J. Fluid Mech. (2001), vol. 434, pp. 23-37. Printed in the United Kingdom

© 2001 Cambridge University Press

 

Modelling Rayleigh-Taylor instability

of a sedimenting suspension of

several thousand circular particles

in a direct numerical simulation

 

By T. W. PAN¹, D. D. JOSEPH² AND R. GLOWINSKI¹

¹Department of Mathematics, University of Houston, Houston, TX 77204, USA

²Aerospace Engineering and Mechanics, University of Minneapolis, 107 Ackerman Hall,

110 Union Street, Minneapolis, MN 55455, USA

(Received 7 September 1999 and in revised form 9 October 2000)

 

In this paper we study the sedimentation of several thousand circular particles in two dimensions using the method of distributed Lagrange multipliers for solid-liquid flow. The simulation gives rise to fingering which resembles Rayleigh-Taylor instabilities. The waves have a well-defined wavelength and growth rate which can be modeled as a conventional Rayleigh-Taylor instability of heavy fluid above light. The heavy fluid is modelled as a composite solid-liquid fluid with an effective composite density and viscosity. Surface tension cannot enter this problem and the characteristic shortwave instability is regularized by the viscosity of the solid-liquid dispersion. The dynamics of the Rayleigh{Taylor instability are studied using viscous potential flow, generalizing work of Joseph, Belanger & Beavers (1999) to a rectangular domain bounded by solid walls; an exact solution is obtained.