Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order

Abstract

The Chapman–Enskog expansion is generalized in order to derive constitutive relations for flows of inelastically colliding spheres in three dimensions – to Burnett order. To this end, the pertinent (nonlinear) Boltzmann equation is perturbatively solved by performing a (double) expansion in the Knudsen number and the degree of inelasticity. One of the results is that the normal stress differences and the ‘temperature anisotropy’, characterizing granular fluids, are Burnett effects. The constitutive relations derived in this work differ, both qualitatively and quantitatively, from those obtained in previous studies. In particular, the Navier–Stokes (order) terms have a different dependence on the degree of inelasticity and the number density than in previously derived constitutive relations; for instance, the expression for the heat flux contains a term which is proportional to ε∇ log n, where ε is a measure of the degree of inelasticity and n denotes the number density. This contribution to the heat flux is of zeroth order in the density; a similar term, i.e. one that is proportional to ε∇n, has been previously obtained by using the Enskog correction but this term is O(n) and it vanishes in the Boltzmann limit. These discrepancies are resolved by an analysis of the Chapman–Enskog and Grad expansions, pertaining to granular flows, which reveals that the quasi-microscopic rate of decay of the temperature, which has not been taken into account heretofore, provides an important scale that affects the constitutive relations. Some (minor) quantitative differences between our results and previous ones exist as well. These are due to the fact that we take into account an isotropic correction to the leading Maxwellian distribution, which has not been considered before, and also because we consider the full dependence of the corrections to the Maxwellian distribution on the (fluctuating) speed.

Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport

“…This can be misleading for granular fluids under conditions where the size of the gradients increase with the degree of dissipation. For such states, strong dissipation can require additional terms in the constitutive equations beyond those of Navier-Stokes order [40,69,70]. This does not mean that the results obtained here are not correct at strong dissipation, only that they must be distinguished carefully from other effects of the same order.…”

Viscous Overstability in Saturn's B-Ring II. Hydrodynamic Theory and Comparison to Simulations

“…They can be calculated in the framework of kinetic theory for dense systems, using Enskog's extension of the Boltzmann equation (Chapman and Cowling 1970), to the desired order in the Chapman-Enskog expansion. For granular gases transport coefficients have been derived in the limit of nearly elastic collisions to Navier-Stokes order (see for instance Jenkins and Richman 1985;Lun et al 1984;Garzó and Dufty 1999) and to Burnett order (Sela and Goldhirsch 1998). However, it is known, that, for instance, the dependence of the viscosity on the density and the temperature in a planetary ring (Goldreich and Tremaine 1978a) differs drastically from that in a sheared granular flow.…”

A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion

“…Rather, this set of equations is also used to predict various phenomena pertinent to granular flows (Sela & Goldhirsch 1998) as well as other relativistic flows (Denicol et al 2010). An appropriate resolution of problems related to Burnett's hydrodynamic-regime equations constitutes an utmost necessity.…”