The granular Blasius problem

Abstract

We consider the steady flow of a granular current over a uniformly sloped surface that is smooth upstream (allowing slip for$x<0$) but rough downstream (imposing a no-slip condition on$x>0$), with a sharp transition at$x=0$. This problem is similar to the classical Blasius problem, which considers the growth of a boundary layer over a flat plate in a Newtonian fluid that is subject to a similar step change in boundary conditions. Our discrete particle model simulations show that a comparable boundary-layer phenomenon occurs for the granular problem: the effects of basal roughness are initially localised at the base but gradually spread throughout the depth of the current. A rheological model can be used to investigate the changing internal velocity profile. The boundary layer is a region of high shear rate and therefore high inertial number$I$; its dynamics is governed by the asymptotic behaviour of the granular rheology for high values of the inertial number. The$\unicode[STIX]{x1D707}(I)$rheology (Jopet al.,Nature, vol. 441 (7094), 2006, pp. 727–730) asserts that$\text{d}\unicode[STIX]{x1D707}/\text{d}I=O(1/I^{2})$as$I\rightarrow \infty$, but current experimental evidence is insufficient to confirm this. We show that this rheology does not admit a self-similar boundary layer, but that there exist generalisations of the$\unicode[STIX]{x1D707}(I)$rheology, with different dependencies of$\unicode[STIX]{x1D707}(I)$on$I$, for which such self-similar solutions do exist. These solutions show good quantitative agreement with the results of our discrete particle model simulations.