Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-20T00:46:24.853Z Has data issue: false hasContentIssue false
  • English
  • Français

Stokes flow past a particle of arbitrary shape: a numerical method of solution

Published online by Cambridge University Press:  29 March 2006

G. K. Youngren  and
A. Acrivos
G. K. Youngren
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, California 94305
A. Acrivos
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, California 94305
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear integral equations of the first kind for a distribution of Stokeslets over the particle surface. The unknown density of Stokeslets is identical with the surface-stress force and can be obtained numerically by reducing the integral equations to a system of linear algebraic equations. This appears to be an efficient way of determining solutions for several external flows past a particle, since it requires that the matrix of the algebraic system be inverted only once for a given particle.

The technique was tested successfully against the analytic solutions for spheroids in uniform and simple shear flows, and was then applied to two problems involving the motion of finite circular cylinders: (i) a cylinder translating parallel to its axis, for which the local stress force distribution and the drag were determined; and (ii) the equivalent axis ratio of a freely suspended cylinder, which was calculated by determining the couple on a stationary cylinder placed symmetrically in two different simple shear flows. The numerical results were found to be consistent with the asymptotic analysis of Cox (1970, 1971) and in excellent agreement with the experiments of Anczurowski & Mason (1968), but not with those of Harris & Pittman (1975).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 69 , Issue 2 , 27 May 1975 , pp. 377 - 403
Copyright
© 1975 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anczurowski, E. & Mason, S. G. 1968 Particle motions in sheared suspensions. IV. Rotation of rigid spheroids and cylinders Trans. Soc. Rheol. 12, 209.Google Scholar
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow J. Fluid Mech. 44, 419.Google Scholar
Bowen, B. D. & Masliyah, J. H. 1973 Drag force on isolated axisymmetric particles in Stokes flow Can. J. Chem. Engng, 51, 8.Google Scholar
Brenner, H. 1964a The Stokes resistance of a slightly deformed sphere. Chem. Engng Sci. 19, 519.Google Scholar
Brenner, H. 1964b The Stokes resistance of an arbitrary particle. III. Shear fields. Chem. Engng Sci. 19, 631.Google Scholar
Brenner, H. 1966 Hydrodynamic resistance of particles Adv. in Chem. Eng. 6, 287.Google Scholar
Chang, Y. P., Kang, C. S. & Chen, D. J. 1973 The use of fundamental Green's functions for the solution of problems of heat conduction in anisotropic media Int. J. Heat Mass Transfer, 16, 1905.Google Scholar
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory J. Fluid Mech. 44, 791.Google Scholar
Cox, R. G. 1971 The motion of long slender bodies in a viscous fluid. Part 2. Shear flow J. Fluid Mech. 45, 625.Google Scholar
Cruse, T. A. 1969 Numerical solutions in three-dimensional elastostatics Int. J. Solids Struct. 5, 1259.Google Scholar
Finlayson, B. A. 1972 The Method of Weighted Residuals and Variational Principles.
Gluckman, M. J., Weinbaum, S. & Pfeffer, R. 1972 Axisymmetric slow viscous flow past an arbitrary convex body of revolution J. Fluid Mech. 55, 677.Google Scholar
Goldsmith, H. L. & Mason, S. G. 1967 The microrheology of dispersions. In Rheology (ed. Eirich), pp. 85–250. Academic.
GÜNTER, N. M. 1967 Potential Theory and Its Application to Basic Problems of Mathematical Physics. Ungar.
Harris, J. B. & Pittman, J. F. T. 1975 Equivalent ellipsoidal axis ratios of slender rod-like particles J. Colloid Interface Sci. 50, 280.Google Scholar
Hunt, B. H. 1968 Numerical solution of an integral equation for flow from a circular orifice J. Fluid Mech. 31, 361.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc. A 102, 161.Google Scholar
Kantorovich, L. V. & Krylov, V. I. 1958 Approximate Methods of Higher Analysis, Interscience.
Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Nir, A. & Acrivos, A. 1973 On the creeping motion of two arbitrary-sized touching spheres in a linear shear field J. Fluid Mech. 59, 209.Google Scholar
Oberbeck, A. J. 1876 Rüber stationräre Flrüssigkeitsbewgungen mit Berücksichtigung der inner Reibung J. reine angew. Math. 81, 62.Google Scholar
O'BRIEN, V. 1968 Form factors for deformed spheroids in Stokes flow A.I.Ch.E. J. 14, 870.Google Scholar
Odqvist, F. K. G. 1930 Rüber die Randwertaufgaben der Hydrodynamik Zräher Flrüssigkeiten Math. Z. 32, 329.Google Scholar
Payne, L. E. & Pell, W. H. 1960 The Stokes flow problem for a class of axially symmetric bodies J. Fluid Mech. 7, 529.Google Scholar
Rosen, A. L. 1972 A computational algorithm for the Stokes problem. I. Methodology J. Inst. Math. Appl. 9, 265.Google Scholar
Sampson, R. A. 1891 On Stokes's current function. Phil. Trans. A 182, 449.Google Scholar
Smith, A. M. O. & Hess, J. L. 1966 Calculation of potential flow about arbitrary bodies Prog. Aero. Sci. 8, 1.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on pendulums Trans. Camb. Phil. Soc. 9, 8.Google Scholar
Taylor, T. D. & Acrivos, A. 1964 The Stokes flow past an arbitrary particle – the slightly deformed sphere Chem. Engng Sci. 19, 445.Google Scholar