Hydrodynamic modes of a three-dimensional sheared granular material

The hydrodynamic modes of a three-dimensional sheared granular flow are determined by solving the linearised Boltzmann equation. The steady state is determined using an expansion in the parameter $\epsilon=(1-e)^{1/2}$, and terms correct to $O(\epsilon^4)$ are retained in the expansion. The distribution function is expressed as the product of a Gaussian distribution and an expansion in Hermite polynomials, and the coefficients in the expansion are determined by solving the Boltzmann equation for the steady flow. A basis set consisting of 14 functions, containing products of Hermite polynomials upto fourth order, were used for calculating the steady distribution function. In order to determine the decay rate of the hydrodynamic modes, small perturbations in the form of Fourier modes in the spatial directions and a Hermite polynomial expansion in the particle velocities, were placed on the base state, and the initial growth rates of these perturbations were determined. The number of solutions for the initial growth rates depend on the number of basis functions used for defining the perturbations. However, it was found that the initial growth rates of the hydrodynamic modes showed small variations when the number of basis functions was increased from 10 to 20. The initial growth rates for the hydrodynamic modes showed unusual behaviour in the flow and the vorticity directions. In the flow directions, the scaling laws previously obtained for a two-dimensional system were recovered in this case as well. In the vorticity direction, it was found that all five growth rates were real, and proportional to m in the limit m→0, where m is the wave number in the vorticity direction. In addition, two of the growth rates are positive, indicating that there are two unstable modes in this direction