Roll waves in mud flow

Abstract

Permanent roll waves on a shallow layer of fluid mud which is modeled as a power-law fluid is studied. Based on long-wave approximation, momentum integral method is applied to derive the averaged continuity and the momentum equations. Linearized instability analysis of a uniform flow is insufficient to suggest a preferred wavelength for the nonlinear roll waves, which are obtained as periodic shocks connected by smooth profiles. Among all wavelengths only those longer than certain threshold correspond to positive energy loss across the shock, suggesting that a roll wave developed spontaneously form infinitesimal disturbances should have the shortest wavelength corresponding to zero dissipation across the shock, though finite dissipation elsewhere. For longer roll waves, no roll wave exist if the uniform flow is linearly stable, when the fluid is slightly non-Newtonian. However when the fluid is highly non-Newtonian, very long roll waves may still exist even if the corresponding uniform flow is stable to infinitesimal disturbances.