Regularization by compressibility of the μ(I) model of dense granular flow

Abstract

The following article deals with the role of compressibility in regularizing the well-known μ(I) model, i.e., eliminating the short-wavelength (Hadamard) instability revealed by Barker et al. [“Well-posed and ill-posed behaviour of the μ(I)-rheology for granular flow,” J. Fluid Mech. 779, 794–818 (2015)]. In particular, we discuss the compressible-flow models proposed in the recent papers by Heyman et al. [“Compressibility regularizes the μ(I)-rheology for dense granular flows,” J. Fluid Mech. 830, 553–568 (2017)] and Barker et al. [“Well-posed continuum equations for granular flow with compressibility and μ(I)-rheology,” Proc. R. Soc. A 473(2201), 20160846 (2017)]. In addition to a critique of certain aspects of their proposed constitutive models, we show that the main effect of their regularizations is to add viscous effects to the shear response in a way that appears unfortunately to eliminate quasi-static yield stress. Another goal of the present work is to show how the development and analysis of visco-plastic constitutive relations are facilitated by dissipation potentials and the dissipative analog of elastic potentials. We illustrate their utility in Sec. IV of this article, where it is shown that a constant non-zero yield stress leads to loss of convexity that can only be restored by substituting viscous effects or else by adding spatial-gradient effects proposed previously by the present authors [Goddard, J. and Lee, J., “On the stability of the μ(I) rheology for granular flow,” J. Fluid Mech. 833, 302–331 (2017)].