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

TL;DR

This paper presents a theoretical model for the erosion threshold of a granular bed under viscous fluid shear, predicting a continuous transition with specific scaling laws and drainage patterns, supported by experimental agreement.

Contribution

Introduces a novel particle interaction model predicting erosion transition behavior and drainage patterns, linking to depinning transition models in superconductors.

Findings

Predicts a linear growth of particle current beyond threshold

Identifies divergence of transient time near the threshold

Describes broad distribution and spatial correlations of local current

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 $t_{\rm conv}$ which diverges near the transition as $t_{\rm conv}\sim |\theta-\theta_c|^{-z}$ 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.