Anomalous behaviour of hydrodynamic modes in the two dimensional shear flow of a granular material

The growth rates of the hydrodynamic modes in the homogeneous sheared state of a granular material are determined by solving the Boltzmann equation. The steady velocity distribution is considered to be the product of the Maxwell–Boltzmann distribution and a Hermite polynomial expansion in the velocity components; this form is inserted into the Boltzmann equation and solved to obtain the coefficients of the terms in the expansion. The solution is obtained using an expansion in the parameter ε=(1−e)1/2, and terms correct to ε4 are retained to obtain an approximate solution; the error due to the neglect of higher terms is estimated at about 5% for e=0.7. A small perturbation is placed on the distribution function in the form of a Hermite polynomial expansion for the velocity variations and a Fourier expansion in the spatial coordinates; this is inserted into the Boltzmann equation and the growth rate of the Fourier modes is determined. It is found that in the hydrodynamic limit, the growth rates of the hydrodynamic modes in the flow direction have unusual characteristics. The growth rate of the momentum diffusion mode is positive, indicating that density variations are unstable in the limit k→0, and the growth rate increases proportional to |k|2/3 in the limit k→0 (in contrast to the k2 increase in elastic systems), where k is the wave vector in the flow direction. The real and imaginary parts of the growth rates corresponding to the propagating also increase proportional to |k|2/3 (in contrast to the k2 and k increase in elastic systems). The energy mode is damped due to inelastic collisions between particles. The scaling of the growth rates of the hydrodynamic modes with the wave vector l in the gradient direction is similar to that in elastic systems.