Bed of polydisperse viscous spherical drops under thermocapillary effects



Viscous flow past an ensemble of polydisperse spherical drops is investigated under thermocapillary effects. We assume that the collection of spherical drops behaves as a porous media and estimates the hydrodynamic interactions analytically via the so- called cell model that is defined around a specific representative particle. In this method, the hydrodynamic interactions are assumed to be accounted by suitable boundary conditions on a fictitious fluid envelope surrounding the representative particle. The force calculated on this representative particle will then be extended to a bed of spherical drops visualized as a Darcy porous bed. Thus, the “effective bed permeability” of such a porous bed will be computed as a function of various parameters and then will be compared with Carman–Kozeny relation. We use cell model approach to a packed bed of spherical drops of uniform size (monodisperse spherical drops) and then extend the work for a packed bed of polydisperse spherical drops, for a specific parameters. Our results show a good agreement with the Carman–Kozeny relation for the case of monodisperse spherical drops. The prediction of overall bed permeability using our present model agrees well with the Carman–Kozeny relation when the packing size distribution is narrow, whereas a small deviation can be noted when the size distribution becomes broader.


Cell model Thermocapillary Polydisperse spherical drops Overall bed permeability 

Mathematics Subject Classification

76D07 76S05 60E05 76A15 82D15 82D30 


  1. 1.
    Choudhuri D., Raja Sekhar G.P.: Thermocapillary drift on a spherical drop in a viscous fluid. Phys. Fluids 25(4), 043104 (2013)CrossRefGoogle Scholar
  2. 2.
    Sharanya V., Sekhar G.R.: Thermocapillary migration of a spherical drop in an arbitrary transient stokes flow. Phys. Fluids (1994-present) 27(6), 063104 (2015)CrossRefMATHGoogle Scholar
  3. 3.
    Hadamard J.S.: Motion of liquid drops (viscous). Comp. Rend. Acad. Sci. Paris 154, 1735–1755 (1911)Google Scholar
  4. 4.
    Rybczynski W.: Uber die fortschreitende bewegung einer flussigen kugel in einem zahen medium. Bull. Acad. Sci. Crac. 1, 40–46 (1911)Google Scholar
  5. 5.
    Hetsroni G., Haber S.: The flow in and around a droplet or bubble submerged in an unbound arbitrary velocity field. Rheol. Acta 9(4), 488–496 (1970)CrossRefMATHGoogle Scholar
  6. 6.
    Nas S., Muradoglu M., Tryggvason G.: Pattern formation of drops in thermocapillary migration. Int. J. Heat Mass transf. 49(13), 2265–2276 (2006)CrossRefMATHGoogle Scholar
  7. 7.
    Young N.O., Goldstein J., Block M.: The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6(03), 350–356 (1959)CrossRefMATHGoogle Scholar
  8. 8.
    Balasubramaniam R., Chai A.: Thermocapillary migration of droplets: an exact solution for small Marangoni numbers. J. Colloid Interface Sci. 119(2), 531–538 (1987)CrossRefGoogle Scholar
  9. 9.
    Anderson J.L.: Droplet interactions in thermocapillary motion. Int. J. Multiph. Flow 11(6), 813–824 (1985)CrossRefMATHGoogle Scholar
  10. 10.
    Keh H.J., Chen L.S.: Droplet interactions in thermocapillary migration. Chem. Eng. Sci. 48(20), 3565–3582 (1993)CrossRefGoogle Scholar
  11. 11.
    Happel J.: Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. AIChE J. 4(2), 197–201 (1958)CrossRefGoogle Scholar
  12. 12.
    Kuwabara S.: The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small reynolds numbers. J. Phys. Soc. Jpn. 14(4), 527–532 (1959)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cunningham, E.: On the velocity of steady fall of spherical particles through uid medium. In: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol. 83 no. 563, pp. 357–365 (1910)Google Scholar
  14. 14.
    Mehta G.D., Morse T.: Flow through charged membranes. J. Chem. Phys. 63(5), 1878–1889 (1975)CrossRefGoogle Scholar
  15. 15.
    Sherwood J.: Cell models for suspension viscosity. Chem. Eng. Sci. 61(20), 6727–6731 (2006)CrossRefGoogle Scholar
  16. 16.
    Prakash J., Sekhar G.R.: Estimation of the dynamic permeability of an assembly of permeable spherical porous particles using the cell model. J. Eng. Math. 80(1), 63–73 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Prakash J., Sekhar G.R., Kohr M.: Stokes flow of an assemblage of porous particles: stress jump condition. Z. Angew. Math. Phys. 62(6), 1027–1046 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gal-Or B., Waslo S.: Hydrodynamics of an ensemble of drops (or bubbles) in the presence or absence of surfactants. Chem. Eng. Sci. 23(12), 1431–1446 (1968)CrossRefGoogle Scholar
  19. 19.
    Haber S., Hetsroni G.: Sedimentation in a dilute dispersion of small drops of various sizes. J. Colloid Interface Sci. 79(1), 56–75 (1981)CrossRefGoogle Scholar
  20. 20.
    Ferreira J., Soares A., Chhabra R.: Hydrodynamic behaviour of an ensemble of encapsulated liquid drops in creeping motion: a fluid-mechanic based model for liquid membranes. Fluid Dyn. Res. 32(5), 201–215 (2003)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kishore N., Chhabra R., Eswaran V.: Drag on ensembles of fluid spheres translating in a power-law liquid at moderate reynolds numbers. Chem. Eng. J. 139(2), 224–235 (2008)CrossRefGoogle Scholar
  22. 22.
    Nield D.A., Bejan A.: Convection in Porous Media. Springer, New York (2006)MATHGoogle Scholar
  23. 23.
    Happel J., Brenner H.: Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media. Springer, New York (1983)MATHGoogle Scholar
  24. 24.
    Carman P.C.: Flow of Gases Through Porous Media. Academic Press, New York (1956)MATHGoogle Scholar
  25. 25.
    Ouchiyama N., Tanaka T.: Estimation of the average number of contacts between randomly mixed solid particles. Ind. Eng. Chem. Fundam. 19(4), 338–340 (1980)CrossRefGoogle Scholar
  26. 26.
    Ouchiyama N., Tanaka T.: Porosity of a mass of solid particles having a range of sizes. Ind. Eng. Chem. Fundam. 20(1), 66–71 (1981)CrossRefGoogle Scholar
  27. 27.
    Li Y., Park C.-W.: Permeability of packed beds filled with polydisperse spherical particles. Ind. Eng. Chem. Res. 37(5), 2005–2011 (1998)CrossRefGoogle Scholar
  28. 28.
    Endo Y., Chen D.-R., Pui D.Y.: Theoretical consideration of permeation resistance of fluid through a particle packed layer. Powder Technol. 124(1), 119–126 (2002)CrossRefGoogle Scholar
  29. 29.
    Subramanian R.S., Balasubramaniam R.: The Motion of Bubbles and Drops in Reduced Gravity. Cambridge University Press, Cambridge (2001)MATHGoogle Scholar
  30. 30.
    Clift R., Grace J.R., Weber M.E.: Bubbles, Drops and Particles. Academic Press, New York (1978)Google Scholar
  31. 31.
    Polyanin A.D., Kutepov A.M., Vyazmin A.V., Kazenin D.A.: Hydrodynamics, Mass and Heat Transfer in Chemical Engineering. CRC Press LLC, Boca Raton (2002)Google Scholar
  32. 32.
    Rosenfeld L., Lavrenteva O.M., Nir A.: Thermocapillary motion of hybrid drops. Phys. Fluids 20(7), 072102 (2008)CrossRefMATHGoogle Scholar
  33. 33.
    Filippov A., Vasin S., Starov V.: modeling of the hydrodynamic permeability of a membrane built up from porous particles with a permeable shell. Colloids Surf. A Physicochem. Eng. Asp. 282, 272–278 (2006)CrossRefGoogle Scholar
  34. 34.
    Dullien F.: Single phase flow through porous media and pore structure. Chem. Eng. J. 10(1), 1–34 (1975)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • V. Sharanya
    • 1
  • G. P. Raja Sekhar
    • 1
  • Christian Rohde
    • 2
  1. 1.Department of MathematicsIndian Institute of TechnologyKharagpurIndia
  2. 2.Institute of Applied Analysis and Numerical SimulationUniversity of StuttgartStuttgartGermany

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