D. Srinivasacharya

Comptes Rendus Mécanique (CRM)

/igated. Theere usedhere andcorrectionnen utilisantulementrieur det éprouvéesr servess someC. R. Mecanique 333 (2005) 612–616http://france.elsevier.com/direct/CRAS2BMotion of a porous sphere in a spherical containerD. SrinivasacharyaDepartment of Mathematics, National Institute of Technology, Warangal – 506004, IndiaReceived 12 March 2005; accepted 14 June 2005Available online 25 August 2005Presented by Évariste Sanchez-PalenciaAbstractThe creeping motion of a porous sphere at the instant it passes the center of a spherical container has been investBrinkman’s model for the flow inside the porous sphere and the Stokes equation for the flow in the spherical container wto study the motion. The stream function (and thus the velocity) and pressure (both for the flow inside the porous spinside the spherical container) are calculated. The drag force experienced by the porous spherical particle and wallfactor is determined.To cite this article: D. Srinivasacharya, C. R. Mecanique 333 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméMouvement d’une sphère poreuse à l’intérieur d’un récipient sphérique. Nous considérons le mouvement rampant d’usphère poreuse à l’instant où elle traverse le centre d’un récipient sphérique. L’étude du mouvement a été faite ele modèle de Brinkman d’un écoulement à l’intérieur d’une sphère poreuse et l’équation de Stokes décrivant l’écodans un récipient sphérique. La fonction de courant (et ainsi la vitesse) et la pression (pour l’écoulement à l’intéla sphère poreuse et à l’intérieur du récipient sphérique) Ont été calculées. La force de résistance au mouvemenpar une particule poreuse sphérique et le facteur de correction dû à l’effet de la paroi sont aussi determinés.Pour citer cetarticle : D. Srinivasacharya, C. R. Mecanique 333 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords: Porous media; Porous sphere; Spherical container; Brinkman model; Drag; Wall correction factorMots-clés : Milieux poreux ; Sphère poreuse ; Récipient sphérique ; Modèle de Brinkman ; Traînée ; Facteur de correction d’une paroi1. IntroductionThe problems of the motion of a particle at the instant it passes the centre of the spherical containeas a model of interaction in multi-particle systems. This class of problems is important because it provideE-mail address: dsc@nitw.ernet.in (D. Srinivasacharya).1631-0721/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2005.07.017D. Srinivasacharya / C. R. Mecanique 333 (2005) 612–616 613spheretion ofe motionwtonianiner [5].gineer-ticulatellets arees indi-tion isused thel con-phericalstudiedphericalicaladiusuid isininformation on wall effects. A number of researchers have considered the creeping motion of rigid or fluidin a spherical or spheroidal container. Cunnigham [1] and Williams [2], independently, considered the moa solid sphere in a spherical container. Haberman and Sayre [3] have made an analogous study for thof an inner Newtonian fluid sphere. Ramkissoon and Rahaman investigated the motion of inner non-Ne(Reiner–Revlin) fluid sphere in a spherical container [4] and a solid spherical particle in a spheroidal contaIt is well known that problems of fluid flow around porous particles are encountered in many important ening applications such as flows of fluids through porous beds (fixed or fluidized), sedimentation of fine parsuspensions, modeling of polymer macromolecule coils in solvent, catalytic reactions where porous paused, folc settling process etc. A survey of literature regarding the fluid flows past and within porous bodicates that while abundant information is available for flows in an infinite expanse of fluid, very little informaavailable for flows in enclosures.In this Note, we consider the creeping motion of a porous sphere in a spherical container. We haveBrinkman’s model for the flow inside the porous particle and Stokes model for the flow within the sphericatainer. The stream function and the pressure for both the flows inside porous particle and within the scontainer are calculated. The drag experienced by the porous particle is calculated and its variation isnumerically.2. Formulation of the problemConsider a porous spherical particle of radiusa passing the center of a spherical vessel of radiusb containingan incompressible Newtonian viscous fluid. This is equivalent to the inner particle at rest while the outer scontainer moves with a constant velocityU in the negativeZ direction. We assume that the flow within the sphercontainer is Stokesian, and Brinkman’s law [6] governs the flow inside the porous spherical particle.The equations of motion for the region within the spherical container arediv q(1) = 0 (1)gradp(1) + µcurl curlq(1) = 0 (2)whereq(1) is the velocity,µ is the coefficient of viscosity andp(1) is the pressure.For the region inside the porous sphere, the equations of motion are:div q(2) = 0 (3)gradp(2) + µkq(2) + µcurl curlq(2) = 0 (4)whereq(2) is the velocity,p(2) is the pressure,µ is viscosity andk is the permeability of the porous medium.Let (r, θ,φ) denote a spherical polar co-ordinate system with the origin at the centre of the sphere of raand diameter coinciding with the line of motion of the inner sphere as the initial line. Since the flow of the flin the meridian plane and the flow is axially symmetric, all the physical quantities are independent ofφ. Hence, weassume thatq(i) = [u(i)(r, θ)er + v(i)(r, θ)eθ ], i = 1,2 (5)where(er , eθ , eφ) are unit base vectors andh1 = 1, h2 = r andh3 = r sinθ are the corresponding scale factorsthe spherical polar coordinate system.614 D. Srinivasacharya / C. R. Mecanique 333 (2005) 612–616ionless3. Solution of the problemIntroducing the stream functionsψ(i)(r, θ), i = 1,2, throughu(i) = − 1r2 sinθ∂ψ(i)∂θ, v(i) = 1r sinθ∂ψ(i)∂r, i = 1,2 (6)in Eqs. (1)–(4) and eliminating the pressure from the resulting equations, we get the following dimensequations forψ(i), i = 1,2,E4ψ(1) = 0 (7)andE2(E2 − α2)ψ(2) = 0 (8)whereα2 = a2/k andE2 denotes the Stokes stream function operator given byE2 =[∂2∂r2+ 1r2∂2∂θ2− cotθr2∂∂θ](9)The boundary conditions are:– The normal velocity component is continuous at the boundary of the sphere i.e.,u(1)(r, θ) = u(2)(r, θ) on r = a (10)– The tangential velocity component is continuous at the boundary of the sphere i.e.,v(1)(r, θ) = v(2)(r, θ) on r = a (11)– Continuity of tangential stresses components at the boundary of the sphere i.e.,τ(1)rθ (r, θ) = τ (2)rθ (r, θ) on r = a (12)– Continuity of the pressure distributions at the boundary of the sphere i.e.,p(1)(r, θ) = p(2)(r, θ) on r = a (13)On the outer sphere, the condition of impenetrability leads tou(1)(r, θ) = U cosθ and v(1)(r, θ) = −U sinθ on r = b (14)and the condition that velocity and pressure must have no singularities anywhere in the flow field.The boundary conditions (10)–(14) in terms of the stream function areψ(1)(r, θ) = 12r2 sin2 θ, ψ(1)r (r, θ) = r sin2 θ on r = 1/η (15)andψ(1)(r, θ) = ψ(2)(r, θ), ψ(1)r (r, θ) = ψ(2)r (r, θ)ψ(1)rr (r, θ) = ψ(2)rr (r, θ), p(1)(r, θ) = p(2)(r, θ) on r = 1 (16)whereη = a/b.The solutions of (7) and (8), which are nonsingular everywhere in the flow region, areψ(1) =∞∑[Anrn + Bnr1−n + Cnrn+2 + Dnr3−n]ϑn(ζ ) (17)n=2D. Srinivasacharya / C. R. Mecanique 333 (2005) 612–616 615helare seenandψ(2) =∞∑n=2[Enrn + Fn√rIn−1/2(αr)]ϑn(ζ ) (18)whereAn, Bn, Cn, Dn, En, andFn, are arbitrary constants,In−1/2(αr) denote the modified Bessel function of tfirst kind andϑn(ζ ) is the Gegenbauer function of the first and second kind of ordern and degree−1/2.Using (17) in (2) and integrating the resulting equations, we get the pressure distributionp(1) within the sphericacontainer asp(1) = −∞∑n=2[(4n + 2)rn−1n − 1 Cn −6− 4nnDnr−n]Pn−1(ζ ) (19)Similarly, the pressure distributionp(2) within the porous sphere is given byp(2) = α2∞∑n=2Enrn−1n − 1Pn−1(ζ ) (20)Using the boundary conditions (15) and (16), the constants appearing in the solutions of the problemto beA2 ={[(−9η5 + 5η3 + 4)α5 + (−126η5 + 30η3 + 6)α3 − 270η5α]cosh(α)+ 3[90η4 + 5α4(3η2 − 1)η3 + 2α2(36η5 − 5η3 − 1)]sinh(α)}/X1 (21)B2 = −2α2{α[15η3 + (η3 − 1)α2 − 6]cosh(α) + 3[−5η3 + α2(1− 2η3) + 2]sinh(α)}/X1 (22)C2 = 3α2η3{α[6η2 + (η2 − 1)α2]cosh(α) + [α2(1− 3η2) − 6η2]sinh(α)}/X1 (23)D2 = 6α2{α[15η5 + (η5 − 1)α2]cosh(α) + [α2(1− 6η3) − 15η5]sinh(α)}/X1 (24)E2 = −6{α[45η5 + (6η5 − 5η3 − 1)α2]cosh(α) + [α2(1+ 5η3 − 21η5) − 45η5]sinh(α)}/X1 (25)F2 = −3√2πα7/2[3η5 − 5η3 + 2]/X1 (26)whereX1 = α[{−270η5 + α4(η − 1)4(4η2 + 7η + 4) + 6α2(10η6 − 21η5 + 10η3 + 1)}cosh(α)− 3{−90η5 + α4(η − 1)3(8η2 + 9η + 3)η + 2α2(10η6 − 36η5 + 10η3 + 1)}sinh(α)] (27)andAn = Bn = Cn = Dn = En = Fn = 0 for n  3 (28)4. Drag on the body and wall effectsThe drag experienced by the inner spherical particle is given by [7]F = µππ∫0 3∂∂r(E2ψ 2)r dθ (29)where = r sinθ . On carrying out the integration, it is found to beDrag = 24µπaUα2{α[15η5 + (η5 − 1)α2]cosh(α) + [α2(1− 6η3) − 15η5]sinh(α)}/X1 (30)616 D. Srinivasacharya / C. R. Mecanique 333 (2005) 612–616ntlosurereeability(1910)Basin7–34.Fig. 1. The variation of drag coefficient withα2 for various valuesof η.Fig. 2. The variation of wall correction factor withη for various valuesof α2.The variation of drag coefficientDN = Drag/(24µπaU) with respect toα2 for various values ofη is shown inFig. 1. It can be observed from the figure that, as the inner porous particle size (η) increases, the drag coefficiedecreases and this is expected. The drag coefficient is decreasing as the permeability parameterα2 is increasing.As b → ∞, we get the drag on a porous sphere in the case of streaming in an unbounded medium,12πµUaα2 {−α cosh(α) + sinh(α)}α(3+ 2α2)cosh(α) − 3 sinh(α) (31)which agrees with the drag on the porous sphere case derived Brinkman [6], Neale et al. [8].The wall correction factorWc is defined as the ratio of the actual drag experienced by the particle in the encand the drag on a particle in an infinite expanse of fluid. With the aid of Eqs. (30) and (31) this becomesWc = 2{α[15η5 + (η5 − 1)α2]cosh(α) + [α2(1− 6η3) − 15η5]sinh(α)}× {α(3+ 2α2)cosh(α) − 3 sinh(α)}/{−α cosh(α) + sinh(α)}X1 (32)The variation ofWc againstη for various values ofα2 is shown in Fig. 2. It can be observed from the figuthat, asη increases, the wall correction factor increases. The wall correction factor is increasing as the permparameterα2 is increasing.References[1] E. Cunningham, On the velocity of steady fall of spherical particles through fluid medium, Proc. Roy. Soc. London Ser. A 83357–369.[2] W.E. Williams, On the motion of a sphere in a viscous fluid, Philos. Mag. 29 (1915) 526–550.[3] W.L. Haberman, R.M. Sayre, Wall effects for rigid and fluid spheres in slow motion with a moving liquid, David Taylor model,Report No. 1143, Washington, DC, 1958.[4] H. Ramkissoon, K. Rahaman, Non Newtonian fluid sphere in a spherical container, Acta Mech. 149 (2001) 239–245.[5] H. Ramkissoon, K. Rahaman, Wall effects on a spherical particle, Int. J. Engrg. Sci. 41 (2003) 283–290.[6] H.C. Brinkman, A calculation of viscous force exerted by a flowing fluid on dense swarm of particles, Appl. Sci. Res. A 1 (1957) 2[7] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, 1965.[8] G. Neale, N. Epstein, W. Nader, Creeping flow relative to permeable spheres, Chem. Engrg. Sci. 28 (1973) 1865–1874.

Motion of a porous sphere in a spherical container

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Motion of a porous sphere in a spherical container

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