Modelling of phase transitions in granular flows

Abstract

Abstract. We present in this work a system for unidimensional granular flows first mentioned in a paper of A. Lefebvre-Lepot and B. Maury (2011), which captures the transitions between compressible and incompressible phases. This model exhibits in the incompressible regions some memory effects through an additional variable called adhesion potential. We derive this system from compressible Navier-Stokes equations with singular viscosities and pressure, the singular limit between the two systems can then be seen as an analogue of the low Mach number limit for fluid with pressure dependent viscosity. It answers in a sense to the problem of transition between suspension flows and immersed granular flows identified by B. Andreotti, Y. Forterre and O. Pouliquen (2011). We illustrate the singular limit by numerical simulations.

A remark on memory effects in constrained fluid systems

“…The results presented in [4], [21] and [22] justify in particular the limit from (5) towards the adhesion potential equation in system 3:…”

“…We briefly review in this section the recent results obtained in [16], [24] and [4], our goal here is to highlight two aspects of the notion of adhesion potential: on the one hand the adhesion potential seen as a residual effect of singular lubrication forces; on the other hand the adhesion potential seen as a result of a projection of a free or spontaneous dynamics onto the set of admissible dynamics for the maximal density constraint. As said before, these results have been already presented in the recent synthesis [23], and we refer to it (and papers [16], [22], [24] and [4]) for the precise statements of the results and more details concerning the technical tools that are involved in the mathematical analysis.…”

Compression effects in heterogeneous media

“…In the present paper, we characterize precisely the respective effects of pressure and bulk viscosity. At the limit on the hard congestion system, we cover in particular the two cases introduced in [33] and [32] where pressure effects or memory effects are activated.…”

“…We consider the same regularized system (34) with truncated pressure (32) and bulk viscosity (33) as in the previous section. Recall that we ensure from Lemma 3.2 the following properties on ν:…”

One-Dimensional Granular System with Memory Effects

“…The mathematical difficulty of this singular limit relies in the lack of compactness of the non-linear term ρ ε u 2 ε . This kind of singular limit has nevertheless been proved in [20] (see also [21] for a result in dimension 2) on an augmented system where an additional physical dissipation is taken into account.…”