Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-29T00:49:40.626Z Has data issue: false hasContentIssue false
  • English
  • Français

A strong-interaction theory for the motion of arbitrary three-dimensional clusters of spherical particles at low Reynolds number

Published online by Cambridge University Press:  21 April 2006

Qaizar Hassonjee ,
Peter Ganatos  and
Robert Pfeffer
Qaizar Hassonjee
Affiliation:
Spraylat Corporation, 716 S. Columbus Ave., Mt. Vernon, NY 10550, USA
Peter Ganatos
Affiliation:
The City College of The City University of New York, New York, NY 10031, USA
Robert Pfeffer
Affiliation:
The City College of The City University of New York, New York, NY 10031, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper contains an ‘exact’ solution for the hydrodynamic interaction of a three-dimensional finite cluster at arbitrarily sized spherical particles at low Reynolds number. The theory developed is the most general solution to the problem of an assemblage of spheres in a three-dimensional unbounded media. The boundary-collocation truncated-series solution technique of Ganatos, Pfeffer & Weinbaum (1978) for treating planar symmetric Stokes flow problems has been extensively modified to treat the non-symmetric multibody problem. The orthogonality properties of the eigenfunctions in the azimuthal direction are used to satisfy the no-slip boundary conditions exactly on entire rings on the surface of each particle rather than just at discrete points.

Detailed comparisons with the exact bipolar solutions for two spheres show the present theory to be accurate to five significant figures in predicting the translational and angular velocity components of the particles at all orientations for interparticle gap widths as close as 0.1 particle diameter. Convergence of the results to the exact solution is rapid and systematic even for unequal-sized spheres (a1/a2 = 2). Solutions are presented for several interesting and intriguing configurations involving three or more spherical particles settling freely under gravity in an unbounded fluid or in the presence of other rigidly held particles. Advantage of symmetry about the origin is taken for symmetric configurations to reduce the collocation matrix size by a factor of 64. Solutions for the force and torque on three-dimensional clusters of up to 64 particles have been obtained, demonstrating the multiparticle interaction effects that arise which would not be present if only pair interactions of the particles were considered. The method has the advantage of yielding a rather simple expression for the fluid velocity field which is of significance in the treatment of convective heat and mass transport problems in multiparticle systems.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 197 , December 1988 , pp. 1 - 37
Copyright
© 1988 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

Davis, A. M. J., O'Neill, M. E., Dorrepaal, J. M. & Ranger, K. B. 1976 Separation from the surface of two equal spheres in Stokes flow. J. Fluid Mech. 77, 625–644.Google Scholar
Davis, M. H.1969 The slow translation and rotation of two unequal spheres in a viscous fluid. Chem. Engng Sci. 24, 1769–1776.Google Scholar
Dagan, Z., Weinbaum, S. & Pfeffer, R.1983 Theory and experiment on the three-dimensional motion of a freely suspended spherical particle at the entrance to a pore at low Reynolds number. Chem. Engng Sci. 38, 583–596.Google Scholar
Dean, W. R. & O'Neill, M. E.1963 A slow motion of viscous liquid caused by the rotation of a solid sphere. Mathematica 10, 13–26.Google Scholar
Durlofsky, L., Brady, J. F. & Bossis, G.1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21–49.Google Scholar
Ganatos, P., Pfeffer, R. & Weinbaum, S.1978 A numerical-solution technique for three-dimensional Stokes flows, with application to the motion of strongly interacting spheres in a plane. J. Fluid Mech. 84, 79–111.Google Scholar
Ganatos, P., Weinbaum, S. & Pfeffer, R.1980a A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion. J. Fluid Mech. 99, 739–753.Google Scholar
Ganatos, P., Pfeffer, R. & Weinbaum, S.1980b A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 2. Parallel motion. J. Fluid Mech. 99, 755–783.Google Scholar
Ganatos, P., Weinbaum, S. & Pfeffer, R.1982 Gravitational and zero-drag motion of a sphere of arbitrary size in an inclined channel at low Reynolds number. J. Fluid Mech. 124, 27–43.Google Scholar
Gluckman, M. J., Pfeffer, R. & Weinbaum, S.1971 A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids. J. Fluid Mech. 50, 705–740.Google Scholar
Gluckman, M. J., Weinbaum, S. & Pfeffer, R.1972 Axisymmetric slow viscous flow past an arbitrary convex body of revolution. J. Fluid Mech. 55, 677–709.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H.1966 The slow motion of two identical arbitrarily oriented spheres through a viscous fluid. Chem. Engng Sci. 21, 1151–1170.Google Scholar
Happel, J. & Brenner, H.1973 Low Reynolds Number Hydrodynamics, 2nd edn. Noordhoff.
Hasimoto, M.1959 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, 317–328.Google Scholar
Hocking, L. M.1964 The behaviour of clusters of spheres falling in a viscous fluid. Part 2. Slow motion theory. J. Fluid Mech. 20, 129–139.Google Scholar
Jayaweera, K. O. L. F., Mason, B. J. & Slack, G. W.1964 The behaviour of clusters of spheres falling in a viscous fluid. Part 1. Experiment. J. Fluid Mech. 20, 121–128.Google Scholar
Kim, S. & Mifflin, R. T.1985 The resistance and mobility functions of two equal spheres in low-Reynolds-number flow. Phys. Fluids 28, 2033–2045.Google Scholar
Lamb, H.1945 Hydrodynamics, 6th edn. Dover.
Liao, W. H. & Krueger, D. A.1980 Multipole expansion calculation of slow viscous flow about spheroids of different sizes. J. Fluid Mech. 96, 223–241.Google Scholar
Sangani, A. S. & Acrivos, A.1982 Slow flow through periodic arrays of spheres. Intl J. Multiphase Flow 8, 343–360.Google Scholar
Stimson, M. & Jeffery, G. B.1926 The motion of two spheres in a viscous fluid. Proc. R. Soc. Lond. A 111, 110–116.Google Scholar
Wacholder, E. & Sather, N. F.1974 The hydrodynamic interaction of two unequal spheres moving under gravity through quiescent viscous fluid. J. Fluid Mech. 65, 417–437.Google Scholar
Wakiya, S.1967 Slow motions of a viscous fluid around two spheres. J. Phys. Soc. Japan 22, 1101–1109.Google Scholar
Yan, Z. Y., Weinbaum, S., Ganatos, P. & Pfeffer, R. 1987 The three-dimensional hydrodynamic interaction of a finite sphere with a circular orifice at low Reynolds number. J. Fluid Mech. 174, 39–68.Google Scholar
Zick, A. A. & Homsy, G. M.1982 Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115, 13–26.Google Scholar
Supplementary material: PDF

Hassonjee et al. supplementary material

Supplementary Appendices

Download Hassonjee et al. supplementary material(PDF)
PDF 1.5 MB