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

© 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. PAN^{1}, D. D. JOSEPH^{2} AND R. GLOWINSKI^{1}

^{1}Department of Mathematics,

^{2}Aerospace Engineering and Mechanics,

(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.