Roll waves on a shallow layer of a dilatant fluid

This paper is about the free surface instabilities of granular flows, usually called roll waves. A shallow layer of shear-thickening fluid (τ=a(∂u/∂y)nτ=a(∂u/∂y)n with n=2n=2) is considered to study finite-amplitude permanent roll waves down a slope, simplified by Karman’s momentum integral approach. The existence of conditions of a periodic discontinuous solution is derived, as smooth profiles with depth increasing monotonically between periodic shocks. Energy dissipation in the body of the stream and in the discontinuity is analysed and discussed. Two conditions are derived. The first is related to the physically acceptable shape of the smooth profiles, and the second is related to positive energy loss across the shock. These conditions can be converted into a limiting discharge, viewed in the fixed frame, and in a limiting flow thickness (or limiting Froude number), for the permanent periodic roll wave to exist without further conditions. A minimum-length roll wave (MLRW) is defined as the periodic permanent roll waves with zero energy dissipation in the shock. The MLRW also requires a limiting value of the Froude number to exist.