Modeling Rayleigh-Taylor Instability of a Sedimenting Suspension of Several Thousand Circular Particles in Direct Numerical Simulation

In this paper we study the sedimentation of several thousand circular particles in 2D using the method of distributed Lagrange multipliers for solid-liquid ow. The simulation gives rise to ngering which resembles Rayleigh-Taylor instabilities. The waves have a well de ned wavelength and growth rate which can be modeled as a conventional Rayleigh-Taylor instability of heavy uid above light. The heavy uid is modeled as a composite solid{liquid uid with an e ective composite density and viscosity. Surface tension cannot enter this problem and the characteristic short wave instability is regularized by the viscosity of the solid liquid dispersion. The dynamics of the Rayleigh-Taylor instability are studied using viscous potential ow generalizing work of Joseph, Belanger, and Beavers (1999) to a rectangular domain bounded by solid walls; an exact solution is obtained. The data in this paper is generated by the direct numerical simulation of solid{liquid ow using a distributed Lagrange multiplier/ ctitious domain method (see Glowinski, Pan, Hesla & Joseph 1999, Glowinski, Pan, Hesla, Joseph & Periaux 2000). The calculation is carried on xed triangular mesh on which uid equations are satis ed everywhere. Rigid motions of the portions of the uid occupied by solids are accomplished by a strategic choice of a Lagrange multiplier eld there. The method has a certain elegance in that the rigid motion constraint on the uid is associated with a multiplier eld in a manner analogous to the way in which the pressure in an incompressible ow is a multiplier eld associated with