The dam-break problem for Herschel-Bulkley viscoplastic fluids down steep flumes
In this paper we investigate the dam-break problem for viscoplastic (Herschel- Bulkley) fluids down a sloping flume: a fixed volume of fluid initially contained in a reservoir is released onto a slope and flows driven by gravitational forces until these forces are unable to overcome the fluid’s yield stress. Like in many earlier investigations, we use lubrication theory and matched asymptotic expansions to de- rive the evolution equation of the flow depth, but with a different scaling for the flow variables, which makes it possible to study the flow behavior on steep slopes. The evolution equations takes on the form a nonlinear diffusion-convection equation. To leading order, this equation simplifies into a convection equation and reflects the balance between gravitational forces and viscous forces. After presenting analytical and numerical results, we compare theory with experimental data obtained with a long flume. We explore a fairly wide range of flume inclinations from 6° to 24° , while the initial Bingham number lies in the 0.07–0.26 range. Good agreement is found at the highest slopes, where both the front position and flow-depth profiles are properly described by theory. In contrast, at the lowest slopes, theoretical pre- dictions substantially deviate from experimental data. Discrepancies may arise from the formation of unsheared zones or lateral levees that cause slight flow acceleration.