes out as

F ¼ 8pl1U�ðln cþ 1Þ þ 2c2=ð1þ c2Þ ; ð29Þ

and the drag coefficient CD will become as

CD ¼16p

Re½�ðln cþ 1Þ þ 2c2=ð1þ c2Þ� : ð30Þ

This result of the drag force agrees with the earlier result as reported by Happel [6].

6. Evaluation of hydrodynamic permeability of a membrane consist of porous particles

Hydrodynamic permeability of a membrane,eL11 representing one of the elements of the Onzager’s matrix, is defined asthe ratio of the uniform flow rate U to the cell gradient pressure F/V [3]:

eL11 ¼U

F=V; ð31Þ

where V = pc2 is the volume of the cell of the unit length.Substituting the value of F from Eq. (24) and the value of V from above in Eq. (31), we have

eL11 ¼b2

4l1cD1¼ L11

b2

l1; ð32Þ

where L11 ¼ 14cD1

is the dimensionless hydrodynamic permeability of a membrane. The value of D1 is given in Appendix. Thehydrodynamic permeability is a function of three parameters, i.e. L11(‘,r,c).

6.1. Limiting cases

(1) If a ? 0, i.e. ‘ = a/b ? 0 then above model will reduce to a system of porous cylindrical particles, each of radius b. Inthis case, the expression for hydrodynamic permeability of a membrane is

L11 ¼r2ðc2 � 1Þ½rI1ðrÞ � 2I2ðrÞ� þ ð8� r2 ln cÞ½rð1þ c2ÞI1ðrÞ � 4c2I2ðrÞ�

8cr2½rI1ðrÞð1þ c2Þ � 4c2I2ðrÞ�: ð33Þ

(2) When permeability k vanishes, i.e. permeability parameter r ?1, and a ? 0, i.e. ‘ = a/b ? 0 then the above modelbehave like a system of solid cylindrical particles of radius b, and the expression for permeability of membrane is givenas

L11 ¼�ðln cþ 1Þ þ 2c2=ð1þ c2Þ

8c: ð34Þ

(3) If a ? 0, i.e. ‘ = a/b ? 0 and c = 1 then above model will reduce to porous single particle in an unbounded liquidmedium. In this case, the expression for hydrodynamic permeability of a membrane is

L11 ¼1r2 : ð35Þ

(4) If c = 1 then above model will reduce to a solid cylinder covered with a porous layer in a homogeneous fluid flow. Inthis case, the expression for hydrodynamic permeability of membrane is

L11 ¼ ½�16þ r2ð�2‘2I2ðr‘Þð2K0ðrÞ þ rK1ðrÞÞ þ 4‘2I0ðrÞK2ðr‘Þ þ rððI1ðrÞ þ I3ðrÞÞK0ðr‘Þ � 2‘2I1ðrÞK2ðr‘Þþ I0ðr‘ÞðK1ðrÞ þ K3ðrÞÞÞÞ�=D2; ð36Þ

where

D2 ¼ r2½�16þ rðr2I0ðr‘Þ þ ‘2ð8þ r2ÞI2ðr‘ÞÞK1ðrÞ þ r3I3ðrÞðK0ðr‘Þ þ ‘2K2ðr‘ÞÞ þ rI1ðrÞðr2K0ðr‘Þþ ‘2ð8þ r2ÞK2ðr‘ÞÞ þ r3ðI0ðr‘Þ þ ‘2I2ðr‘ÞÞK3ðrÞ�: ð37Þ

1336 S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343

6.2. Velocity components, pressure, vorticity and stress tensor

Using the values of w(1)(r,h) and w(2)(r,h) from Eqs. (14) and (15), in Eq. (6), we have the following expressions for thevelocity components for the outside and inside regions of porous cylindrical shell as

v ð1Þr ¼ U½A1 þ B1ðr0Þ2 þ C1ð1=r0Þ2 þ D1 lnðr0Þ� cos h; ð38Þv ð1Þh ¼ �U½A1 þ 3B1ðr0Þ2 � C1ð1=r0Þ2 þ D1ð1þ lnðr0ÞÞ� sin h; ð39Þv ð2Þr ¼ U½A2 þ B2ð1=r0Þ2 þ C2ð1=r0ÞI1ðrr0Þ þ D2ð1=r0ÞK1ðrr0g� cos h; ð40Þv ð2Þh ¼ �U½A2 � B2ð1=r0Þ2 þ C2frI0ðrr0Þ � ð1=r0ÞI1ðrr0Þg þ D2frK0ðrr0Þ þ ð1=r0ÞK1ðrr0Þg� sin h: ð41Þ

Again, substituting the values of velocity components v ðiÞr ;v ðiÞh ; i ¼ 1;2, from (38)–(41) in Eqs. (12) and (13) and then integrat-ing the resulting equations:

dpðiÞ ¼ @pðiÞ

@rdr þ @pðiÞ

@hdh; i ¼ 1;2; ð42Þ

we obtain,

pð1Þ ¼ 2l1Ub

4r0B1 �1r0

D1

� �cos h; ð43Þ

pð2Þ ¼ l2r2Ub

�r0A2 þ1r0

B2

� �cos h; ð44Þ

where p(1) and p(2) are the pressures in the outside and inside regions of porous cylindrical shell, respectively.The vorticity x(1) and x(2) for outside and inside regions of the porous cylindrical shell can be expressed in terms of veloc-

ity components as

xðiÞ ¼ @vðiÞh

@r� 1

r@v ðiÞr

@hþ v ðiÞh

r¼ �r2wðiÞ; i ¼ 1;2: ð45Þ

Therefore,

xð1Þ ¼ �2Ub½4r0B1 þ ð1=r0ÞD1� sin h; ð46Þ

xð2Þ ¼ �Ur2

b½I1ðrr0ÞC2 þ K1ðrr0ÞD2� sin h: ð47Þ

Components of the stress tensor for the outside region of the porous cylindrical shell are given by Eqs. (22) and (23) and thecomponents of the stress tensor for the inside of the porous cylindrical shell can be obtained by using the Eqs. (10) and (11)which comes out as

Tð2Þrh ¼l2U

b½�4ð1=r0g3B2 þ fð2r=r0ÞI0ðrr0Þ � ðr2 þ 4=r02ÞI1ðrr0ÞgC2 � fð2r=r0ÞK0ðrr0Þ þ ðr2 þ 4=r02ÞK1ðrr0ÞgD2� sin h;

ð48Þ

Tð2Þrr ¼l2U

b½r2r0A2 � ðr2 þ 4=r02Þ=r0ÞB2 þ frI0ðrr0Þ � ð2=r0ÞI1ðrr0Þgð2=r0ÞC2 � frK0ðrr0Þ þ ð2=r0ÞK1ðrr0Þgð2=r0ÞD2� cos h:

ð49Þ

7. Presentation of results and discussions

Stream lines and drag force are two important fluid-flow properties that are practically useful for flow through porousmedia. The stream line patterns are important for finding the flow rate through the internal part of the porous particleand drag force is needed for calculation of settling time of the porous particle in the fluid. In this section we discuss the var-iation of drag coefficient with permeability parameter and particle volume fraction for the Happel’s cell model. Apart fromthis, analysis of the flow pattern has been done by plotting stream lines. We have also discussed the specific case of porouscylinder-in-cell with no cylindrical core. The dependence of hydrodynamic permeability of the membrane which plays animportant role in filtration process, on different parameters is discussed and some interesting results are reported. Alongwith this, a comparative study of the results for both cases has been done. Here is a brief description of the results.

7.1. Variation of ReCD with different parameters in Happel’s cell

ReCD slightly increases with increasing permeability parameter r for low values of particle volume fraction c (c < 0.4).However, for large values of particle volume fraction c(c > 0.4) a significant increase in ReCD is observed with r. Except

Fig. 2. Variation of ReCD versus particle volume fraction c and permeability parameter r for the porous cylindrical shell when ‘ = 0.5.

Fig. 3. Variation of ReCD with particle volume fraction c for r = 100 and ‘ = 0.5 H ? Happel’s result, P ? Present result, D ? Deo’s result.

S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343 1337

for very low values of permeability parameter r(r < 5), ReCD increases to asymptotic value with particle volume fraction(Fig. 2). It is seen that the effect of permeability is to reduce the drag on the porous cylinder.

The comparison of results for ReCD with the existing results has been made through analytical expressions. Fig. 3 repre-sents the graphical comparison of our results with the previous results reported by Happel [6] and Deo [17]. We observe thatour result lies between the results of Happel [6] and Deo [17].

7.2. Analysis of flow pattern by stream lines in Happel’s cell

Fig. 4a and b shows that for a fixed permeability parameter, the flow through porous region increases with increasingparticle volume fraction. For low value of c and with the increasing r the rotation around solid core and in the cell region

Fig. 4. Stream lines for flow past a solid cylinder covered with porous layer and confined within a hypothetical cell. (black ? solid core, lightblack ? porous region, white ? clear fluid region)

1338 S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343

is reported (Fig. 4c and d). Fig. 4e and f shows that the rotation in the cell region shifts away from the porous region for high cand with the increasing r. Also for high c no rotation is observed around solid core. For completely porous particles, we

Fig. 5. Variation of natural logarithm of L11 versus particle volume fraction c for different values of permeability parameter r for the porous cylindrical shellwhen ‘ = 0.5.

Fig. 7. Variation of L11 versus particle volume fraction c for different values of permeability parameter r for the porous cylindrical particle.

Fig. 6. Variation of L11 versus permeability parameter r for different values of particle volume fraction c for the porous cylindrical shell when ‘ = 0.5.

S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343 1339

Fig. 8. Variation of L11 versus permeability parameter r for different values of particle volume fraction c for the porous cylindrical particle.

Fig. 9. Variation of L11 versus particle volume fraction c for solid cylindrical particle.

Fig. 10. Variation of L11 versus permeability parameter r for porous single particle in an unbounded liquid medium.

1340 S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343

Fig. 11. Variation of L11 versus permeability parameter r for a solid cylinder covered with a porous layer in a homogeneous fluid flow, when ‘ = 0.5

S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343 1341

observe that on increasing r, the flow through porous region become difficult and the rotational tendency shifts towards theporous region (Fig. 4g and h).

7.3. Variation of the hydrodynamic permeability of the membrane with different parameters

With increase in c, a significant increase followed by a consistent decrease is reported for hydrodynamic permeability L11.Since this variation is unaltered for different values of r, so we conclude that variation of L11 with c is independent of r(Fig. 5). From Fig. 6, we conclude that after certain values of r, the hydrodynamic permeability L11 rapidly increases withr and with lower the value of c, higher will be rate of increase of L11.

For a system of porous cylindrical particles of radius b, the variation of hydrodynamic permeability L11 with r and c isshown in Figs. 7 and 8. We observe that L11 decreases with increasing c. However, this decrease becomes more rapid for highvalues of r. A similar variation of hydrodynamic permeability L11 with r is observed for different values of c.

For the limiting case of r ?1, and a ? 0 or ‘ = a/b ? 0 i.e. a system of solid cylindrical particles, each of radius b, thedependence of hydrodynamic permeability L11 on c is shown in Fig. 9. A rapid decrease followed by steady decrease withc is observed for L11.

For a ? 0 or ‘ = a/b ? 0 and c = 1, i.e. porous single particle in an unbounded liquid volume, we observe that after a sig-nificant decrease for (r < 3), the hydrodynamic permeability L11 steadily decreases with increasing r (Fig. 10).

For c = 1, i.e. a solid cylinder covered with a porous layer in a homogeneous fluid flow, we observe that the hydrodynamicpermeability L11 decreases with increasing r. However, this decrease is less rapid comparative to the previous case (Fig. 11).

Acknowledgements

The first author is thankful to the Department of Science and Technology, Govt. of India for providing the financial assis-tance under its Project No. SR/FTP/MS-07/2004 during the work. The third author acknowledge with thanks to UGC, NewDelhi for the award of JRF No. 10-2 (5)/2005(i)-E.U.II for undertaking this research work.

Appendix A

A1 ¼ �1� ln mD1;

B1 ¼ ð1=4Þ½�4C1 þ 4D1 � r2A2 þ ð4þ r2ÞB2 � ð2rI0ðrÞ � 4I1ðrÞÞC2 þ ð2rK0ðrÞ þ 4K1ðrÞÞD2�;

C1 ¼ �m4B1;

D1 ¼ 2r½rI0ðr‘Þf4ð1þ ‘2ÞrK0ðrÞ þ ð8þ ð1þm4Þr2 þ ‘2ð16þ ð1þm4Þr2ÞÞK1ðrÞ � 8‘K1ðr‘Þg þ I1ðrÞf�8rK0ðrÞþ rð8þ ð1þm4Þr2 þ ‘2ð16þ ð1þm4Þr2ÞÞK0ðr‘Þ þ 2‘ð16þ ð1þm4Þr2ÞK1ðr‘Þg � 2f‘I1ðr‘Þf4rK0ðrÞþ 4rK0ðr‘Þ þ ð16þ ð1þm4Þr2ÞK1ðrÞg þ 2rI0ðrÞðð1þ ‘2ÞrK0ðr‘Þ þ 2K1ðrÞ þ 2‘K1ðr‘ÞÞg�=D;

1342 S. Deo et al. / Applied Mathematical Modelling 34 (2010) 1329–1343

A2 ¼ ðr=2Þ½�I0ðr‘ÞC2 þ K0ðr‘ÞD2�;

B2 ¼ A1 þ B1 þ C1 � A2 � I1ðrÞC2 � K1ðrÞD2;

C2 ¼ ð1=f2rI0ðrÞ � ð4þ r2ÞI1ðrÞgÞ½�4B1 � 4C1 þ 4B2 þ f2rK0ðrÞ þ ð4þ r2ÞK1ðrÞgD2�;

D2 ¼ ð�1=ðrK0ðrÞ þ K1ðrÞÞÞ½A1 þ 3B1 � C1 þ D1 � A2 þ B2 � ðrI0ðrÞ � I1ðrÞÞC2�;

D ¼ I1ðrÞ½�4K0ðrÞf16þ ð�1þm4Þr2 þ 4r2 ln mg þ 2‘rK1ðr‘Þf�16� ð�1þm4Þr2 þ 2ð16þ ð1þm4Þr2Þ ln mg

þ K0ðr‘Þf64þ 4ð3þm4 � 4‘2Þr2 � ð�1þm4Þð1þ ‘2Þr4 þ 2r2ð8þ ð1þm4Þr2 þ ‘2ð16þ ð1þm4Þr2ÞÞ ln mg�

þ I0ðr‘Þ½�4‘K1ðr‘Þf16þ ð�1þm4Þr2 þ 4r2 ln mg þ 2rK0ðrÞf16þ r2 �m4r2 þ ‘2ð�16þ r2 �m4r2Þ

þ 4ð1þ ‘2Þr2 ln mg þ K1ðrÞf64þ 4ð3þm4 � 4‘2Þr2 � ð�1þm4Þð1þ ‘2Þr4 þ 2r2ð8þ ð1þm4Þr2‘2

� ð16þ ð1þm4Þr2ÞÞ ln mg� þ 2½‘I1ðr‘Þf2K0ðrÞð16þ ð�1þm4Þr2 � 4r2 ln mÞ � 2K0ðr‘Þ

� ð16þ ð�1þm4Þr2 þ 4r2 ln mÞ � rK1ðrÞð�16� ð�1þm4Þr2 þ 2ð16þ ð1þm4Þr2Þ ln mÞg

þ I0ðrÞfrK0ðr‘Þð�16� r2 þm4r2 þ ‘2ð16þ ð�1þm4Þr2Þ � 4ð1þ ‘2Þr2 ln mÞ

� 2ð‘K1ðr‘Þð�16� ð�1þm4Þr2 þ 4r2 ln mÞ þ K1ðrÞð16þ ð�1þm4Þr2 þ 4r2 ln mÞÞg�;

where ‘ = a/b and m = c/b are dimensionless variables.Also, we have used the following recurrence relations

I0nðrr0Þ ¼ In�1ðrr0Þ � ðn=rr0ÞInðrr0Þ;

I00nðrr0Þ ¼ ð1þ n2=ðr2r02ÞÞInðrr0Þ � ð1=rr0ÞI0nðrr0Þ;

K 0nðrr0Þ ¼ �½Kn�1ðrr0Þ þ ðn=rr0ÞKnðrr0Þ�;

K 00nðrr0Þ ¼ ð1þ n2=ðr2r02ÞÞKnðrr0Þ � ð1=rr0ÞK 0nðrr0Þ;

In�1ðrr0Þ ¼ Inþ1ðrr0Þ þ ð2n=rr0ÞInðrr0Þ;

Kn�1ðrr0Þ ¼ Knþ1ðrr0Þ � ð2n=rr0ÞKnðrr0Þ;

where the primes in the functions denote the differentiation with respect to their argument, r0 is dimensionless variable andn being an integer.

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