The segregation instability of a sheared suspension film

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 {\it Phys. Fluids} {\bf 13} (12), p. 3517 (2001), suggesting an explanation of the experiments of Tirumkudulu {\it et al.}, Phys. Fluids {\bf 11}, 507-509 (1999); {\it ibid.} {\bf 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.