Integration through transients approach to the $μ(\mathcal{I})$ rheology

Abstract

This work generalises the granular integration through transients formalism introduced by Kranz et al. [Phys. Rev. Lett. 121, 148002 (2018)] to the determination of the pressure. We focus on the Bagnold regime, and provide theoretical support to the empirical µ(I) rheology laws, that have been successfully applied in many granular flow problems. In particular, we confirm that the interparticle friction is irrelevant in the regime where the µ(I) laws apply.

Granular rheology: a tale of three time scales

“…In a recent study [37], it has been shown that the Granular Integration Through Transients (GITT) formalism provides a set of fundamental equations which, once numerically solved, yield results showing a satisfactory agreement with the predictions of the µ(I) law, with parameter values compatible with the experimental results. Here, we show that the fundamental GITT equations can be reduced to a simpler toy-model in which the rheology of granular liquids is explained as a competition between three time scales associated to the relevant physical processes at play in the system (diffusion, shear advection and structural relaxation).…”

“…The expression of those quantities are not needed in our derivation. For details, the reader is referred to the previous papers on GITT [37,44,45].…”

“…Note that in this picture the overall S q factor does not play any major role. Finally, Φ q (t) is related to rheological quantities through the Integration Through Transients (ITT) formalism [47,48], that can be used to express the shear stress σ and the pressure P in the out-of-equilibrium steady state as integrals over the values of Φ q (t) at former times (see [37,45] for more details):…”

“…All in all, in GITT the rheology of granular liquids is described in terms of integrals over the advected dynamical structure factor Φ q(t) (t), whose dynamical evolution is described by MCT. This method has proven successful to describe the rheology of dry granular liquids [37]. However, the involved structure of the equations makes it difficult to understand precisely how the underlying physical processes at play impact the end result.…”

“…From Eq. (3), it consists of two types of terms: the unsheared pressure P ( γ = 0) which does not depend on advection and is therefore a mere constant (denoted P 0 ) in our toy-model, and the ITT correction given by the two next terms (see [37] for more details). As discussed before, since the q-structure has been reduced to mere constant prefactors, both terms have the form of K 1 given by Eq.…”