Shallow viscoplastic flow on an inclined plane

Evolving viscoplastic flows upon slopes are an important idealization of many flows in a variety of geophysical situations where yield stress is thought to play a role. For such models, asymptotic expansions suitable for slowly moving shallow fluid layers (lubrication theory) reduce the governing equations to a simpler problem in terms of the fluid thickness. We consider the version of the theory for fluids described by the Herschel-Bulkley constitutive law, and provide a variety of solutions to the reduced equation, both numerical and analytical. For extruded inclined domes, we derive the characteristic temporal behaviour of measures of the dome's dimensions.