A Herschel–Bulkley model for mud flow down a slope

Abstract

The spreading and sediment deposit of a two-dimensional, unsteady, laminar mud flow from a constant-volume source on a relatively steep slope is studied theoretically and experimentally. The mud under consideration has the rheological properties of a Herschel–Bulkley fluid. The flow is of low-Reynolds-number type and has a well-formed wave front moving a substantial distance downslope. Due to the nonlinear rheological characteristics, a set of nonlinear partial differential equations is needed for this transient problem. Depth-integrated continuity and momentum equations are derived by applying von Kármán's momentum integral method. A matched-asymptotic perturbation method is implemented analytically to get asymptotic solutions for both the outer region away from, and the inner region near, the wave front. The outer solution gives accurate results for spreading characteristics, while the inner solution, which is shown to agree well with experimental results of Liu & Mei (1989) for a Bingham fluid, predicts fairly well the free-surface profile near the wave front. A composite solution uniformly valid over the whole spreading length is then achieved through a matching of the inner and outer solutions in an overlapping region. The range of accuracy of the solution and the size of the inner and overlapping regions are quantified by physical scaling analyses. Rheological and dynamic measurements are obtained through laboratory experiments. Theoretical predictions are compared with experimental results, showing reasonable agreement. The impact of shear thinning on the runout characteristics, free-surface profiles and final deposit of the mud flow is examined. A mud flow with shear thinning spreads beyond the runout distance estimated by a Bingham model, and has a long and thin deposit.