Computation of periodic Green’s functions of Stokes flow

  • C. Pozrikidis

Abstract

Methods of computing periodic Green’s functions of Stokes flow representing the flow due to triply-, doubly-, and singly-periodic arrays of three-dimensional or two-dimensional point forces are reviewed, developed, and discussed with emphasis on efficient numerical computation. The standard representation in terms of Fourier series requires a prohibitive computational effort for use with singularity and boundary-integral-equation methods; alternative representations based on variations of Ewald’s summation method involving various types of splitting between physical and Fourier space with partial sums that decay in a Gaussian or exponential manner, allow for efficient numerical computation. The physical changes undergone by the flow in deriving singly- and doubly- periodic Green’s functions from their triply-periodic counterparts are considered.

Keywords

Fourier Series Base Vector Null Point Stoke Flow Periodic Array 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge: The University Press (1992) 259 pp.CrossRefMATHGoogle Scholar
  2. 2.
    H.A. Lorentz, A general theorem concerning the motion of a viscous fluid and a few consequences derived from it. Collected Papers, Vol. IV, 7–14. The Hague: Martinus Nijhoff (1937).Google Scholar
  3. 3.
    C. Pozrikidis, On the transient motion of ordered suspensions of liquid drops. J. Fluid Mech. 246 (1993) 301–320.ADSCrossRefMATHGoogle Scholar
  4. 4.
    X. Li, H. Zhou and C. Pozrikidis, A numerical study of the shearing motion of emulsions and foams. J. Fluid Mech. 286 (1995) 379–404.ADSCrossRefMATHGoogle Scholar
  5. 5.
    X. Li, R. Charles and C. Pozrikidis, Shear flow of suspensions of liquid drops. J. Fluid Mech. (1995) Submitted.Google Scholar
  6. 6.
    A. S. Sangani and C. Yao, Transport processes in random arrays of cylinders. II: Viscous flow. Phys. Fluids 31 (1988) 2435–2444.ADSCrossRefMATHGoogle Scholar
  7. 7.
    A. Sangani and S. Behl, The planar singular solutions of Stokes and Laplace equations and their application to transport processes near porous surfaces. Phys. Fluids A 1 (1989) 21–37.ADSCrossRefMATHGoogle Scholar
  8. 8.
    H. Hasimoto, On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (1959) 317–328.ADSCrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    K. Ishii, Viscous flow past multiple planar arrays of small spheres. J. Phys. Soc. Jpn. 46 (1979) 675–680.ADSCrossRefGoogle Scholar
  10. 10.
    C.W.J. Beenakker, Ewald sums of the Rotne-Prager tensor J. Chem. Phys. 85 (1986) 1581–1582.ADSCrossRefGoogle Scholar
  11. 11.
    Van de Vorst, Integral formulation to simulate the viscous sintering of a two-dimensional lattice of periodic unit cells J. Eng. Math. 30 (1996) 97–118.ADSCrossRefMATHGoogle Scholar
  12. 12.
    J. Hautman and M.L. Klein, An Ewald summation method for planar surfaces and interfaces. Molec. Phys. 75 (1992) 379–395.ADSCrossRefGoogle Scholar
  13. 13.
    I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products. New York: Academic Press (1980) 1204 pp.MATHGoogle Scholar
  14. 14.
    M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. New York: Dover (1972) 1046 pp.MATHGoogle Scholar
  15. 15.
    B.R.A. Nijboer and F.W. De Wette, On the calculation of lattice sums. Physica 23 (1957) 309–321.ADSCrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    A.P. Prudmikov, Y.A. Brychkov, and O.I. Mariche, Integrals and Series, Vol. I. New York: Gordon and Breach (1986).Google Scholar
  17. 17.
    C. Pozrikidis, Creeping flow in two-dimensional channels. J. Fluid Mech. 180 (1987) 495–514.ADSCrossRefGoogle Scholar
  18. 18.
    F.K. Lehner, Plane potential flows past doubly periodic arrays and their connection with effective transport properties. J. Fluid Mech. 162 (1986) 35–51.ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • C. Pozrikidis
    • 1
  1. 1.Department of Applied Mechanics and Engineering SciencesUniversity of California at San DiegoLa JollaUSA

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