Slow flow past a weakly permeable spheroidal particle in a hypothetical cell

• Autorzy: Bucha Tina , Madasu Krishna Prasad

Identyfikatory

DOI

10.24425/ame.2021.137044

• Języki publikacji: EN

Abstrakty

EN

The perspective of the current analysis is to represent the incompressible viscous flow past a low permeable spheroid contained in a fictitious spheroidal cell. Stokes approximation and Darcy’s equation are adopted to govern the flow in the fluid and permeable zone, respectively. Happel’s and Kuwabara’s cell models are employed as the boundary conditions at the cell surface. At the fluid porous interface, we suppose the conditions of conservation of mass, balancing of pressure component at the permeable area with the normal stresses in the liquid area, and the slip condition, known as Beavers-Joseph-Saffman-Jones condition to be well suitable. A closed-form analytical expression for hydrodynamic drag on the bounded spheroidal particle is determined and therefore, mobility of the particle is also calculated, for both the case of a prolate as well as an oblate spheroid. Several graphs and tables are plotted to observe the dependence of normalized mobility on pertinent parameters including permeability, deformation, the volume fraction of the particle, slip parameter, and the aspect ratio. Significant results that influence the impact of the above parameters in the problem have been pointed out. Our work is validated by referring to previous results available in literature as reduction cases.

Słowa kluczowe

EN

permeable spheroid; cell model; Stokes law; Darcy’s law; BJSJ condition

PL

sferoida przepuszczalna; model komórki; prawo Stokesa; prawo Darcy'ego; warunek BJSJ

• Wydawca: Komitet Budowy Maszyn PAN
• Czasopismo: Archive of Mechanical Engineering
• Rocznik: 2021
• Tom: LXVIII, nr 2
• Strony: 119--146
• Opis fizyczny: Bibliogr. 55 poz., rys., tab., wykr.

Twórcy

autor

Bucha Tina

• Department of Mathematics, National Institute of Technology, Raipur, Chhattisgarh, India

• https://orcid.org/0000-0002-9709-3025

autor

Madasu Krishna Prasad

• Department of Mathematics, National Institute of Technology, Raipur-492010, Chhattisgarh, India

• https://orcid.org/0000-0002-1819-3651

Bibliografia

• [1] D.A. Nield and A. Bejan. Convection in Porous Media. Springer, New York, 2006.
• [2] H.P.G. Darcy. Les Fontaines Publiques de la Ville de Dijon. Victor Delmont, Paris, 1856.
• [3] H.C. Brinkman. A calculation of viscous force exerted by flowing fluid on dense swarm of particles. Applied Science Research, 1:27-34, 1949. doi: 10.1007/BF02120313.
• [4] D.D. Joseph and L.N. Tao. The effect of permeability on the slow motion of a porous sphere. Journal of Applied Mathematics and Mechanics, 44(8-9):361-364, 1964. doi: 10.1002/zamm.19640440804.
• [5] D.N. Sutherland and C.T. Tan. Sedimentation of a porous sphere. Chemical Engineering Science, 25(12):1948-1950, 1970. doi: 10.1016/0009-2509(70)87013-0.
• [6] M.P. Singh and J.L. Gupta. The effect of permeability on the drag of a porous sphere in a uniform stream. Journal of Applied Mathematics and Mechanics, 51(1):27-32, 1971. doi: zamm.19710510103.
• [7] I.P. Jones. Low Reynolds number flow past a porous spherical shell. Mathematical Proceedings of the Cambridge Philosophical Society, 73(1):231-238, 1973. doi: 10.1017/S0305004100047642.
• [8] G. Neale, N. Epstein, and W. Nader. Creeping flow relative to permeable spheres. Chemical Engineering Science, 28(10):1865-1874, 1973. doi: 10.1016/0009-2509(73)85070-5.
• [9] V.M. Shapovalov. Viscous fluid flow around a semipermeable particle. Journal of Applied Mechanics and Technical Physics, 50(4):584-588, 2009. doi: 10.1007/s10808-009-0079-x.
• [10] G.S. Beavers and D.D. Joseph. Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics, 30(1):197-207, 1967. doi: 10.1017/S0022112067001375.
• [11] P.G. Saffman. On the boundary condition at the surface of a porous medium. Studies in Applied Mathematics, 50(2):93-101, 1971. doi: 10.1002/sapm197150293.
• [12] S. Khabthani, A. Sellier, and F. Feuillebois. Lubricating motion of a sphere towards a thin porous slab with Saffman slip condition. Journal of Fluid Mechanics, 867:949-968, 2019. doi: 10.1017/jfm.2019.169.
• [13] M.C. Lai, M.C. Shiue, and K.C. Ong. A simple projection method for the coupled Navier-Stokes and Darcy flows. Computational Geosciences, 23:21-33, 2019. doi: 10.1007/s10596-018-9781-1.
• [14] J. Happel and H. Brenner. Low Reynolds Number Hydrodynamics. Englewood Cliffs New Jork, Prentice-Hall, 1965.
• [15] J. Happel. Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. American Institute of Chemical Engineers Journal, 4(2):197-201, 1958. doi: 10.1002/aic.690040214.
• [16] S. Kuwabara. The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers. Journal of the Physical Society of Japan, 14(4):527-532,1959. doi: 10.1143/JPSJ.14.527.
• [17] S.B. Chen and X. Ye. Boundary effect on slow motion of a composite sphere perpendicular to two parallel impermeable plates. Chemical Engineering Science, 55(13):2441-2453, 2000. doi: 10.1016/S0009-2509(99)00509-6.
• [18] D. Srinivasacharya. Motion of a porous sphere in a spherical container. Comptes Rendus Mecanique, 333(8):612-616, 2005. doi: 10.1016/j.crme.2005.07.017.
• [19] S.I. Vasin, A.N. Fillipov, and V.M. Starov. Hydrodynamic permeability of membranes built up by particles covered by porous shells: Cell models. Advances in Colloid Interface Science, 139(1-2):83-96, 2008. doi: 10.1016/j.cis.2008.01.005.
• [20] P.K. Yadav, A. Tiwari, S. Deo, A. Filippov, and S. Vasin. Hydrodynamic permeability of membranes built up by spherical particles covered by porous shells: effect of stress jump condition. Acta Mechanica, 215:193-209, 2010. doi: 10.1007/s00707-010-0331-8.
• [21] J. Prakash, G.P. Raja Sekhar, and M. Kohr. Stokes flow of an assemblage of porous particles: stress jump condition. Zeitschrift für angewandte Mathematik und Physik, 62:1027-1046, 2011. doi: 10.1007/s00033-011-0123-6.
• [22] E.I. Saad. Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition. Meccanica, 48:1747-1759, 2013. doi: 10.1007/s11012-013-9706-y.
• [23] J. Prakash and G.P. Raja Sekhar. Estimation of the dynamic permeability of an assembly of permeable spherical porous particle using cell model. Journal of Engineering Mathematics, 80:63-73, 2013. doi: 10.1007/s10665-012-9580-y.
• [24] M.K. Prasad and T. Bucha. Creeping flow of fluid sphere contained in a spherical envelope: magnetic effect. SN Applied Science, 1(12):1594, 2019. doi: 10.1007/s42452-019-1622-x.
• [25] M.K. Prasad and T. Bucha. Magnetohydrodynamic creeping flow around a weakly permeable spherical particle in cell models. Pramana - Journal of Physics, 94(1):1-10, 2020. doi: 10.1007/s12043-019-1892-2.
• [26] M.K. Prasad and T. Bucha. MHD viscous flow past a weakly permeable cylinder using Happel and Kuwabara cell models. Iranian Journal of Science and Technology Transaction A: Science, 44:1063-1073, 2020. doi: 10.1007/s40995-020-00894-4.
• [27] D. Khanukaeva. Filtration of micropolar liquid through a membrane composed of spherical cells with porous layer. Theoretical and Computational Fluid Dynamics, 34(3):215-229, 2020. doi: 10.1007/s00162-020-00527-x.
• [28] M.K. Prasad. Boundary effects of a nonconcentric semipermeable sphere using Happel and Kuwabara cell models. Applied and Computational Mechanics, 15:1-12, 2021. doi: 10.24132/acm.2021.620.
• [29] G.G. Stokes. On the effect of the internal friction of fluids on the motion of pendulums. Proceedings of Cambridge Philosophical Society, 9:8-106, 1851.
• [30] C.R. Reddy and N. Kishore. Momentum and heat transfer phenomena of confined spheroid particles in power-law liquids, Industrial and Engineering Chemical Research, 53(2):989-998, 2014. doi: 10.1021/ie4032428.
• [31] A. Acrivos and T.D. Taylor. The Stokes flow past an arbitrary particle: the slightly deformed sphere. Chemical Engineering Science, 19(7):445-451, 1964. doi: 10.1016/0009-2509(64)85071-5.
• [32] H. Ramkissoon. Stokes flow past a slightly deformed fluid sphere, Journal of Applied Mathematics and Physics, 37:859-866, 1986. doi: 10.1007/BF00953677.
• [33] D. Palaniappan. Creeping flow about a slightly deformed sphere. Zeitschrift für angewandte Mathematik und Physik, 45:832-838, 1994. doi: 10.1007/BF00942756.
• [34] G. Dassios, M. Hadjinicolaou, F.A. Coutelieris, and A.C. Payatakes. Stokes flow in spheroidal particle-in-cell models with Happel and Kuwabara boundary conditions. International Journal of Engineering Science, 33(10):1465-1490, 1995. doi: 10.1016/0020-7225(95)00010-U.
• [35] H. Ramkissoon. Slip flow past an approximate spheroid. Acta Mechanica, 123:227-233, 1997. doi: 10.1007/BF01178412.
• [36] T. Zlatanovski. Axi-symmetric creeping flow past a porous prolate spheroidal particle using the Brinkman model. The Quarterly Journal of Mechanics and Applied Mathematics, 52(1):111-126, 1999. doi: 10.1093/qjmam/52.1.111.
• [37] S. Deo and S. Datta. Slip flow past a prolate spheroid. Indian Journal of Pure and Applied Mathematics, 33(6):903-909, 2002.
• [38] P. Vainshtein, M. Shapiro, and C. Gutfinger. Creeping flow past and within a permeable spheroid. International Journal of Multiphase Flow, 28(12):1945-1963, 2002. doi: 10.1016/S0301-9322(02)00106-4.
• [39] H. Ramkissoon and K. Rahaman. Wall effects on a spherical particle. International Journal of Engineering Science, 41(3-5), 283-290, 2003. doi: 10.1016/S0020-7225(02)00209-4.
• [40] S. Senchenko and H.J. Keh. Slipping Stokes flow around a slightly deformed sphere. Physics of Fluids, 18(8):088104, 2006. doi: 10.1063/1.2337666.
• [41] D. Srinivasacharya. Flow past a porous approximate spherical shell, Zeitschrift für angewandte Mathematik und Physik, 58, 646-658, 2007. doi: 10.1007/s00033-006-6003-9.
• [42] Y.C. Chang and H.J. Keh. Translation and rotation of slightly deformed colloidal spheres experiencing slip. Journal of Colloid and Interface Science, 330:201-210, 2009. doi: 10.1016/j.jcis.2008.10.055.
• [43] E.I. Saad. Translation and rotation of a porous spheroid in a spheroidal container. Canadian Journal of Physics, 88(9):689-700, 2010. doi: 10.1139/P10-040.
• [44] E.I. Saad. Stokes flow past an assemblage of axisymmetric porous spheroidal particle in cell models. Journal of Porous Media, 15(9):849-866, 2012. doi: /10.1615/JPorMedia.v15.i9.40.
• [45] D. Srinivasacharya and M.K. Prasad. Axisymmetric creeping motion of a porous approximate sphere with an impermeable core. The European Physics Journal Plus, 128(1):9, 2013. doi: 10.1140/epjp/i2013-13009-1.
• [46] D. Srinivasacharya and M.K. Prasad. Creeping motion of a porous approximate sphere with an impermeable core in a spherical container. European Journal of Mechanics - B/Fluids, 36:104-114, 2012. doi: 10.1016/j.euromechflu.2012.04.001.
• [47] D. Srinivasacharya and M.K. Prasad. Axisymmetric motion of a porous approximate sphere in an approximate spherical container. Archive of Mechanics, 65(6):485-509, 2013.
• [48] K.P. Chen. Fluid extraction from porous media by a slender permeable prolate-spheroid. Extreme Mechanics Letter, 4:124-130, 2015. doi: 10.1016/j.eml.2015.06.001.
• [49] M. Rasoulzadeh and F.J. Kuchuk. Effective permeability of a porous medium with spherical and spheroidal vug and fracture inclusions. Transport in Porous Media, 116:613-644, 2017. doi: 10.1007/s11242-016-0792-x.
• [50] P.K. Yadav, A. Tiwari, and P. Singh. Hydrodynamic permeability of a membrane built up by spheroidal particles covered by porous layer. Acta Mechanica, 229:1869-1892, 2018. doi: 10.1007/s00707-017-2054-6.
• [51] M.K. Prasad and T. Bucha. Steady viscous flow around a permeable spheroidal particle. International Journal of Applied and Computational Mathematics, 5:109, 2019. doi: 10.1007/s00707-017-2054-6.
• [52] M.K. Prasad and T. Bucha. Effect of magnetic field on the slow motion of a porous spheroid: Brinkman's model. Archive of Applied Mechanics, 91:1739-1755, 2021. doi: 10.1007/s00419-020-01852-7.
• [53] J.D. Sherwood. Cell models for suspension viscosity. Chemical Engineering Science, 61(10):6727-6731, 2006. doi: 10.1016/j.ces.2006.07.016.
• [54] A. Tiwari, P.K. Yadav, and P. Singh. Stokes flow through assemblage of non homogeneous porous cylindrical particle using cell model technique. National Academy of Science Letters, 41(1):53-57, 2018. doi: 10.1007/s40009-017-0605-y.
• [55] H.H. Sherief, M.S. Faltas, and E.I. Saad. Slip at the surface of an oscillating spheroidal particle in a micropolar fluid. ANZIAM Journal, 55(E):E1-E50, 2013. doi: 10.21914/anziamj.v55i0.6813.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).