Average stress in a Stokes suspension of disks

The ensemble-average velocity and pressure in an unbounded quasi-random suspension of disks (or aligned cylinders) are calculated in terms of average multipoles allowing for the possibility of spatial non-uniformities in the system. An expression for the stress due to the suspended particles is deduced from these results. It is found that spatial non-uniformity can induce an antisymmetric component in this stress even when no external couple acts on the particles. This component has the same order of magnitude as the term responsible for the difference between the effective viscosity of the suspension and that of the pure fluid. General considerations and a simple cell model suggest that the antisymmetric component will induce a flow in the presence of gradients of the particle volume fraction or of the relative interphase velocity, for example in a sedimenting suspension with a horizontally non-uniform particle distribution. While the derivation assumes Stokes flow conditions for the local flow around the particles, the Reynolds number of the mean macroscopic flow is unrestricted. In addition to illustrating the general nature of the particle stress, this work is a necessary prerequisite for the development of a closed suspension model on the basis of direct numerical simulations.