Transport coefficients for an inelastic gas around uniform shear flow: Linear stability analysis

The inelastic Boltzmann equation for a granular gas is applied to spatially inhomogeneous states close to uniform shear flow. A normal solution is obtained via a Chapman-Enskog-like expansion around a local shear flow distribution. The heat and momentum fluxes are determined to first order in the deviations of the hydrodynamic field gradients from their values in the reference state. The corresponding transport coefficients are determined from a set of coupled linear integral equations which are approximately solved by using a kinetic model of the Boltzmann equation. The main new ingredient in this expansion is that the reference state $f^{\left(0\right)}$ (zeroth-order approximation) retains all the hydrodynamic orders in the shear rate. In addition, since the collisional cooling cannot be compensated locally for viscous heating, the distribution $f^{\left(0\right)}$ depends on time through its dependence on temperature. This means that in general, for a given degree of inelasticity, the complete nonlinear dependence of the transport coefficients on the shear rate requires analysis of the unsteady hydrodynamic behavior. To simplify the analysis, the steady-state conditions have been considered here in order to perform a linear stability analysis of the hydrodynamic equations with respect to the uniform shear flow state. Conditions for instabilities at long wavelengths are identified and discussed.

Figure 1

Fourth-degree velocity moment $⟨c^4⟩$ relative to its local equilibrium value as a function of the coefficient of restitution. The solid line is the prediction of the kinetic model while the symbols are simulation results [28].

Figure 2

Plot of the reduced coefficients $\left(a\right)$ ${\mu }_{yy}^{*}$ and $\left(b\right)$ ${\mu }_{xy}^{*}$ as a function of the coefficient of restitution $\alpha$.

Figure 3

Plot of the reduced coefficients $\left(a\right)$ ${\kappa }_{yy}^{*}$ and $\left(b\right)$ ${\kappa }_{xy}^{*}$ as a function of the coefficient of restitution $\alpha$.

Figure 4

Plot of the reduced coefficients $\left(a\right)$ ${\eta }_{yyyy}^{*}$, $\left(b\right)$ ${\eta }_{xyyy}^{*}$, $\left(c\right)$ ${\eta }_{zzyy}^{*}$, and $\left(d\right)$ ${\eta }_{xxyy}^{*}$ as a function of the coefficient of restitution, $\alpha$.

Figure 5

Dependence of $C\left(\alpha \right)$ on the coefficient of restitution, $\alpha$.

Figure 6

Dispersion relations for a granular gas with $\alpha =0.8$ in the case of perturbations along the velocity gradient direction. Only the real parts of the eigenvalues are plotted.

Figure 7

Stability lines $k_s^{*}\left(\alpha \right)$ corresponding to the perturbation along the vorticity direction. The solid line corresponds to the results derived here while the dashed line refers to the results obtained from the Navier-Stokes approximation. The region above the curve corresponds to the stable domain, while the region below the curve corresponds to the unstable domain.