Roll waves on a layer of a muddy fluid flowing down a gentle slope—A Bingham model

Abstract

To study rapid and laminar flow of a mud layer down a gentle slope, a shallow layer of Bingham fluid is considered. Attention is focused on long waves of finite amplitude. Convective inertia is kept in the approximate equations which are simplified by Kármán’s momentum integral approach with an assumed velocity profile. A linearized instability analysis is first carried out for periodic disturbances. The nonlinear development of unstable disturbances is computed numerically to study the formation of periodic shocks, or roll waves. Nonlinear motion caused by external influx to an otherwise steadily flowing mud layer is also investigated.