A Mean Field Model of Layering Instability in Shearing Suspensions

Concentrated suspensions may shear-thin when the suspended particles form planar sheets that slide over one another with less friction than if the particles are randomly distributed. In a na\"ive model the suspension is described by a mean effective viscosity, and particles that collide with each other redistribute the mean density in the shearing direction. This leads to a diffusion equation for the particle density. If the viscosity in the unthinned state is a steeply increasing function of particle density the effective diffusion coefficient is negative and the diffusion equation, meaningful only on scales larger than the particle separation, is ill-posed. This singularity corresponds to the formation of planar sheets of particles and defines a critical particle density for the onset of shear thinning.