Constitutive relations and linear stability of a sheared granular flow

Abstract

The constitutive relation for the granular flow of smooth, nearly elastic particles ($(1 - e) \ll 1$) is derived in the adiabatic limit, where the length scale for conduction, $\lambda / (1 - e)^{1/2}$, is small compared to the macroscopic scale. Here, $\lambda$ is the mean free path and $e$ is the coefficient of restitution. In this case, energy is convected by the mean flow, and the rate of change of the energy in a Lagrangian reference frame is determined by a balance between the rate of production (due to the shear) and the rate of dissipation (due to inelastic collisions). A Gaussian approximation is used for the velocity distribution function, and the velocity variance is determined from a balance equation for the second moment of the velocity distribution using an asymptotic expansion in the small parameter $\epsilon = (1 - e)^{1/2}$. The stress tensor is then determined from the velocity variance. It is found that the leading-order, $O(\epsilon)$ and $O(\epsilon^{2})$ contributions to the stress tensor are identical in form to those in the Euler, Navier–Stokes and Burnett approximations, and the numerical values of these coefficients are close to those calculated using the Enskog procedure. The stress tensor is used to obtain analytical expressions for the growth rates of the hydrodynamic modes in a linear shear flow in the limit where the wavelength is long compared to the length scale of conduction. In the plane of flow, transverse momentum perturbations are found to be stable, while perturbations in the density and longitudinal momenta grow exponentially at short times and decay in the long-time limit. It is found that the Navier–Stokes approximation captures the leading behaviour of the growth rate in the small-wavenumber limit for perturbations in the plane of flow. In the vorticity direction, it is found that the Navier–Stokes approximation is not adequate to capture the leading behaviour in the small-wavenumber limit, and dominant terms in the growth rates of the transverse modes depend on the Burnett coefficients. Perturbations in this direction are unstable at low density, but become stable as the density is increased. When the wave length is small compared to the conduction length, the rate of conduction of energy is large compared to the rate of dissipation, and the hydrodynamic modes are identical to those for a gas of elastic particles at equilibrium. The hydrodynamic modes are all stable in this case. The transition between these two regimes is examined for a dilute granular flow, and a transition from unstable to stable modes is predicted as the wavenumber is increased. The analysis indicates that the minimal model which accurately captures the dynamics in both limits is one in which the constitutive relation for the stress incorporates the strain-rate-dependent Burnett terms in the stress equation (neglecting the temperature-dependent Burnett terms), and the constitutive relation for the heat flux is the Fourier law for heat conduction (neglecting all the Burnett terms).