Flow of a yield-stress fluid past a topographical feature

Free-surface flows of yield-stress fluids down inclined planes are modelled under the assumptions that they are shallow and sustained by a uniform oncoming stream to determine the steady state that emerges as the flow passes topographic features. In general, the flow may surmount the topography and be deflected around it depending on the thickness of the oncoming flow, the lateral extent and elevation of the mound, the inclination of the plane, and the magnitude of the yield stress relative to the gravitational stress of the flowing layer. Flows deepen upstream of mounds, with amplitude increasing with increasing yield stress. In the absence of a yield stress, flows around isolated mounds exhibit a maximum thickness at a location that is displaced laterally and downstream of the mound due to flow diversion. However, the location of the maximum thickness differs for yield-stress fluids: with increasing yield stress, the flow thickens immediately upstream of the mound and the deflected flux is diminished, leading to a sharp transition in the location of the maximum. Larger amplitude mounds may not be surmounted at all, leading to 'dry zones' downstream into which no fluid flows. It is shown that the steady shape of the dry zone is dependent on the initial condition, because the transient evolution towards it depends upon the plug at its margin, which is not unique. The results are computed by numerical integration of the governing equations and through their asymptotic analysis in various flow regimes to draw out the interplay of the dynamical processes.