The granular Blasius problem

Jonathan Michael Foonlan Tsang

doi:10.1017/jfm.2019.357

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The granular Blasius problem

2019-06-14

Jonathan Michael Foonlan Tsang *

Stuart B. Dalziel

Nathalie Vriend

DOI：10.1017/jfm.2019.357

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Abstract

En 中文

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 D 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 mu I / rheology (Jop et al., Nature, vol. 441 (7094), 2006, pp. 727-730) asserts that d= dI D O. 1 = I2 / as mu 1, 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 mu I / rheology, with different dependencies of mu yI / 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.

Keywords:

granular media

rheology

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