Erosion of a granular bed driven by laminar fluid flow
  Motivated by examples of erosive incision of channels in sand, we investigate the motion of individual grains in a granular bed driven by a laminar fluid to give us new insights into the relationship between hydrodynamic stress and surface granular flow. A closed cell of rectangular cross-section is partially filled with glass beads and a constant fluid flux $Q$ flows through the cell. The refractive indices of the fluid and the glass beads are matched and the cell is illuminated with a laser sheet, allowing us to image individual beads. The bed erodes to a rest height $h_r$ which depends on $Q$. The Shields threshold criterion assumes that the non-dimensional ratio $\theta$ of the viscous stress on the bed to the hydrostatic pressure difference across a grain is sufficient to predict the granular flux. Furthermore, the Shields criterion states that the granular flux is non-zero only for $\theta >\theta_c$. We find that the Shields criterion describes the observed relationship $h_r \propto Q^{1/2}$ when the bed height is offset by approximately half a grain diameter. Introducing this offset in the estimation of $\theta$ yields a collapse of the measured Einstein number $q^*$ to a power-law function of $\theta - \theta_c$ with exponent $1.75 \pm 0.25$. The dynamics of the bed height relaxation are well described by the power law relationship between the granular flux and the bed stress. 
A. E. Lobkovsky, A. V. Orpe, R. Molloy, A. Kudrolli and D. H. Rothman