The shape of a maximal pile of yield-stress fluid supported by an obstruction

Viscoplastic fluid flows that occur in nature, such as lava, mud or debris flows, frequently interact with natural or constructed barriers leaving static mounds of material. Using Charpit's method, we present an analytical solution to the shape of the static mound supported by a square barrier on an inclined plane in the lubrication approximation. Expressions for the physical characteristics of the mound, such as its maximal height and total weight, are found from the solution. The calculated shape represents the maximal static mound supported by the barrier, and thus is independent of the original upstream flux source. A single dimensionless parameter describes the shape of the mound, the ratio of the yield stress to the slope-induced stress, and hence it is independent of the constitutive relation beyond the yield stress. A new rheometric technique is suggested based on our solution, where the yield stress is determined using a simple experimental setup. We demonstrate that the method generalises to other barrier shapes, such as circular and rhomboidal barriers. Comparison with preliminary experimental results are discussed.