Applied Mathematics and Mechanics

, Volume 23, Issue 5, pp 514–534 | Cite as

The three-dimensional fundamental solution to Stokes flow in the oblate spheroidal coordinates with applications to multiples spheroid problems

  • Zhuang Hong
  • Yan Zong-yi
  • Wu Wang-yi


A new three-dimensional fundamental solution to the Stokes flow was proposed by transforming the solid harmonic functions in Lamb's solution into expressions in terms of the oblate spheroidal coordinates. These fundamental solutions are advantageous in treating flows past an arbitrary number of arbitrarily positioned and oriented oblate spheroids. The least squares technique was adopted herein so that the convergence difficulties often encountered in solving three-dimensional problems were completely avoided. The examples demonstrate that present approach is highly accurate, consistently stable and computationally efficient.

The oblate spheroid may be used to model a variety of particle shapes between a circular disk and a sphere. For the first time, the effect of various geometric factors on the forces and torques exerted on two oblate spheroids were systematically studied by using the proposed fundamental solutions. The generality of this approach was illustrated by two problems of three spheroids.

Key words

Stokes flow fundamental solution three-dimension oblate spheroid multipole collocation least squares method low Reynolds number multiple particles 

CLC number



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  1. [1]
    Weinbaum S, Ganatos P, YAN Zong-yi. Numerical multipole and boundary integral equation techniques in Stokes flow[J].Ann Rev Fluid Mech, 1990,22: 275–316.CrossRefGoogle Scholar
  2. [2]
    Gluckman MJ, Pfeffer R, Weinbaum S. A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids[J].J Fluid Mech, 1971,50(4): 705–740.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Guckman MJ, Weinbaum S, Pfeffer R. Axisymmetric slow viscous flow past an arbitrary convex body of revolution[J].J Fluid Mech, 1972,55(4): 677–709.CrossRefGoogle Scholar
  4. [4]
    Ganatos P, Pfeffer R, Weinbaum S. A numerical-solution technique for three-dimensional Stokes flows, with application to the motion of strongly interacting spheres in a plane[J].J Fluid Mech, 1978,84(1): 79–111.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Yan ZY, Weinbaum S, Ganatos P, et al. The three-dimensional hydrodynamic interaction, of a finite sphere with a circular orifice at low Reynolds number[J].J Fluid Mech, 1987,174: 39–68.zbMATHCrossRefGoogle Scholar
  6. [6]
    Hassonjee Q, Ganatos P, Pfeffer R. A strong-interaction, theory for the motion of arbitrary three-dimensional clusters of spherical particles at low Reynolds number[J].J Fluid Mech, 1988,197: 1–37.zbMATHCrossRefGoogle Scholar
  7. [7]
    Leichtberg S, Peffer R, Weinbaum S. Stokes flow past finite coaxial clusters of spheres in a circular cylinder[J].Int J Multiphase Flow, 1976,3(2): 147–169.CrossRefGoogle Scholar
  8. [8]
    Ganatos P, Weinbaum S, Pfeffer R. A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion[J].J Fluid Mech, 1980,99(4): 739–753.zbMATHCrossRefGoogle Scholar
  9. [9]
    Ganatos P, Pfeffer R, Weinbaum S. A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 2. Parallel motion[J].J Fluid Mech, 1980,99(4): 755–783.zbMATHCrossRefGoogle Scholar
  10. [10]
    Dagan Z, Weinbaum S, Pfeffer R. General theory for the creeping motion of a finite sphere along the axis of a circular orifice[J].J Fluid Mech, 1982,117: 143–170.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Yoon BJ, Kim S. A boundary collocation method for the motion of two spheroids in Stokes flow: hydrodynamic and colloidal interactions[J].Int J Multiphase Flow, 1990,16(4): 639–649.zbMATHCrossRefGoogle Scholar
  12. [12]
    Hsu R, Ganatos P. The motion of a rigid body in viscous fluid boundary by a plane wall[J].J Fluid Mech, 1989,207: 29–72.zbMATHCrossRefGoogle Scholar
  13. [13]
    SHI Cang-chueng, WANG Wei-guo, WU Wang-yi. The axisymmetric creeping flow in an infinite long circular cone with a sphere moving along the axis of symmetry[J].Acta Scientiarum Naturalium Universitatis Pekinensis, 1988,24(1): 85–94. (in Chinese with English abstract)Google Scholar
  14. [14]
    Lamb H.Hydrodynamics[M]. 6th edn. New York: Dover, 1945.Google Scholar
  15. [15]
    Happel J, Brenner H.Low Reynolds Number Hydrodynamics[M]. 2nd edn. The Hague: Martinus Noordhoff Publishers, 1973.Google Scholar
  16. [16]
    Goldman AJ, Cox RG, Brenner H. The slow motion of two identical arbitrarily oriented spheres through a viscous fluid[J].Chem Eng Sci., 1966,21(12): 1151–1170.CrossRefGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics All rights reserved 1980

Authors and Affiliations

  • Zhuang Hong
    • 1
    • 2
  • Yan Zong-yi
    • 1
  • Wu Wang-yi
    • 1
  1. 1.Department of Mechanics and Engineering SciencePeking UniversityBeijingP R China
  2. 2.4837-267-605, Pharmaceutical SciencesKalamazooUSA

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