Plane Poiseuille flow of a sedimenting suspension of Brownian hard-sphere particles: Hydrodynamic stability and direct numerical simulations

Abstract

The pressure driven flow of a suspension of sedimenting Brownian hard-sphere particles in a plane channel is considered. The balance of gravity and Brownian forces leads to a stationary state where a concentration profile of the particles is established in the channel, with a transition from a viscous sediment to clear fluid. The hydrodynamic stability of the flow and the nonlinear evolution of unstable disturbances are studied numerically by spectral element/Fourier expansion techniques. Two modes of instability with different characteristics are identified. The first is of the Tollmien-Schlichting type, similar to the one present in single-fluid parallel shear flows. This instability appears at much lower Reynolds numbers than for a single fluid when the transition in viscosity is gradual and the sediment is receptive to the fluid motion in the bulk, that is, for relatively small colloidal particles well into the submicron range. An interesting feature, observed through three-dimensional numerical simulations, is the formation of longitudinal striation patterns in the sediment, reminiscent of drag reducing surfaces with organized roughness known as riblets. The second type of instability is similar to the interfacial instability in stratified shear flows with a jump in viscosity. This type of instability appears also at low Reynolds numbers, when the transition from sediment to clear fluid is sharper, that is, for relatively larger particles. This instability results in slow waves traveling with velocities characteristic of those in the sediment, and gives rise to a significant resuspension and formation of low concentration regions in the vicinity of the sediment. An interesting feature is the formation of patterns in the sediment reminiscent of sand ripples.