A fast numerical method for solving the three-dimensional stokes' equations in the presence of suspended particles

A new fast numerical method for solving the three-dimensional Stokes' equations in the presence of suspended particles is presented. The fluid dynamics equations are solved on a lattice. A particle is represented by a set of points each of which moves at the local fluid velocity and is not constrained to lie on the lattice. These points are coupled by forces which resist deformation of the particle. These forces contribute to the force density in the Stokes' equations. As a result, a single set of fluid dynamics equations holds at all points of the domain and there are no internal boundaries. Particle size, shape, and deformability may be prescribed. Computational work increases only linearly with the number of particles, so large numbers (500-1000) of particles may be studied efficiently. The numerical method involves implicit calculation of the particle forces by minimizing an energy function and solution of a finite-difference approximation to the Stokes' equations using the Fourier-Toeplitz method. The numerical method has been implemented to run on all CRAY computers: the implementation exploits the CRAY's vectorized arithmetic, and on machines with insufficient central memory, it performs efficient disk 1/O while storing most of the data on disk. Applications of the method to sedimentation of one-, two-, and many-particle systems are described. Trajectories and settling speeds for two-particle sedimentation, and settling speed for multiparticle sedimentation from initial distributions on a cubic lattice or at random give good quantitative agreement with existing theories.