**Granular flow: Particle Size Segregation and New Constitutive Laws**

The segregation of particles of different sizes can be achieved by vibration or by shear flow. I describe a nonlinear conservation law that captures the main features of segregation under shear flow, such as occurs in an avalanche. Predictions of the theory are tested against experiments in a Couette cell. The partial differential equation has interesting mathematical properties that are explored through analysis, explicit solutions and numerical simulation. In the second half of the talk I introduce constitutive laws for two-dimensional time dependent flow that lead to linear well-posedness. The new theory, known as the CIDR rheology, specifies a yield condition and flow rule related to the mu(I) rheology, but including compressibility that is crucial to establishing well-posedness.