The dam-break problem for Herschel–Bulkley viscoplastic fluids down steep flumes

Abstract

a b s t r a c tIn 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 derive 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 equation takes on the form of 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 flowdepth profiles are properly described by theory. In contrast, at the lowest slopes, theoretical predictions substantially deviate from experimental data. Discrepancies may arise from the formation of unsheared zones or lateral levees that cause slight flow acceleration.

Experimental investigation of the spreading of viscoplastic fluids on inclined planes

“…The channels were almost symmetric about the centerline and aligned with the flow direction. These structures were not observed when the mass was confined in a flume [41]. This may be due to surface instabilities, but except for this scenario, we failed to find any reasonable explanation why regular patterns formed on the free surface.…”

“…Section 3 is devoted to experimental results. There we present the data obtained with the inclined plane; a companion paper [41] includes experimental data related to channelized flows. A few conclusions are drawn in the last section.…”

A Well-balanced Finite Volume-Augmented Lagrangian Method for an Integrated Herschel-Bulkley Model

“…We note that it is clear, from the literature on the subject (see [1] and references therein), that it is very dificult to postulate a precise constitutive relation for the stress tensor in terms of a deformation measure that correctly describes avalanche behavior encountered in natural environments (landslides, debris flow, snow avalanches, etc). Herschel-Bulkley model has attracted growing attention both from the experimental and theoretical viewpoints (see for instance [2] and references therein).…”

“…Nevertheless, they can be extended to the case of steep slopes, as it was pointed out by Balmforth et al in [4]. Some other works on this issue, and in the context of the Herschel-Bulkley model, can be found for instance in [3,2,22]. On the other hand, for thicker flows, shallow water approaches consist in deriving governing equations by averaging the local mass and momentum equations across the stream depth.…”

Comparing thixotropic and Herschel–Bulkley parameterizations for continuum models of avalanches and subaqueous debris flows

“…Liu and Mei (1989) reported good agreement for theory and experiment with a Bingham plastic model and a homogeneous mud flow that provides a steady front propagation speed (necessarily long after the initiation phase). The Herschel-Bulkley model has also been used to simulate debris flow along a slope, but reported results have discrepancies with experimental data, especially in the early stages (Ancey and Cochard, 2009;Balmforth et al, 2007). Bovet et al (2010) Their results for the subaerial debris flows were in a reasonable agreement with laboratory data, but their subaqueous simulations showed a significant discrepancy with measurements.…”