On the dynamics of particles in continuum suspensions. I. Low Reynolds number

Recent work by a few different groups has aimed to develop a general continuum framework for describing the dynamics of suspensions. In one method, moments of the force distributions from the suspended particles are ensemble-averaged to derive the effective stress tensor, while a separate method computed the effective stress tensor using the spatial average of a series of spherical Green's functions. Here, we begin by showing the equivalence of these methods. We then focus on the Green's function approach and consider in more detail and generality what this method predicts for the equations of motion for the suspended particles. Specifically, by treating the volume fraction and particle force distributions as slowly varying fields in the vicinity of a suspended particle, we show that this formalism naturally leads to the presence of the divergence of the particle stress tensor in the equation of motion for the particles, consistent with what other authors have found, albeit with a different resistive drag than is often assumed. Furthermore, by considering the pair distribution in the presence of shear, we show that the particle stress contribution to the particle dynamic equation leads to shear-induced hydrodynamic diffusion and particle transport driven by normal stress differences. Consequently, the results presented here expand our Green's function formalism for the dynamics of continuum suspensions to account for the direct effect of the particle-induced stress on the dynamics of the suspended particles, which leads to a more complete description of shear-induced transport in suspensions.