On the dynamics of particles in continuum suspensions. II. Unsteady flow

Since the pioneering theoretical investigation by Einstein on how suspended particles alter the effective viscosity of a fluid, researchers have worked to develop a general continuum framework for the dynamics of suspensions. Recently, a promising approach has been developed that links the continuum dynamics of the fluid to moments of the force distributions from the suspended particles. These models have primarily considered the dynamics of the particles in a low Reynolds number limit. Here, we show how to modify this approach to include the unsteady motion of the suspended particles. Our results include a more complete description of the inertial terms than was previously derived in other works on the continuum dynamics of suspensions at finite Reynolds number. In addition, we show that the inertial terms can also naturally account for the Brownian motion of the suspended particles. Indeed, our results are not only consistent with the prior classical work on diffusion in suspensions but also provide a direct way to compute the diffusion coefficient that does not rely on independently computing the volume fraction dependence of the hindered settling function and chemical potential. The leading-order dependence of the diffusion coefficient on volume fraction found using this method also agrees with experimental data. Consequently, our results not only provide a way to include some inertial behavior into the continuum dynamics of suspensions but also provide a general means for accounting for the thermal motions of the particles in these suspensions.