The granular Blasius problem. (14th June 2019)

• Record Type: Journal Article
• Authors: Tsang, Jonathan Michael Foonlan; Dalziel, Stuart B.; Vriend, N. M.
• Is Part Of: Journal of fluid mechanics. Volume 872(2019)
• Page Start: 784
• Page End: 817
• Publication Date: 2019-06-14
• Subjects: granular media, -- rheology; Fluid mechanics -- Periodicals; 532.005
• DOI: 10.1017/jfm.2019.357
• Languages: English
• ISSNs: 0022-1120

Abstract

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 (Jop et 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.