Wall-bounded shear flow and channel flow of suspensions of liquid drops

Abstract The wall-bounded shear flow and the plane Poiseuille channel flow of monodisperse suspensions of liquid drops are considered by theory and numerical simulation. First, the motion of an individual drop in infinite or semi-infinite shear flow is discussed in the limit of small volume fractions. An expression for the flux of the drops normal to the streamlines of the unperturbed flow is derived in terms of (a) the migration velocity of the drop away from the wall, and (b) the net displacement of a drop’s center after interception with another drop. In the case of two-dimensional infinite shear flow, in the limit of infinite dilution, and in the context of Stokes flow, the self- and gradient-diffusivity are found to diverge, and this underlines the importance of fluid inertia and the necessity to perform renormalization by requiring global constraints. Numerical simulations of pairwise drop interceptions in semi-infinite shear flow above a plane wall reveal that the capillary number, expressing the drop deformability, and the distance of the drop pair from the wall, play an important role in determining not only the magnitude, but also the direction of the net displacement of the drop center after recession. Dynamic simulations of the expansion of a periodic bed of drops distributed randomly within a layer next to a wall illustrate explicitly the formation of a particle-free zone near the wall and the thickening of the bed due to hydrodynamic interceptions at a rate that is a strong function of the capillary number. Results of dynamic simulations of the pressure-driven flow of a two-dimensional suspension in a channel confined between two parallel walls are presented illustrating the effect of the capillary number and of the ratio of the viscosity of the drop and suspending fluid on (a) the time required for the suspension to reach statistical equilibrium, (b) the distribution of the drop number density across the channel width, (c) the mean velocity profile, and (d) the effective viscosity of the suspension. The general features of the flow are found to be in good agreement with published laboratory observations.