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Steady Viscous Flow Around a Permeable Spheroidal Particle

  • Krishna Prasad MadasuEmail author
  • Tina Bucha
Original Paper
  • 9 Downloads

Abstract

Stokes incompressible viscous fluid flow through a permeable spheroidal particle which is a bit deformed from the shape of a sphere is studied and solved analytically. It consists of two regions, porous region which obeys Darcy’s law and liquid region in which Stokes approximation is used. Boundary conditions used at the interface are mass conservation, balance of normal stress, and Beavers–Joseph–Saffman–Jones condition. Expression for drag which acts on the spheroid is obtained and well known results are deduced in the limiting cases. Variation of drag coefficient with various parameters like deformation, slip, permeability, no slip are shown by graphs.

Keywords

Permeable spheroid Stokes flow Darcy law Drag force 

Mathematics Subject Classification

76D07 

Notes

References

  1. 1.
    Darcy, H.P.G.: Les fontaines publiques de la ville de dijon. Proc. R. Soc. Lond. Ser. 83, 357–369 (1910)CrossRefGoogle Scholar
  2. 2.
    Brinkman, H.C.: A calculation of viscous force exerted by a flowing fluid on dense swarm of particles. Appl. Sci. Res. A 1(1), 27–34 (1947)CrossRefGoogle Scholar
  3. 3.
    Leonov, A.I.: The slow stationary flow of a viscous fluid about a porous sphere. J. Appl. Maths. Mech. 26, 564–566 (1962)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Joseph, D.D., Tao, L.N.: The effect of permeability on the slow motion of a porous sphere. Z. Angew. Math. Mech. 44, 361–364 (1964)CrossRefGoogle Scholar
  5. 5.
    Sutherland, D.N., Tan, C.T.: Sedimentation of a porous sphere. Chem. Eng. Sci. 25, 1948–1950 (1970)CrossRefGoogle Scholar
  6. 6.
    Neale, G., Epstein, N.: Creeping flow relative to permeable spheres. Chem. Eng. Sci. 28, 1865–1874 (1973)CrossRefGoogle Scholar
  7. 7.
    Jones, I.P.: Low Reynolds number flow past a porous spherical shell. Proc. Camb. Philos. Soc 73, 231 (1973)CrossRefGoogle Scholar
  8. 8.
    Feng, Z.G., Michaelides, E.E.: Motion of a permeable sphere at finite but small Reynolds numbers. Phys. Fluid. 10, 6 (1998)CrossRefGoogle Scholar
  9. 9.
    Jager, W., Mikelic, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Vainshtein, P., Shapiro, M., Gutfinger, C.: Creeping flow past and within a permeable spheroid. Int. J. Multiph. 28, 1945–1963 (2002)CrossRefGoogle Scholar
  11. 11.
    Srinivasacharya, D.: Flow past a porous approximate spherical shell. Z. Angew. Math. Phys. 58, 646–658 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Beavers, G.S., Joseph, D.D.: Boundary condition at a naturally permeable wall. J. Fluid Mech. 30, 197 (1967)CrossRefGoogle Scholar
  13. 13.
    Urquiza, J.M., N’Dri, D., Garon, A., Delfour, M.C.: Coupling Stokes and Darcy equations. Appl. Numer. Math. 58, 525–538 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shapovalov, V.M.: Viscous fluid flow around a semipermeable sphere. J. Appl. Mech. Tech. Phys. 50(4), 584–588 (2009)CrossRefGoogle Scholar
  15. 15.
    Cao, Y., Gunzburger, M., Hua, F., Wang, X.: Coupled Stokes–Darcy model with Beavers–Joseph interface boundary condition. Commun. Math. Sci. 8(1), 1–25 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Vereshchagin, A.S., Dolgushev, S.V.: Low velocity viscous incompressible fluid flow around a hollow porous sphere. J. Appl. Mech. Techn. Phys. 52(3), 406–414 (2011)CrossRefGoogle Scholar
  17. 17.
    Prakash, J., Raja Sekhar, G.P., Kohr, M.: Stokes flow of an assemblage of porous particles: stress jump condition. Z. Angew. Math. Phys. 62, 1027–1046 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Prakash, J., Raja Sekhar, G.P.: Estimation of the dynamic permeability of an assembly of permeable spherical porous particles using the cell model. J. Eng. Math. 80, 63–73 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Saffman, P.G.: On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50, 93 (1971)CrossRefGoogle Scholar
  20. 20.
    Saad, E.I.: Stokes flow past an assemblage of axisymmetric porous spheroidal particle in cell models. J. Porous Media. 15(9), 849–866 (2012)CrossRefGoogle Scholar
  21. 21.
    Chen, P.C.: Fluid extraction from porous media by a slender permeable prolate-spheroid. Extreme Mech. Lett. 4, 124–130 (2015)CrossRefGoogle Scholar
  22. 22.
    Rasoulzadeh, M., Kuchuk, F.J.: Effective permeability of a porous medium with spherical and spheroidal vug and fracture inclusion. Transp. Porous Med. 116, 613–644 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tiwari, A., Yadav, P.K., Singh, P.: Stokes flow through assemblage of nonhomogeneous porous cylindrical particle using cell model technique. Natl. Acad. Sci. Lett. 4(1), 53–57 (2018)CrossRefGoogle Scholar
  24. 24.
    Khabthani, S., Sellier, A., Feuillebois, F.: Lubricating motion of a sphere towards a thin porous slab with Saffman slip condition. J. Fluid Mech. 867, 949–968 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lai, M.C., Shiue, M.C., Ong, K.C.: A simple projection method for the coupled Navier–Stokes and Darcy flows. Comput. Geosci. 23, 21–33 (2019)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics. Prentice-Hall, Englewood Cliffs, NJ (1965)zbMATHGoogle Scholar
  27. 27.
    Nield, D.A., Bejan, A.: Convection in Porous Media. Studies in Applied Mathematics, 3rd edn. Springer, New York (2006)zbMATHGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyRaipurIndia

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