Skip to main content
SpringerLink
Log in

A boundary integral method for the numerical computation of the forces exerted on a sphere in viscous incompressible flows near a plane wall

  • Original Papers
  • Published:
Zeitschrift für angewandte Mathematik und Physik ZAMP Aims and scope Submit manuscript

Abstract

Slow uniform flows of a viscous, incompressible fluid past a rigid sphere near a plane wall are considered. The drag and lateral forces exerted on the sphere by the fluid are computed. The numerical results are compared with existing theoretical and experimental data for these and related fluid flows. They are based on a boundary element method for various linearized boundary value problems.

Zusammenfassung

Es werden langsame, gleichförmige Strömungen einer zähen, inkompressiblen Flüssigkeit um eine starre Kugel in der Nähe einer ebenen Wand betrachtet. Die von der Flüssigkeit auf die Kugel ausgeübten Widerstands- und Querkräfte werden berechnet. Die numerischen Ergebnisse werden mit bestehenden, theoretischen und experimentellen Daten für diese und verwandte Strömungen verglichen. Sie basieren auf einer Randelementmethode für verschiedene, linearisierte Randwertprobleme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Brenner,Hydrodynamic resistance of particles at small Reynolds numbers. Advances in Chemical Engineering (T. B. Drew et al. ed.), Vol. 6, 287. Academic Press, New York 1966.

    Google Scholar 

  2. H. Buggisch,Langsame Relativbewegung von festen Partikeln in Strömungsfeldern-Anwendungsbeispiele aus der mechanischen Verfahrenstechnik. ZAMM64, T3 (1984).

    Google Scholar 

  3. R. G. Cox and H. Brenner,The lateral migration of solid particles in Poiseuille flow-I Theory. Chem. Engng. Sci.23, 147 (1968).

    Google Scholar 

  4. R. G. Cox and S. K. Hsu,The lateral migration of solid particles in a laminar flow near a plane. Int. J. Multiphase Flow3, 201 (1977).

    Google Scholar 

  5. W. R. Dean and M. E. O'Neill,A slow motion of viscous liquid caused by the rotation of a solid sphere. Mathematika10, 13 (1963).

    Google Scholar 

  6. H. Faxén,Einwirkung der Gefäβwände auf den Widerstand gegen die Bewegung einer kleinen Kugel in einer zähen Flüssigkeit. Thesis, Uppsala 1921.

  7. T. M. Fischer,An integral equation procedure for the exterior three-dimensional slow viscous flow. Integral Equations Operator Theory5, 490 (1982).

    Google Scholar 

  8. T. M. Fischer,Über die langsame Bewegung eines starren Körpers in einer zähen, inkompressiblen Flüssigkeit längs einer ebenen Wand. Thesis, Techn. Hochschule Darmstadt 1983.

  9. T. M. Fischer,Wall effects on the slow steady motion of a particle in a viscous incompressible fluid. Math. Meth. Appl. Sci.8, 23 (1986).

    Google Scholar 

  10. T. M. Fischer, G. C. Hsiao and W. L. Wendland,On the exterior three-dimensional slow viscous flow problem. ZAMM64, T 276 (1984).

    Google Scholar 

  11. T. M. Fischer, G. C. Hsiao and W. L. Wendland,Singular perturbations for the exterior three dimensional slow viscous flow problem. J. Math. Anal. Appl.110, 583 (1985).

    Google Scholar 

  12. A. J. Goldman, R. G. Cox and H. Brenner,Slow viscous motion of a sphere parallel to a plane wall-I Motion through a quiescent fluid. Chem. Engng. Sci.22, 637 (1967).

    Google Scholar 

  13. A. J. Goldman, R. G. Cox and H. Brenner,Slow viscous motion of a sphere parallel to a plane wall-II Couette flow. Chem. Engng. Sci.22, 653 (1967).

    Google Scholar 

  14. H. L. Goldsmith and S. G. Mason,The microrheology of dispersions. Rheology, Theory and Applications (F. R. Eirich ed.), Vol. 4, 85. Academic Press, New York 1967.

    Google Scholar 

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

    Google Scholar 

  16. F.-K. Hebeker,A theorem of Faxén and the boundary integral method for three-dimensional viscous incompressible fluid flows. Math. Meth. Appl. Sci., to appear.

  17. A. Heertsch,Experimente zur Entmischung kugelförmiger Teilchen in Mikrokanälen. Max-Planck-Institut für Strömungsforschung, Göttingen, Bericht 20/1980.

  18. B. P. Ho and L. G. Leal,Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech.65, 365 (1974).

    Google Scholar 

  19. G. C. Hsiao, P. Kopp and W. L. Wendland,Some applications of a Galerkin-collocation method for integral equations of the first kind. Math. Meth. Appl. Sci.6, 280 (1984).

    Google Scholar 

  20. G. C. Hsiao and R. C. MacCamy,Solution of boundary value problems by integral equations of the first kind. SIAM Rev.15, 687 (1973).

    Google Scholar 

  21. G. C. Hsiao and W. L. Wendland,A finite element method for some integral equations of the first kind. J. Math. Anal. Appl.58, 449 (1977).

    Google Scholar 

  22. R. C. Jeffrey and J. R. A. Pearson,Particle motion in laminar vertical tube flow. J. Fluid Mech.22, 721 (1965).

    Google Scholar 

  23. S. Kaplun and P. A. Lagerstrom,Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers. J. Math. Mech.6, 585 (1957).

    Google Scholar 

  24. H. A. Lorentz,Ein allgemeiner Satz, die Bewegung einer reibenden Flüssigkeit betreffend, nebst einigen Anwendungen desselben. Abhandlungen über Theoretische Physik I, 23. Teubner, Leipzig 1907.

    Google Scholar 

  25. F. K. G. Odqvist,Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten. Math. Z.32, 329 (1930).

    Google Scholar 

  26. D. R. Oliver,Influence of particle rotation on radial migration in the Poiseuille flow of suspensions. Nature194, 1269 (1962).

    Google Scholar 

  27. M. E. O'Neill,A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika11, 67 (1964).

    Google Scholar 

  28. C. W. Oseen,Neuere Methoden und Ergebnisse in der Hydrodynamik. Akad. Verlagsgesellschaft, Leipzig 1927.

    Google Scholar 

  29. I. Proudman and J. R. A. Pearson,Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech.2, 237 (1957).

    Google Scholar 

  30. S. I. Rubinow and J. B. Keller,The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech.11, 447 (1961).

    Google Scholar 

  31. P. G. Saffman,The lift on a small sphere in a slow shear flow. J. Fluid Mech.22, 385 (1965).

    Google Scholar 

  32. G. Segré and A. Silberberg,Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 1 (Determination of local concentration by statistical analysis of particle passages through crossed light beams). J. Fluid Mech.14, 115 (1962).

    Google Scholar 

  33. G. Segré and A. Silberberg,Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2 (Experimental results and interpretation). J. Fluid Mech.14, 136 (1962).

    Google Scholar 

  34. A. H. Stroud,Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs 1971.

    Google Scholar 

  35. M. Tachibana,On the behaviour of a sphere in the laminar tube flows. Rheol. Acta12, 58 (1973).

    Google Scholar 

  36. P. Vasseur and R. G. Cox,The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech.78, 385 (1976).

    Google Scholar 

  37. P. Vasseur and R. G. Cox,The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech.80, 561 (1977).

    Google Scholar 

  38. W. L. Wendland,On applications and the convergence of boundary integral methods. Treatment of Integral Equations by Numerical Methods (C. Baker and G. Miller ed.), 463. Academic Press, London 1982.

    Google Scholar 

  39. G. K. Youngren and A. Acrivos,Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech.69, 377 (1975).

    Google Scholar 

  40. J. Zhu,A boundary integral equation method for the stationary Stokes problem in 3D. Boundary Elements, 5th Int. Con. Hiroshima (C. A. Brebbia et al. ed.), 283. Springer-Verlag, Berlin 1983.

    Google Scholar 

  41. M. B. Bush and R. I. Tanner,Numerical solution of viscous flows using integral equation methods. Int. J. Num. Meth. Fluids3, 71 (1983).

    Google Scholar 

  42. P. J. Davis and P. Rabinowitz,Numerical Integration. Blaisdell, Waltham 1967.

    Google Scholar 

  43. D. Leighton and A. Acrivos,The lift on a small sphere touching a plane in the presence of a simple shear flow. Z. angew. Math. Phys.36, 174 (1985).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Strömungsmechanik, DFVLR, D-3400, Göttingen

    T. M. Fischer

  2. Fachbereich Mathematik, Technische Hochschule Darmstadt, D-6100, Darmstadt, Germany

    R. Rosenberger

Authors
  1. T. M. Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Rosenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fischer, T.M., Rosenberger, R. A boundary integral method for the numerical computation of the forces exerted on a sphere in viscous incompressible flows near a plane wall. Z. angew. Math. Phys. 38, 339–365 (1987). https://doi.org/10.1007/BF00944955

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00944955

Keywords

Search

Navigation