A model for the erosion onset of a granular bed sheared by a viscous fluid

Abstract

We study theoretically the erosion threshold of a granular bed forced by a viscous fluid. We first introduce a novel model of interacting particles driven on a rough substrate. It predicts a continuous transition at some threshold forcing ${\theta }_c$, beyond which the particle current grows linearly $J\sim \theta -{\theta }_c$, in agreement with experiments. The stationary state is reached after a transient time $t_{\mathrm{conv}}$ which diverges near the transition as $t_{\mathrm{conv}}\sim |\theta -{\theta }_c{|}^{-z}$ with $z\approx 2.5$. The model also makes quantitative testable predictions for the drainage pattern: the distribution $P(\sigma )$ of local current is found to be extremely broad with $P(\sigma )\sim J/\sigma$, spatial correlations for the current are negligible in the direction transverse to forcing, but long-range parallel to it. We explain some of these features using a scaling argument and a mean-field approximation that builds an analogy with $q$-models. We discuss the relationship between our erosion model and models for the depinning transition of vortex lattices in dirty superconductors, where our results may also apply.

5 pages, 6 figures

References in corpus (2)

• A two-phase flow model of sediment transport: transition from bedload to suspended load
• Onset and cessation of motion in hydrodynamically sheared granular beds

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• Flowing emulsions through disorder: Critical depinning and smectic rivers