Stokes Flow Over Composite Sphere: Liquid Core with Permeable Shell B.R

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Journal of Applied Fluid Mechanics , Vol. 8, No. 3, pp. 339-350, 2015. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.18869/acadpub.jafm.67.222.23135

Stokes Flow over Composite Sphere: Liquid Core with Permeable Shell B.R. Jaiswal1† and B.R.Gupta2

1 Department of Mathematics, Jaypee University of Engineering & Technology, Guna, M.P., 473226, India 2 Department of Mathematics, Jaypee University of Engineering & Technology, Guna, M.P., 473226, India

† Corresponding Author Email: [email protected]

(Received May 19 2014; accepted June 19 2014)

ABSTRACT This paper presents an analytical study of an infinite expanse of uniform flow of steady axisymmetric Stokes flow of an incompressible Newtonian fluid around the spherical drop of Reiner-Rivlin liquid coated with the permeable layer with the assumption that the liquid located outside the capsule penetrates into the permeable layer, but it is not mingled with the liquid located in the internal concave of capsule. The flow inside the permeable layer is described by the Brinkman equation. The viscosity of the permeable medium is assumed to be same as pure liquid. The stream function solution for the outer flow field is obtained in terms of modified Bessel functions and Gegenbauer functions, and for the inner flow field, the stream function solution is obtained by expanding the stream function in terms of S. The flow fields are determined explicitly by matching the boundary conditions at the pure liquid-porous interface, porous-Reiner-Rivlin liquid interface, and uniform velocity at infinity. The drag force experienced by the capsule is evaluated, and its variation with regard to permeability parameter α, dimensionless parameter S, ratio of viscosities λ2, and thickness of permeable layer δ is studied and graphs plotted against these parameters. Several cases of interest are deduced from the present analysis. It is observed that the cross-viscosity increases the drag force, whereas the thickness δ decreases the drag on capsule. It is also observed that the drag force is increasing or decreasing function of permeability parameter for λ2 < 1.

Keywords: Brinkman equation; Drag force; Modiﬁed Bessel functions; Reiner-Rivlin liquid; Permeability parameter; Stream functions; Permeable spherical shell.

NOMENCLATURE a radius of the liquid sphere yn(αr),y−n(αr) modiﬁed Bessel functions b radius of the outer permeable shell µ1,µ3 viscosity coefﬁcients D N dimensionless drag µe effective viscosity of permeable region F drag force κ permeability Pn(ζ) Legendre functions ψ(i)(r,ζ) stream functions S dimensionless parameter τrr,τrθ dimensionless stress components r,θ,φ spherical polar coordinates λ2 relative viscosity(µ /µ ) (i) 1 3 u velocity vectors δ thickness of permeable layer µc cross viscosity α permeability parameter

1. INTRODUCTION dustry, hydrology, lubrication problems, etc. The most practical example of physical process of vis- The study of viscous flow through permeable me- cous flow within a permeable spherical region is dia has attracted substantial practical and theoret- the structure of the earth. Due to its broad areas ical interest in science, engineering, and technol- of applications in science, engineering and indus- ogy. The flow through permeable media takes tries, many different theoretical and experimental place generally in geophysical and bio-mechanical models have been used for describing the viscous problems and also has many engineering applica- flow past and through the solids or porous bod- tions, such as, flow in fixed beds, petroleum in- ies. For efficient and effective utilization of bodies B.R. Jaiswal et.al / JAFM, Vol. 8, No. 3, pp. 339-350, 2015. with porous structured in the above mentioned ar- Matsumoto and Suganuma (1977) are theoretically eas, the structure of porous lamina must be consid- verified by Tam (1969), and Lundgren (1972) es- ered and analyzed from all point of views. For ana- tablished the validity of the Brinkman’s equation. lytical study of the fluid flows within porous struc- The problem of creeping flow relative to permeable tured bodies, so called porous media, the two terms: spheres was solved by Neale et al. (1973). Higdon porosity and permeability, play significant and vi- and Kojima (1981) have studied the Stokes flow tal role. The porosity is the measure of how much past porous particles using the Brinkman’s equation of a porous material has tiny spaces in it. More for the flow inside the porous media and derived precisely, it is defined as the ratio of voids’ vol- some asymptotic results in case of low and high per- ume to that of the volume of the material. The- meability by using Greens function formulation of oretically, it seems that if the material has more the Brinkman’s equation. A numerical study was pores (voids), it will allow the fluid to pass through done by Nandkumar and Masliyah (1982) to inves- it easily, but actually it is not so and could be tigate the flow field within and around an isolated understood through the permeability which is de- porous sphere taking into account the low Reynolds fined as the easiness or ability (inter connectivity number using the Brinkman’s equation for the inter- of pores) of the material to allow the fluids to pass nal flow field. The computed hydrodynamic resis- through it. The first model that was initially intro- tances were found to agree with the experiments on duced to analyze the fluid flow in permeable me- settling of porous spheres by Masliyah and Polikar dia was Darcy’s law by Darcy (1856). In his pro- (1980). By using Stokes approximation, Birikh and posed model he stated that the rate of flow in porous Rudakoh (1982) investigated the problem of slow media (through a densely packed bed of fine par- motion of a permeable sphere in a viscous fluid and ticles) is proportional to pressure gradient. This calculated the drag and the flow rate of the fluid. model was broadly used as the basic model in lit- Masliyah et al. (1987) solved analytically the prob- erature. Several studies of the flow field within and lem of creeping flow past a sphere having solid outside porous spheres are limited mainly to low core with porous shell and found that the results are Reynolds numbers. Joseph and Tao (1964) inves- in excellent agreement with the experimental ones. tigated the problem of creeping flow past a porous Using this Brinkman’s equation for the flow inside spherical shell immersed in a uniform steady, vis- the permeable sphere, Qin and Kaloni (1988) ob- cous, and incompressible fluid using Darcy’s law tained a Cartesian-tensor solution for the flow of for the flow inside the porous region and Stokes incompressible viscous fluid past a porous sphere equations for the fluid outside the sphere using no- and evaluated the hydro dynamical force experi- slip condition for tangential velocity component at enced by a porous sphere and also discussed and the surface of the sphere and reported that the drag laid emphasis on the merits of Brinkman’s equation on the porous sphere is the same as that of a rigid over Darcy’s equation. Bhatt and Sacheti (1994) sphere with minimized radius. Padmavathi et al. have studied the problem of viscous flow past a (1994) have solved the problem of creeping flow porous spherical shell using the Brinkman’s model past a porous sphere immersed in a uniform vis- and they evaluated the drag force experienced by cous incompressible fluid using Darcy’s law tak- the shell. Barman (1996) has investigated the prob- ing into account the Saffman boundary condition lem of a Newtonian fluid past an impervious sphere in place of no-slip boundary condition at the sur- embedded in a constant permeable medium by ap- face of the sphere. Whereas, Sutherland and Tan plying Brinkman’s model and he obtained an ex- (1970) assumed continuity of the tangential veloc- act solution to the governing equations in terms of ity component at the sphere surface. However, stream function setting forth constant velocity far Darcy’s law seems to be inadequate for the flows from the sphere. Using the stream function for- with high porosity. Most of the early authors like mulation, Zlatanovski (1999) obtained a conver- Boutros et al. (2006), Mukhopadhyay and Layek gent series solution to the Brinkman’s equation for (2009) etc. applied various types of extended Darcy the problem of creeping flow past a porous pro- models on convection in porous media. During late spheroidal particle and recovered the solution nineteenth century, after the Darcy’s work, flow for flow past a porous spherical particle as a limit- through porous media was influenced and emulated ing case of the spheroidal particle. Srinivasacharya by questions emerging in practical problems. To (2003) has discussed the creeping flow of a viscous study the flows with high porosity media and large fluid past and through a porous approximate sphere shear rates, Brinkman (1947) proposed a modifica- neglecting inertia terms, and deduced the result tion to Darcy’s law which was assumed to be gov- for an oblate spheroid in an unbounded medium. erned by a swarm of homogeneous spherical par- The motion of a porous sphere in a spherical con- ticles by appending a Laplacian term in velocity tainer using Brinkman’s model in the porous re- and is commonly known as Brinkman’s equation. gion was studied by Srinivasacharya (2005). Deo Experimental results of Ooms et al. (1970), and and Gupta (2009) investigated the problem of sym-

340 B.R. Jaiswal et.al / JAFM, Vol. 8, No. 3, pp. 339-350, 2015. metrical creeping flow of an incompressible vis- zero boundary condition for micro-rotation and cous fluid past a swarm of porous approximately found that spin parameter τ decreases the drag on spheroidal particles with Kuwabara boundary con- the body. And later, the same problem was re- dition with vanishing vorticity on the boundary. Re- investigated by Jaiswal and Gupta (2014b), Jaiswal cently, Deo and Gupta (2010) have evaluated the and Gupta (2015), respectively, for the case of non- drag force on a porous sphere embedded in another Newtonian liquid spheroid in spherical container, porous medium. Yadav et al. (2010) have evalu- and non-Newtonian liquid sphere embedded in a ated the hydrodynamic permeability of membranes porous medium saturated with Newtonian fluid. All built up by spherical particles covered by porous these investigations mentioned above, motivated us shells. The problem of the flow around particles to study the slow viscous flow past and within a with porous shell, and the filtration of viscous liq- spherical shell encapsulated by non-Newtonian liquid in the porous medium with complex internal uid. structure are closely interconnected. In this paper, we consider the problem of steady At present, authors are paying much attention on axisymmetric creeping flow of Newtonian fluid keen and thorough study of microcapsules. As around the capsule that contains the non-Newtonian these microcapsules are used for supplying of drugs liquid which can not flow out of the permeable and reagents. Modern microcapsules represent layer, but the liquid outside the capsule can per- porous shells that may contain either desired par- meates into the porous layer. Both, the internal ticles or liquids. The properties of porous media and external, flow fields are determined by using differ from the properties of a dispersion medium. the stokes approximation and expanding the inter- These capsules are of very great importance for nal stream function in powers of a dimensionless modern nanotechnologies and are characterized by parameter S. The flow within the permeable region diverse values of their parameters. Hence, the the- of the capsule is governed by Brinkman’s equa- oretical prediction of the hydrodynamic behavior tion. The stream functions have been determined of microcapsules presents not only mathematical by matching the solutions. Some well known re- interest, but can also be useful for the formation sults are also deduced from the present study. The and application of encapsulated media. The mo- drag force experienced by capsule is calculated and tion of capsules in the flow of liquid is of great its variation with respect to the fluid parameters is applied and theoretical interest. There are devel- studied numerically and graphically as well. opments that involve the encapsulation of anticor- rosive additives for paint and lacquer coatings. In 2. MATHEMATICAL FORMULATION AND all of these cases, the motion of the drop-shell sys- SOLUTIONOFTHEPROBLEM tem relative to the external flow occurs either at the We consider steady, axis-symmetric, creeping flow material production stage or in the practical use of of a viscous and incompressible Newtonian fluid capsules. Sekhar and Bhattacharyya (2005) used (region I) of viscosity µ1 with a constant uniform stress jump boundary condition to study the Stokes velocity U at infinity relative to the spherical cap- flow of a viscous fluid inside a sphere with internal sule with radius b comprises the drop of a Reiner- singularities enclosed by a porous spherical shell, Rivlin liquid (region III) of viscosity µ3 with ra- and concluded that the fluid velocity at a porous- dius a covered by a permeable layer (region II) with liquid interface varies with the stress jump coeffi- thickness δ, porosity ε, and permeability κ with the cient and plays an important role in describing the assumption that the fluid outside the capsule pen- flow field associated with porous medium. Vasin etrates into the permeable layer but is not mingled and Kharitonova (2011a) investigated the problem with the liquid in the region III. The flow field due of a viscous liquid flow past a spherical porous cap- to the motion of the uniform stream of a steady, sule. Stokes flow of an assemblage of porous par- incompressible Newtonian fluid past the encapsu- ticles was studied by Prakash et al. (2011). Vasin lated drop of radius b is to be determined in the and Kharitonova (2011b) solved the problem of the absence of body forces and couples. We take into flow around the encapsulated drop of Newtonian consideration that the inner flow (region III) is also liquid. In these right above discussed problems, au- steady and incompressible. Further, we assume that thors have taken into consideration the tangential the capsule to be stationary having its center at the stress jump condition on the porous - pure liquid origin of the spherical co-ordinates system (R,θ,φ) interface. Ramkissoon and Rahaman (2001) solved and taking θ = 0 as an axis in the direction of the the problem of Non-Newtonian fluid sphere in a free stream flow directed along the + ve z-direction spherical container and found that cross-viscosity as depicted in Fig.1. The parameters pertaining to decreases the wall effects. Recently, Jaiswal and the exterior, the permeable layer, and the interior of Gupta (2014a) solved a problem of Reiner-Rivlin the liquid capsule to be distinguished by the index liquid sphere placed in micro-polar fluid with non- in the superscripts under the bracket of an entity

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uR τ˜i j is the stress tensor, d˜i j is the rate of strain (defor- U (3) θ mation) tensor,p ˜ is an arbitrary hydrostatic pres- U sure, µ3 is the coefficient of viscosity and µc is the coefficient of cross-viscosity of Reiner-Rivlin liq- U δ uθ uid in region III. II Taking in view of the axial symmetry flow in which III θ b all the flow functions are independent of φ, the ve- o = (u ,u , ) a Z locity vector u˜ ˜R ˜θ 0 can be expressed by introducing the stream function through I U −1 ∂ψ˜ 1 ∂ψ˜ u˜ = , u˜ = . (6) R R2 sinθ ∂θ θ Rsinθ ∂r Approaching Newtonian fluid U In order to non-dimensionalize the quantities and operators appearing in the governing equations, we Fig. 1. Schematic representation of flow insert the following non-dimensional variables U R = br, u˜R = Uur, u˜θ = Uuθ, τ˜i j = µi τi j, χ(i), i = 1,2,3 related to the region I (i = 1), region b U U II (i = 2) and region III (i = 3) respectively. d˜ = d , p˜ = µ p, ψ˜ = Ub2ψ, i j b i j i b Within the unimpeded fluid in the region I, outside the capsule, the Stokes and continuity equations where U and b represent some typical velocity and characterize the prevailing flowfield as follows: length of the flow field, respectively. 2 (1) (1) µ1∇˜ u˜ = ∇˜ p˜ , b ≤ R < ∞ (1) Introducing the stream function, we write for the internal flow within the liquid drop (region III) of and radius a as follows ˜ (1) ∇ · u˜ = 0, b ≤ R < ∞ (2) (3) 2 ψ = ψ0 + ψ1S + ψ2S + ....., (3) 2 and the corresponding flowfield equations govern- p = p0 + p1S + p2S + ...... , (7) ing the motion of the fluid within the permeable re- where S is the dimensionless number µcU assumed gion are described by the Brinkman and continuity µ3b equations with inertial terms omitted, respectively, to be small. It can be shown by Ramkissoon (1989) as that the Stokes approximation of momentum equation for Reiner-Rivlin fluid provides α2 1 ˜ 2 (2) (2) ˜ (2) 4 4 2 ∇ u˜ − u˜ = ∇p˜ , a ≤ R ≤ b (3) E ψ0 = 0, E ψ1 = 8r sin θcosθ, b µe 32 E4ψ = r2 sin2 θ, (8) and 2 3 ∇˜ · u˜ (2) = 0, a ≤ R ≤ b, (4) where 2 2 ∂ sinθ ∂ 1 ∂ α2 = µ1b is the dimensionless parameter pertaining E2 = + . (9) µeκ ∂r2 r2 ∂θ sinθ ∂θ to the permeable region, µ1 is the viscosity of the clear fluid, µe is an effective viscosity of the permeable shell and κ is the permeability of the permeable Particular solutions of the equations (8) are given shell. The velocity vector and the pressure for re- by (i) gion I and II are, respectively, denoted by u˜ and 2 (i) ψ = (r4 − r2)sin2 θ, ψ = r5 sin2 θcosθ, p˜ , i = 1,2. 0 1 21 The constitutive equation for isotropic non- 2 6 2 ψ2 = r sin θ. (10) Newtonian Reiner-Rivlin liquid in the region III 63 takes the form Eliminating the pressures from the Eqs. (1) and (3), (3) τ˜i j = −p˜ δi j + 2µ3d˜i j + µcd˜ikd˜k j, (5) the stream function formulation of these equations for regions I and II get reduced, respectively, into where the following non-dimensional forms 1 (3) (3) d˜ = (u˜ + u˜ ), 4 (1) i j 2 i, j j,i E ψ = 0, (11)

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1 1 and r cosh(αr), y2(αr) = αcosh(αr) − r sinh(αr), and G (ζ) is Gegenbauer function deﬁned in 2 2 2 (2) 2 E (E − α )ψ = 0, (12) Abramowitz and Stegun (1970). where the second order differential operator is While for ﬂow within the Reiner-Rivlin drop in re- given in Eq. (9). gion III, we may take Ramkissoon (1989)

The surviving velocity components are (3) 2 ψ = ψ0 + ψ1S + ψ2S −1 ∂ψ(i) 1 ∂ψ(i) ∞ (i) (i) + [a rn + b rn+2]G ( ). ur = 2 , uθ = . (13) ∑ n n n ζ (20) r sinθ ∂θ r sinθ ∂r n=2

The dimensionless expression of tangential and With the aid of the Eq.(10), we can now write the normal stresses for region I and II, respectively, are Eq. (20) explicitly in the form: given by (3) 2 4 4 2 " # ψ (r,ζ) = [(a2 − 2)r + (b2 + 2)r + S 1 ∂2ψ(i) 2 ∂ψ(i) 63 τ(i) = − − rθ r sinθ ∂r2 r ∂r 6 3 4 5 r ]G2(ζ) + [a3r + (b3 + S)r ]G3(ζ) " !# 21 1 sinθ ∂ 1 ∂ψ(i) ∞ − , (14) + [a rn + b rn+2]G ( ), r ≤ l. (21) r sinθ r2 ∂θ sinθ ∂θ ∑ n n n ζ n=4

" (i) # 3. BOUNDARYCONDITIONSANDDE- (i) (i) 2µ1 2 ∂ψ τrr = −p + + TERMINATIONOFARBITRARYCON- r2 sinθ r ∂θ STANTS " # µ 2 (i) 2 1 ∂ ψ The unknown appearing in Eqs. (18), (19) and (21) + 2 , i = 1,2. (15) r sinθ ∂r∂θ to be determined by the following boundary conditions: Also, the dimensionless pressures in both the re- On the outer shell surface r = 1 gions may be evaluated by integrating the following relations (i). Continuity of velocity components implies ∂p(1) 1 ∂ that we may take = − (E2ψ(1)); ∂r r2 sinθ ∂θ (1) (2) ∂p(1) 1 ∂ (1) (2) ∂ψ ∂ψ = (E2ψ(1)), (16) ψ = ψ and = . (22) ∂θ r sinθ ∂r ∂r ∂r (ii). Also, we assume that the tangential and nor- ∂p(2) µ /µ ∂ = − e 1 [(E2 − α2)ψ(2)], mal components of stresses are continuous ∂r r2 sinθ ∂θ across the surface, so we can take (2) (1) (2) (1) (2) ∂p µe/µ1 ∂ 2 2 (2) = [(E − α )ψ ]. (17) τrθ = τrθ and τrr = τrr (23) ∂θ r sinθ ∂r On the inner shell surface r = l(a/b) In case of axisymmetric incompressible creeping ﬂow, the particular regular solutions of Stokes equation (11) by Happel and Brenner (1983), and (iii). Impenetrability at the inner shell surface re- Brinkman equation (12) by Zlatanovski (1999) on quires that the symmetry axis in spherical polar coordinates, (2) = , (3) = . respectively, may be taken as ψ 0 ψ 0 (24)

(1) ? −1 2 ? (iv). Continuity of tangential velocity requires that ψ (r,ζ) = [A2r +r +C2 r]G2(ζ),1 ≤ r < ∞,(18) ∂ψ(2) ∂ψ(3) and = . (25) ∂r ∂r (2) −1 2 ψ (r,ζ) = [A2r + B2r +C2y−2(αr)+ (v). We further assume that the theory of interfa- + D2y2(αr)]G2(ζ), l ≤ r ≤ 1 (19) cial tension is applicable to our problem. This ? ? where l = a/b, and A2, C2 , A2, B2, C2, D2 are means that the presence of interfacial tension constants to be determined, y−2(αr) = αsinh(αr)− only produces a discontinuity in the normal

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n−4 n−2 stress τrr and does not in any way affect tan- n(n − 3)l an + (n − 1)(n + 2)l bn (2) (3) gential stress τrθ. i.e. τrθ = τrθ which can be = 0, n ≥ 4. (39) shown to be equivalent to Solving the above system of linear Eqs. (27) to ! ! ∂ 1 ∂ψ(2) ∂ 1 ∂ψ(3) (39), we have determined all the unknowns ap- µ = µ . (26) 1 ∂r r2 ∂r 3 ∂r r2 ∂r pearing in the stream functions (18), (19) and (21) which are given in the appendix A. As a result of the employment of boundary condi- Hence, the dimensionless stream functions for the tions (Eqs. (22)–(26) ) , we are led to the following ﬂow ﬁelds corresponding to the regions I, II and III linear equations: are now known, and they are given, respectively, as ? ? A2 +C2 − A2 − B2 −C2y−2(α) − D2y2(α) + = , (1) ? −1 2 ? 1 0 (27) ψ (r,ζ) = [A2r +r +C2 r]G2(ζ),1 ≤ r < ∞,(40)

? ? 2 −A2 +C2 + A2 − 2B2 − [α y−1(α) − y−2(α)]C2 (2) −1 2 ψ (r,ζ) = [A2r + B2r +C2y−2(αr)+ 2 − [α y1(α) − y2(α)]D2 + 2 = 0, (28) + D2y2(αr)]G2(ζ), l ≤ r ≤ 1 (41)

? 2 2 and 6A2 − 6A2 − [(α + 6)y−2(α) − 2α y−1(α)]C2 (3) 2 4 2 2 ψ (r,ζ) = [(a2 − 2)r + (b2 + 2)r + − [(α + 6)y2(α) − 2α y1(α)]D2 = 0, (29) 4 + S2r6]G (ζ), r ≤ l (42) 63 2 2 2 ? ? µeα µeα 6A2 + 3C2 − + 6 A2 + B2+ ? ? 2µ1 µ1 where the values of arbitrary constants A2, C2 , A2, 2 B2, C2, D2, a2, and b2 are listed in the appendix A. +2[ − 3y−2(α) + α y−1(α)]C2+ 2 + 2[−3y2(α) + α y1(α)]D2 = 0, (30) 4. EVALUATION OF DRAG ON CAPSULE The drag force F to the capsule by external ﬂuid 2 4 2 4 4 2 6 is evaluated by integrating the normal and tangen- l a2 + l b2 − 2l + 2l + S l = 0, (31) 63 tial stresses over the surface by Happel and Brenner (1965) as follows −1 2 l A2 + l B2 +C2y−2(αl) + D2y2(αl) = 0, (32) Z π 2 (1) (1) F = 2πb τrr cosθ − τrθ sinθ sinθdθ.(43) 0 r=1 −2 −1 2 −l A2 + 2lB2 + [−l y−2(αl) + α y−1(αl)]C2 On evaluating dimensional stress-components, we + [−l−1y (αl) + α2y (αl)]D − 2la 2 1 2 2 get 8 Uµ 3C? 6A? − 4l3b + 4l − 8l3 − S2l5 = 0, (33) τ(1) = 1 2 + 2 cosθ, 2 21 rr b r2 r4

and ? 2 −5 2 −2 2 −2 2 −4 (1) 3Uµ1A 4λ l A2 − 2λ l B2 + λ [(l α + 4l )y−2(αl) 2 τrθ = 4 sinθ. 2 −3 2 −2 2 −4 br − 2α l y−1(αl)]C2 + λ [(l α + +4l )y2(αl) (1) (1) 2 −3 −2 −2 Substituting the values of τrr and τ in Eq. (43) − 2α l y1(αl)]D2 + 2l a2 − 4b2 − 4l rθ and integrating w.r.t. ‘θ’, we obtain 8 2 2 − 8 − S l = 0, (34) 2 3C? 6A? 4A? 7 F = 2πbUµ 2 + 2 − 2 , 1 3 r2 r4 r4 n n+1 ? anl + bnl = 0, (35) = 4πbUµ1C2 , at r = 1. (44) The ultimate expression for F is obtained by sub- 4 3a l2 + 5 b + S l4 = 0, (36) stituting for C?, and is presented in by Eq. (A.2) 3 3 21 2 of the appendix A. The algebraic calculation lead- ing to Eq. (A.2) is although unsophisticated but ex- n−1 n+1 nanl + (n + 2)bnl = 0, (37) tremely cumbersome. However, the genuineness of ? C2 may be examined by considering few limiting 4 cases for which the analytical solutions already ex- 10 b + S l = 0, (38) 3 21 ist in the literature. 344 B.R. Jaiswal et.al / JAFM, Vol. 8, No. 3, pp. 339-350, 2015.

5. RESULTS AND DISCUSSION follows At the outset, it is instructive to consider some lim- F DN = iting situations of the drag force as discussed below: −6bπUµ1 2 As a → b (for a liquid sphere of radius b): = C?, (49) 3 2 In this case, encapsulated drop reduces into the liq- ? uid sphere of radius b, and drag in this case comes here C2 is given in Eq. (A.2).The expression (49) out as therefore gives the drag force in non-dimensional form experienced by a capsule when a Newtonian ﬂuid stream past it. 2bπUµ1 2 32 2 F = − 6λ + 9 + S . (45) The drag coefﬁcient D is plotted for numerous val- 3(1 + λ2) 63 N ues of thickness of permeable layer(δ), ratio of vis- 2 2 cosities (λ ), dimensionless parameter S character- Where λ = µ1/µ3. This result was previously obtained by Ramkissoon (1989). izing the cross viscosity of the Reiner-Rivlin liquid and permeability parameter (α) and its variation is As a → 0 (for a homogeneous permeable sphere of depicted in Figs.2–7. Dependence of force DN radius b): The drag on the permeable sphere of radius b is obtained by letting the inner radius a to zero as follows

2 tanhα 12bπUα 1 − µ1 F = − α , (46) 2 tanhα 3α + 3 1 − α this result agrees with a well-known result earlier reported by Brinkman (1947), Ooms et al. (1970), Neale et al. (1973), Masliyah et al. (1987), Qin and Kaloni (1988), Qin and Kaloni (1993), Vasin and Kharitonova (2011a), Vasin and Kharitonova (2011b) and later Yadav and Deo (2012) for the drag force experienced by a permeable sphere in an Fig. 2. Variations in dimensionless force D with unbounded clear fluid. N respect to the parameter δ at λ2 = 1.5,α = 2 for As µ3 → ∞,S = 0 (permeable sphere with solid various values of S. core): on the dimensionless thickness of permeable layer In this case, the drag on the permeable sphere with δ at the various values of parameter S are shown solid core is obtained as in Fig.2. It is evident from the figure that the non- dimensional drag D decreases with increasing val- 3 3 2 N F = − 2bπUµ1[567(2 + l )α cosh(δα) + 3(−378α ues of δ between 0 and 1 and increases with the − 189l3α2 + 567l2(−1 + α2))sinh(δα)]/[189 increasing values of S. At δ = 0, we deal with the liquid drop, and the drag on liquid drop is differ- × (α(3 + 3l2 + 2α2 + l3α2)cosh(δα) ent for different values of S. It is obvious from − 3(2lα + (1 − l2α2)sinh(δα)))] (47) the Fig.2 that the drag on Newtonian liquid drop (S = 0) is less than the drag on non-Newtonian liq- here δ = 1 − l. This is the same result Earlier re- uid drop(S 6= 0). It is also clear that for small val- ported byMasliyah et al. (1987). ues of S(< 4) the decrements in DN is slow and for higher values of S(≥ 4) drag decreases rapidly. At As κ → 0, when l → 1(δ = 0) and µ3 → ∞ (Stokes’ δ → 1, the encapsulated drop becomes absolutely force on a sphere of radius b): permeable and drag in this case becomes almost same and constant for all the values of S. F = −6bπUµ1, (48) 2 The variation of DN with λ is depicted in Fig.3 for this is the well known Stokes force on a solid sphere various values of α which shows that corresponding as required. to very low permeability parameter, drag decreases vary rapidly and as the permeability parameter in- NON-DIMENSIONALDRAG creases, there takes place a slight decrease in drag Dimensionless force DN is defined as the ratio of and then it becomes almost constant with increasing 2 the force F to the Stokes force Fst = −6bπUµ1 as λ . It is interesting to note that when the perme-

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Fig. 5. The variation of non-dimensional drag Fig. 3. Variations in drag coefﬁcient D versus N D versus S for λ2 = 0.5 at δ = 0.2. viscosity ratio λ2 at S = 0.5,δ = 0.3 for various N values of permeability parameter α. ability parameter α is very high, i.e. permeability is very low, the drag throughout is almost constant.

Fig. 6. The variation of non-dimensional drag 2 DN versus S for λ = 1.0 at δ = 0.2.

ter S increases and increasing permeability param- Fig. 4. Variations in drag coefficient DN versus permeability parameter α at S = 1,δ = 0.2 for eter α as well, and it becomes almost constant for various values of viscosity ratio λ2. all values of permeability parameter α as S → 5 for viscosity ratio λ2 = 1. Whereas Fig.7 exhibits that The variation of the drag coefficient DN with re- the value of DN on the permeable sphere increases spect to permeability parameter α for various val- differently for increasing permeability parameter α ues of viscosity ratio λ2 is shown in Fig.4. It can for λ2 = 1.5 at δ = 0.2. This variation of the drag be observed from the Fig.4 that the drag DN in- coefficient DN with S for the specified values of rel- creases with increasing permeability parameterα , ative viscosity λ2 can also be viewed in tables 1–3. i.e., it decreases with permeability, for various val- Table 1 Numerical values of DN with respect to ues of relative viscosity λ2 and initially increases parameter S for various values of permeability very rapidly and then decreases very slowly with parameter α;δ = 0.2,λ2 = 0.5 increasing viscosity ratio λ2. Figures5–7 depict S Drag coefficient → the variation in non-dimensional drag with regard ↓ α = 0.5 α = 3.5 α = 6.5 α = 9.5 2 to S for λ = 0.2,0.5,0.7 respectively. Figure5 0 0.7100 0.7715 0.8422 0.8874 shows that the value of non-dimensional drag DN 1 0.7227 0.7803 0.8471 0.8899 on the permeable sphere increases with the increase 2 0.7608 0.8070 0.8617 0.8976 of permeability parameter α for S < 4.25, and then 3 0.8243 0.8514 0.9103 0.9103 it becomes almost constant for all the values of per- 4 0.9132 0.9135 0.9202 0.9282 meability parameter α, when 4.15 ≤ S < 4.35; but 5 1.0274 0.9934 0.9640 0.9511 the value of DN increases with decrease of permeability parameter α, when S ≥ 4.35. The numerical values of non-dimensional drag DN 2 2 2 Figure6 shows that the value of DN on the perme- for λ = 0.5, λ = 1.0, and λ = 1.5 are presented, able sphere increases as the dimensionless parame- respectively, in Tables 1, 2 and 3, and the cor-

346 B.R. Jaiswal et.al / JAFM, Vol. 8, No. 3, pp. 339-350, 2015.

Various useful results are obtained from the solution, particularly the closed form expression for the drag force and the dependence of the dimensionless drag on the various fluid parameters. It has been found that an increase in the thickness of the permeable layer δ decreases the drag force experienced by the encapsulated drop. It is also observed that hydrodynamic drag, in general, is increasing or decreasing function of the permeability parameter α when the relative viscosity λ2 < 1.The effect of the dimensionless parameter S on the drag for fixed values of the remaining fluid parameters is also studied and found that it increases the drag on the encapsulated drop. Fig. 7. The variation of non-dimensional drag 2 DN versus S for λ = 1.5 at δ = 0.2. REFERENCES Abramowitz, M. and I. Stegun (1970). Hand- responding variations are shown in Figs.5–7. It book of Mathematical Functions. New York: may be seen from the tables that for the relative Dover Publications. 2 viscosity(λ < 1), the numerical values of the drag, Barman, B. (1996). Flow of a Newtonian fluid for S > 4, decreases as permeability parameter in- past an impervious sphere embedded in creases which is in good agreement with the graph- a porous medium. Indian J. Pure Appl. ical representation of the drag, which may also be Math. 27(12), 1244–1256. further checked experimentally. These values are useful in estimating of optimum drag profiles in low Bhatt, B. and N. Sacheti (1994). Flow past a Reynolds number flows also. porous spherical shell using the Brinkman model. J. Phys D, Appl. Phys. 27(1), 37–41.

Table 2 Numerical values of DN with respect to Birikh, R. and R. Rudakoh (1982). Slow mo- parameter S for various values of permeability tion of a permeable sphere in a viscous fluid. parameter α;δ = 0.2,λ2 = 1.0 Fluid Dynamics 17(5), 792–793. S Drag coefficient → Boutros, Y., M. A. el Malek, N. Badran, and ↓ α = 0.5 α = 3.5 α = 6.5 α = 9.5 H. Hassan (2006). Lie-group method of solution for steady two dimensional boundary 0 0.6618 0.7498 0.8362 0.8858 layer stagnation point flow towards a heated 1 0.6716 0.7560 0.8394 0.8874 stretching sheet placed in a porous medium. 2 0.7008 0.7748 0.8489 0.8922 Meccanica 41(6), 681–691. 3 0.7494 0.8062 0.8648 0.9000 4 0.8176 0.8500 0.8871 0.9111 Brinkman, H. (1947). A calculation of viscous 5 0.9052 0.9064 0.9157 0.9253 force exerted by a flowing fluid on dense swarm of particles. Appl. Sci. Res.A1 1(1), 27–34. Table 3 Numerical values of DN with respect to parameter S for various values of permeability Darcy, H. (1856). Les Fontaines Publiques De parameter α;δ = 0.2,λ2 = 1.5 La Ville De Dijon. Paris: Victor Dalmont. S Drag coefficient → Deo, S. and B. Gupta (2009). Stokes flow past ↓ α = 0.5 α = 3.5 α = 6.5 α = 9.5 a swarm of porous approximately spheroidal 0 0.6319 0.7498 0.8362 0.8851 par-ticles with Kuwabara boundary condi- 1 0.6398 0.7560 0.8394 0.8863 tion. Acta Mecheanica 203(3-4), 241–254. 2 0.6635 0.7748 0.8489 0.8897 Deo, S. and B. Gupta (2010). Drag on a porous 3 0.7029 0.8062 0.8648 0.8954 sphere embedded in another porous medium. 4 0.7582 0.8500 0.8871 0.9034 Journal of Porous Media 13(11), 1009– 5 8292 0.9064 0.9157 0.9137 1016. Happel, J. and H. Brenner (1965). Low Reynolds 6. CONCLUSION Number Hydrodynamics. Englewood Cliffs, The expressions for stream function solutions to the NJ: Prentice-Hall. flow field equations for the steady axisymmetric Happel, J. and H. Brenner (1983). Low Reynolds Stokes flow of Newtonian fluid around the encap- Number Hydrodynamics. Hague: Martinus sulated drop of Reiner-Rivlin liquid are obtained. Nijhoff, Publishers.

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Higdon, J. and M. Kojima (1981). On the cal- spherical particle. Archives of Mechan- culation of Stokes flow past porous particles. ics 46(3), 191–199. Int. J. Multiphase Flow 7(6), 719–727. Prakash, J., G. P. R. Sekhar, and M. Kohr (2011). Jaiswal, B. and B. Gupta (2014a). Drag on Stokes flow of an assemblage of porous par- Reiner-Rivlin liquid sphere placed in a ticles: Stress jump condition. journal Appl. micro- polar fluid with non-zero boundary Math. Phys. 62(6), 1027–1046. condition for microrotations. Int. J. of Appl. Qin, Y. and P. Kaloni (1988). A Cartesian-tensor Math. and Mech. 10(7), 90–103. solution of Brinkman equation. Journal of Jaiswal, B. and B. Gupta (2014b). Wall ef- Engineering Mathematics 22(2), 177–188. fects on Reiner-Rivlin liquid spheroid. Appl. Qin, Y. and P. Kaloni (1993). Creeping flow past Compt. Mech. 8(2), 157–176. a spherical shell. Journal of Applied Mathe- Jaiswal, B. and B. Gupta (2015). Brinkman flow matics and Mechanics 73(2), 77–84. of a viscous fluid past a Reiner-Rivlin liq- Ramkissoon, H. (1989). Stokes flow past a uid sphere immersed in a saturated porous Reiner-Rivlin fluid sphere. Journal of Ap- medium. Transport in Porous Media 106(3), plied Mathematics and Mechanics 69(8), DOI: 10.1007/s11242–015–0472–2. 259–261. Joseph, D. and L. Tao (1964). The effect of per- Ramkissoon, H. and K. Rahaman (2001). Non- meability on the slow motion of a porous Newtonian fluid sphere in a spherical con- sphere in a viscous liquid. Journal Appl. tainer. Acta Mechanica 149(1-4), 239–245. Math. Mech. 44(8-9), 361–364. Sekhar, G. P. R. and A. Bhattacharyya (2005). Lundgren, T. (1972). Slow flow through station- Stokes flow inside a porous spherical shell: ary random beds and suspensions of spheres. Stress jump boundary condition. journal Journal of Fluid Mechanics 51(2), 273–299. Appl. Math. Phys. 56(3), 475–496. Masliyah, J., G. Neale, K. Malysa, and T. V. Srinivasacharya, D. (2003). Creeping flow past de Ven (1987). Creeping flow over a com- a porous approximate sphere. Journal of Ap- posite sphere: Solid core with porous shell. plied Math. Mech. 83(7), 499–504. Chemical Engineering Science 4(2), 245– Srinivasacharya, D. (2005). Motion of a 253. porous sphere in a spherical container. Masliyah, J. and M. Polikar (1980). Terminal ve- Mecanique 333(8), 612–616. locity of porous spheres. Canadian Journal Sutherland, D. and C. Tan (1970). Sedimenta- of Chemical Engineering 58(3), 299–302. tion of a porous sphere. Chemical Engineer- Matsumoto, K. and A. Suganuma (1977). Set- ing Science 25(12), 1948–1950. tling velocity of a permeable model floc. Tam, C. (1969). The drag on a cloud of spheri- Chemical Engineering Science 32(4), 445– cal particles in a low Reynolds number flow. 447. Journal of Fluid Mechanics 38(3), 537–546. Mukhopadhyay, S. and G. Layek (2009). Radi- Vasin, S. and T. Kharitonova (2011a). Uniform ation effect on forced convective flow and liquid flow around porous spherical shell. heat transfer over a porous plate in a porous Colloid Journal 73(1), 18–23. medium. Meccanica 44(5), 587–597. Vasin, S. and T. Kharitonova (2011b). Uniform Nandkumar, K. and J. Masliyah (1982). Lami- liquid flow around porous spherical shell. nar flow past a permeable sphere. Canadian Colloid Journal 73(3), 297–302. Journal of Chemical Engineering 60 (2), Yadav, P. and S. Deo (2012). Stokes flow past a 202–211. porous spheroid embedded in another porous Neale, G., N. Epstein, and W. Nader (1973). medium. Meccanica 47(6), 1499–1516. Creeping flow relative to permeable spheres. Yadav, P., A. Tiwari, S. Deo, A. Filippov, and Chemical Engineering Science 28(10), S. Vasin (2010). Hydrodynamic permeability 1865–1873. of membranes built up by spherical particles Ooms, G., P. Mijulieff, and H. Becker (1970). covered by porous shells: Effect of stress Frictional force exerted by a flowing fluid jump condition. Acta Mechanica 215(1-4), in a permeable particle with particular refer- 193–209. ence to polymer coils. Journal of Chemical Zlatanovski, T. (1999). Axisymmetric creeping Physics 53(11), 4123–4130. flow past a porous prolate spheroidal particle Padmavathi, B., T. Amarnath, and D. Pala- using the Brinkman model. Q. J. Mech. Appl. niappan (1994). Stokes flow about a porous Math. 52(1), 111–126.

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APPENDIX A:

Determination of arbitrary constants

Solving the Eqs. (27) to (39) for the special case µe = µ1, we obtain the surviving coefﬁcients as given below

? 7 2 2 4 2 2 2 6 2 2 2 4 2 2 (A.1) A2 = − (−2l(16l S α + 32l S (3 + α ) − 189(3 + 2λ )) + 3α(64l S + 84lα λ + 42l α λ + 126l2(3 + 2λ2) − 21l3α2(3 + 2λ2) − 42(6 + α2)(3 + 2λ2))cosh(δα) + 3(32l6S2(1+ + α2) − 21l4α2(−3 + α2)λ2 − 42l(−6 + 3α2 + α4)λ2 + 63l3α2(3 + +2λ2) − 63l2(−3+ + α2)(3 + +2λ2) + 126(2 + α2)(3 + 2λ2))sinh(δα))/(126∆),

? 5 3 2 3 3 3 2 6 2 2 2 (A.2) C2 = − (32l (2 + l )S α + 189(2 + l )α (3 + 2λ )cosh(δ) + α3(32l S (−1 + α ) + 126lα × (−1 + α2)λ2 + 63l4α2(−1 + α2)λ2 − 126α2(3 + 2λ2) − 63l3α2(3 + 2λ2) + 189l2(−1 + α2)(3 + 2λ2))sinh(δα))/(126∆),

3 7 2 2 2 3 3 2 2 4 2 (A.3) A2 = − (l(2α (567 + 16l S + 378λ ) + l α (567 + 64l S + 378λ )cosh(δα) + l(32l S (−3 + 3α2 + 2α4) + 189l2α2(−1 + α2)λ2 − 189lα2(3 + 2λ2) − 567(1 − α2)(3 + 2λ2)) × sinh(δα)))/(63α2∆),

4 2 2 6 2 2 6 2 2 (A.4) B2 = − (lα(567 − 32l S + 378λ ) + α(32l S − 189(3 + 2λ ))cosh(δα) + (32l S α − 189l × (−1 + α2)λ2 + 189(3 + 2λ2))sinh(δα))/(63∆),

7 2 2 4 2 2 2 5 2 2 2 (A.5) C2 = − (lα(32l S α + 32l S (3 + 2α ) + 567(3 + 2λ ))cosh(α) − 3(lα(32l S + 126α λ + + 63l3α2λ2 + 189l(3 + 2λ2))cosh(lα) + l(567 + 32l4S2 + 378λ2)sinh(α) − (32l6S2+ + 126lα2λ2 + 63l4α2λ2 + 189l2(3 + 2λ2) + 126α2(3 + 2λ2) + 63l3α2(3 + 2λ2)) × sinh(lα)))/(63α2∆),

4 2 2 6 2 2 2 4 2 2 2 (A.6) D2 = + (−3l(567 + 32l S + 378λ )cosh(α) + 3(32l S + 126lα λ + 63l α λ + 189l (3+ + 2λ2) + 126α2(3 + 2λ2) + 63l3α2(3 + 2λ2))cosh(δα) + lα((32l7S2α2 + 32l4S2(3+ + 2α2) + 567(3 + 2λ2))sinh(α) − 3(32l5S2 + 126α2λ2 + 63l3α2λ2 + 189l(3 + 2λ2)) × sinh(lα)))/(63α2∆),

2 2 3 2 2 2 5 2 2 2 2 5 2 (A.7) a2 = − (3α(126α λ + 63l α λ + 504l(3 + 2λ ) + 16l S (7 + 2λ )) − 4α(126lα λ + 4l S × α2λ2 + 2l8S2α2λ2 + +189l2(3 + 2λ2) + 63l3α2(3 + 2λ2) + 63(3 + 2α2)(3 + +2λ2)+ + 6l6S2(7 + 2λ2) + 2l7S2α2(7 + 2λ2) + l4(63α2λ2 + 2S2(3 + 2α2)(7 + 2λ2)))cosh(δα)− − (8l8S2α4λ2 + 8l5S2(−3 + 3α2 + 2α4)λ2 + 63l(−3 + 3α2 + 8α4)λ2 − 756(3 + 2λ2)+ + 756l2α2(3 + 2λ2) + 24l6S2α2(7 + 2λ2) − 12l4(−21α42 + 2S2(7 + 2λ2))) × sinh(δα))/(126∆),

2 2 5 2 2 3 2 2 2 2 2 6 2 (A.8) b2 = − (−3α(126α λ + 32l S (5 + 2λ ) + 63l (24 + (16 + α )λ )) + 4l α(126lα λ + 4l S × α2λ2 + 63(3 + 2α2)(3 + 2λ2) + 4l5S2α2(5 + 2λ2) + l3α2(189 + 2(63 + 4S2)λ2) + 3l4 × (21α2λ2 + 4S2(5 + 2λ2)) + l2(189(3 + 2λ2) + 4S2(3 + 2α2)(5 + 2λ2)))cosh(δα)+ + l(16l7S2α4λ2 − 567(−1 + α2)λ2 + 252l2(−3 + 3α2 + 2α4)λ2 + +16l4S2(−3 + 3α2+ + 2α4)λ2 − 756l(3 + 2λ2) + 12l5α2(21α2λ2 + 4S2(5 + 2λ2)) − 12l3(−63α2(3 + 2λ2)+ + 4S2(5 + 2λ2)))sinh(δα))/(126l2∆),

349 B.R. Jaiswal et.al / JAFM, Vol. 8, No. 3, pp. 339-350, 2015. where

∆ = − 6lα(3 + 2λ2) + α(2lα2λ2 + l4α2λ2 + l3α2(3 + 2λ2) + (3 + 2α2)(3 + 2λ2) + l2(9 + 6λ2)) × cosh(δα) + (−9 − 6λ2 + l4α4λ2 + l(−3 + 3α2 + 2α4)λ2 + 3l2α2(3 + 2λ2))sinh(δα), b µ α =√ , δ = 1 − l, λ2 = 1 . κ µ3

APPENDIX B:

The Gegenbauer functions Gn(ζ) of ﬁrst kind are the solutions of the following differential equation, which are the polynomials of degree n

d2ψ (B.1)(1 − ζ2) + n(n − 1)ψ = 0, dζ2 where the Gegenbauer polynomials Gn(ζ) are related to the Legendre polynomials of degree n by the following relation

P (ζ) − P (ζ) (B.2) G (ζ) = n−1 n , n ≥ 2. n 2n − 1 The governing equation in the stream function solution of Brinkman equation takes the following form

d2R (B.3) r2 + [α2r2 + n(n − 1)]R = 0. dr2 √ √ The linearly independent solutions of (B.3) are αrIν(αr) and αrI−ν(αr). To facilitate our calculation, we have utilized the under-mentioned notations in dimensionless form

rπαr rπαr n − 1 (B.4) y (αr) = ανI (αr) and y (αr) = ανI (αr), ν = . n 2 ν −n 2 −ν 2

In particular,

(B.5) y1(αr) = sinh(αr); y−1(αr) = cosh(αr), 1 1 y (αr) = αcosh(αr) − sinh(αr); y (αr) = αsinh(αr) − cosh(αr). 2 r −2 r

350