The segregation instability of a sheared suspension film

Abstract

Starting from the equations of Stokes flow and the mass conservation of particles as determined by shear-induced diffusion, we derive the coupled equations for the dynamics of particle concentration and film thickness for the free-surface flow of a fluid film pulled up by a tilted wall rising from a pool of neutrally buoyant, non-Brownian suspension. We find an instability of the film with respect to axial undulations of film thickness and modulations of particle concentration, and the instability growth-rate increases as a certain combination of the two dimensionless shear induced diffusivities (which determine the particle flux driven by concentration and shear rate gradients) falls below a critical value. This reinforces the conclusions of Phys. Fluids 13 (12), p. 3517 (2001), suggesting an explanation of the experiments of Tirumkudulu et al., Phys. Fluids 11, 507-509 (1999); ibid. 12, 1615 (2000). In addition, we predict a "pile-up" instability in which perturbations that vary in the direction of the wall velocity are amplified; this instability is not driven by shear-induced migration, but is a result of the dependence of the suspension viscosity on the particle concentration.