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

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 $t_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}$ which diverges near the transition as $t_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}\sim \mathrm{\mid }\theta -{\theta }_c{\mathrm{\mid }}^{-z}$ 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 $P(\sigma )\sim J\mathrm{/}\sigma$, 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.