Localized force representations for particles sedimenting in Stokes flow

A finite-valued force multipole expansion is developed to describe the dynamics of spherical particles sedimenting in Stokes flow. The lowest level of the method is a simple force coupling procedure to represent the dynamic coupling between the particle phase and the fluid phase. The length scale associated with the finite force envelope provides an additional parameter that when matched to the particle size gives consistent results for the particle velocity. Sedimentation velocities for particles in a cubic lattice are predicted with good accuracy up to volume fractions of 20% by this approximate method. The calculated kinetic energy dissipation by viscosity is correctly balanced with the release of potential energy. As a full expansion procedure it is possible to calculate exactly the flow surrounding the particles, using standard numerical procedures such as fast Fourier transforms in periodic domains. Examples for the settling of particle pairs and random suspension are also given.