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Journal of Porous Media
Impact-faktor: 1.752 5-jähriger Impact-Faktor: 1.487 SJR: 0.43 SNIP: 0.762 CiteScore™: 2.3

ISSN Druckformat: 1091-028X
ISSN Online: 1934-0508

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Journal of Porous Media

DOI: 10.1615/JPorMedia.2020027173
pages 613-626


R. Selvi
Department of Mathematics, Vellore Institute of Technology, Chennai-600127, India
Pankaj Shukla
Department of Mathematics, Vellore Institute of Technology, Chennai-600127, India
Abhishek Kumar Singh
Department of Mathematics, Vellore Institute of Technology, Chennai-600127, India


Consideration is given to the problem of steady axisymmetric creeping flow of a micropolar fluid around the spherical drop of non-Newtonian liquid shell covered with permeable medium. The field equations of micropolar fluids are presented in terms of velocity vector and microrotation vector. External liquid permeates into the porous layer, but it is not mixed with the liquid located in the internal cavity of a capsule. The flow inside the permeable medium is described by the Brinkman equation. The stream function solution for the external flow field is derived in terms of modified Bessel's function and Gegenbauer's polynomial. The solution is determined by dilating the stream function in terms of the dimensionless parameter S for the internal flow field (Reiner-Rivlin liquid sphere). Analytical expressions for the pressure field, coupling number, microrotation component, viscosity ratio, permeability parameter, and drag force are calculated. The effect of various parameters on the drag force is presented graphically and discussed. It is observed that the drag on a micropolar fluid sphere is more than that on a permeable sphere. Different limiting cases are also considered.


  1. Abramowitz, M. and Stegun, I.A., Hand Book of Mathematical Function, New York: Dover Publications, 1970.

  2. Ansari, I.A. and Deo, S., Magnetohydrodynamic Viscous Fluid Flow past a Porous Sphere Embedded in Another Porous Medium, Spec. Topics Rev. Porous Media, vol. 9, no. 2, pp. 191-200, 2018.

  3. Barman, B., Flow of a Newtonian Fluid past an Impervious Sphere Embedded in a Porous Medium, Indian J. Pure Appl. Math., vol. 27, no. 12, pp. 1244-1256, 1996.

  4. Bhatti, M.M. and Rashidi, M.M., Effects of Thermo-Diffusion and Thermal Radiation on Williamson Nanofluid over a Porous Shrinking/Stretching Sheet, J. Mol. Liq., vol. 221, pp. 567-573, 2016.

  5. Birikh, R. and Rudakoh, R., Slow Motion of a Permeable Sphere in a Viscous Fluid, Fluid Dyn., vol. 17, no. 5, pp. 792-793, 1982.

  6. Brinkman, H.C., A Calculation of Viscous Force Exerted by a Flowing Fluid on Dense Swarm of Particles, Appl. Sci. Res., vol. A1, pp. 27-34, 1947.

  7. Darcy, H., Les Fontaines Publiques De La Ville De Dijion, Paris: Victor Dalmont, 1856.

  8. Deo, S., Shukla, P., and Gupta, B.R., Drag on a Fluid Sphere Embedded in a Porous Medium, Adv. Theor. Appl. Mech, vol. 3, no. 1, pp. 45-52, 2010.

  9. Ellahi, R., Bhatti, M.M., Riaz, A., and Sheikholeslami, M., Effects of Magnetohydrodynamics on Peristaltic Flow of Jeffery Fluid in a Rectangular Duct through a Porous Medium, J. Porous Media, vol. 17, no. 2, pp. 143-157, 2014.

  10. Eringen, A.C., Simple Micropolar Fluids, Int. J. Eng. Sci., vol. 2, pp. 205-217, 1964.

  11. Eringen, A.C., Theory of Micropolar Fluids, J. Math. Mech., vol. 16, no. 1, pp. 1-18, 1966.

  12. Gupta, B.R. and Deo, S., Axisymmetric Creeping Flow of a Micropolar Fluid over a Sphere Coated with a Thin Fluid Film, J. Appl. Fluid Mech., vol. 6, no. 2, pp. 149-155, 2013.

  13. Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamics, Englewood Cliffs, NJ: Prentice-Hall, 1965.

  14. Iyengar, T.K.V. and Srinivasacharya, D., Stokes Flow of an Incompressible Micropolar Fluid past an Approximate Sphere, Int. J. Eng. Sci., vol. 31, no. 1, pp. 115-123, 1993.

  15. Jaiswal, B.R. and Gupta, B.R., Drag on Reiner-Rivlin Liquid Sphere Placed in a Micropolar Fluid with Non-Zero Boundary Conditions for Microrotations, Int. J. Appl. Math. Mech., vol. 10, no. 7, pp. 90-103, 2014.

  16. Jaiswal, B.R. and Gupta, B.R., Brinkman Flow of a Viscous Fluid past a Reiner-Rivlin Liquid Sphere Immersed in a Saturated Porous Medium, Transp. Porous Media, vol. 106, no. 3, pp. 907-925, 2015.

  17. Jaiswal, B.R. and Gupta, B.R., Stokes Flow over a Non-Newtonian Encapsulated Drop of Another Liquid: Effect of Stress Jump, J. Porous Media, vol. 20, no. 9, pp. 807-821, 2017.

  18. Joseph, D. and Tao, L., The Effect of Permeability on the Slow Motion of a Porous Sphere in a Viscous Liquid, J. Appl. Math. Mech., vol. 44, nos. 8-9, pp. 361-364, 1964.

  19. Juncu, G., A Numerical Study of the Flow past an Impermeable Sphere Embedded in a Porous Medium, Transp. Porous Media, vol. 108, no. 3, pp. 555-579, 2015.

  20. Lukaszewicz, G., Micropolar Fluids: Theory and Applications, Basel, Switzerland: Birkhauser, 1999.

  21. Lundgren, T., Slow Flow through Stationary Random Beds and Suspensions of Spheres, J. Fluid Mech., vol. 51, no. 2, pp. 273-299, 1972.

  22. Masliyah, J. and Polikar, M., Terminal Velocity of Porous Spheres, Can. J. Chem. Eng., vol. 58, no. 3, pp. 299-302, 1980.

  23. Masliyah, J., Neale, G., Malysa, K., and de Ven, T.V., Creeping Flow over a Composite Sphere: Solid Core with Porous Shell, Chem. Eng. Sci., vol. 4, no. 2, pp. 245-253, 1987.

  24. Matsumoto, K. and Suganuma, A., Settling Velocity of a Permeable Sphere, Chem. Eng. Sci., vol. 32, no. 4, pp. 445-447, 1977.

  25. Nandkumar, K. and Masliyah, J., Laminar Flow past a Permeable Sphere, J. Chem. Eng, vol. 60, no. 2, pp. 202-211, 1982.

  26. Neale, G., Epstein, N., and Nader, W., Creeping Flow Relative to Permeable Spheres, Chem. Eng. Sci., vol. 28, no. 10, pp. 1865-1873, 1973.

  27. Niefer, R. and Kaloni, P.N., On the Motion of a Micropolar Fluid Drop in a Viscous Fluid, J. Eng. Math., vol. 14, no. 2, pp. 107-116, 1980.

  28. Ooms, G., Mijulieff, P., and Becker, H., Frictional Force Exerted by a Flowing Fluid in a Permeable Particle with Particular Reference to Polymer Coils, J. Chem. Phys., vol. 53, no. 11, pp. 4123-4130, 1970.

  29. Qin, Y. and Kaloni, P.N., A Cartesian-Tensor Solution of Brinkman Equation, J. Eng. Math., vol. 22, no. 2, pp. 177-188, 1988.

  30. Ramkissoon, H., Flow of Micropolar Fluid past a Newtonian Fluid Sphere, Z. Angew. Math. Mech., vol. 65, no. 12, pp. 635-637, 1985.

  31. Ramkissoon, H., Stokes Flow past a Reiner-Rivlin Fluid Sphere, J. Appl. Math. Mech., vol. 69, no. 8, pp. 259-261, 1989.

  32. Ramkissoon, H. and Majumadar, S.R., Drag on the Axially Symmetric Body in the Stokes Flow of Micropolar Fluid, Phys. Fluids, vol. 19, no. 1, pp. 16-21, 1976.

  33. Rao, S.K.L. and Rao, P.B., Slow Stationary Flow of a Micropolar Fluid past a Sphere, J. Eng. Math., vol. 4, no. 3, pp. 209-217, 1970.

  34. Rathna, S.L., Slow Motion of a Non-Newtonian Liquid past a Sphere, Quart. J. Mech. Appl. Math., vol. 15, no. 4, pp. 427-434, 1989.

  35. Saad, E.I., Motion of a Spherical Particle in a Micropolar Fluid Contained in a Spherical Envelope, Can. J. Phys., vol. 86, no. 9, pp. 1039-1056, 2008.

  36. Saad, E.I., Cell Models for Micropolar Fluid past a Viscous Fluid Sphere, Meccanica, vol. 47, no. 8, pp. 2055-2068, 2012.

  37. Sharma, H.G., Creeping Motion of Non-Newtonian Fluid past a Sphere, Int. J. Pure Appl. Math., vol. 10, no. 12, pp. 1565-1575, 1979.

  38. Sheikholeslami, M. and Bhatti, M.M., Active Method for Nanofluid Heat Transfer Enhancement by Means of EHD, Int. J. Heat Mass Transf., vol. 109, pp. 115-122, 2017a.

  39. Sheikholeslami, M. and Bhatti, M.M., Forced Convection of Nanofluid in Presence of Constant Magnetic Field Considering Shape Effects of Nano Particles, Int. J. Heat Mass Transf., vol. 111, pp. 1039-1049, 2017b.

  40. Srinivasacharya, D., Creeping Flow past a Porous Approximate Sphere, ZAMM, vol. 83, no. 7, pp. 499-504, 2003.

  41. Srinivasacharya, D. and Kumar, P.V., Mixed Convection over an Inclined Wavy Surface in a Nanofluid Saturated Non-Darcy Porous Medium with Radiation Effect, Int. J. Chem. Eng., vol. 7, pp. 1-15, 2015.

  42. Srinivasacharya, D. and Murthy, J., Flow past an Axisymmetric Body Embedded in a Saturated Porous Medium, C. R. Mecanique, vol. 330, no. 6, pp. 417-423, 2002.

  43. Srinivasacharya, D. and Prasad, M., Slow Steady Rotation of a Porous Sphere in a Spherical Container, J. Porous Media, vol. 15, no. 12, pp. 1105-1110, 2012.

  44. Srinivasacharya, D. and Rajyalakshmi, I., Creeping Flow of Micropolar Fluid past a Porous Sphere, Appl. Math. Comput., vol. 153, no. 4, pp. 843-854, 2004.

  45. Stokes, G.G., On the Effects of Internal Friction of Fluids on Pendulums, J. Trans. Camb. Philos. Soc., vol. 9, pp. 8-106, 1851.

  46. Tam, C., The Drag on a Cloud of Spherical Particles in a Low Reynolds Number Flow, J. Fluid Mech., vol. 38, no. 3, pp. 537-546, 1969.

  47. Tien, C., Granular Filtration of Aerosols and Hydrosols, Boston: Butterworths, 1989.

  48. Verma, P.D. and Bhatt, B.S., Low Reynolds Number Flow past a Heterogeneous Porous Sphere Using Asymptotic Technique, Appl. Sci. Res, vol. 32, no. 1, pp. 61-72, 1976.

  49. Yadav, P.K., On the Slow Viscous Flow through a Swarm of Solid Spherical Particles Covered by Porous Shells, Appl. Math., vol. 1, no. 2, pp. 112-121, 2011.

  50. Zlatanovski, T., Axisymmetric Creeping Flow past a Porous Prolate Spheroidal Particle Using the Brinkman Model, Q. J. Mech. Appl. Mech, vol. 52, no. 1, pp. 111-126, 1999.

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