Viscoplastic slumps supported by a barrier

The shape of a free-surface slump of viscoplastic material supported by an oblique barrier on an inclined plane is investigated theoretically and experimentally. The barrier is sufficiently tall that it is not surmounted by the viscoplastic fluid, and a focus of this study is the largest volume of rigid viscoplastic fluid that can be supported upstream of it. A lubrication model is integrated numerically to determine the transient flow as the maximal rigid shape is approached. Away from the region supported by the barrier, the viscoplastic layer attains a uniform thickness in which the gravitational stresses are in balance with the yield stress of the material. However, closer to the barrier, the layer thickens and the barrier bears the additional gravitational loading. An exact solution for the rigid shape of the viscoplastic material is constructed from the steady force balance and computed by integrating Charpit's equations along characteristics that emanate from the barrier wall. The characteristics represent the late-time streamlines of the flow as it approaches the rigid shape. The exact solution depends on a single dimensionless group, which incorporates the slope inclination, the barrier width and the fluid's yield stress. It is shown that the shape is insensitive to the transient flow from which it originates. The force exerted by the slump is calculated for different barrier shapes. The results of new laboratory experiments are reported; these show that although convergence to the final rigid state is slow, there is good agreement with the experimental measurements at long times.