Transport coefficients of a granular gas of inelastic rough hard spheres

The Boltzmann equation for inelastic and rough hard spheres is considered as a model of a dilute granular gas. In this model, the collisions are characterized by constant coefficients of normal and tangential restitution, and hence the translational and rotational degrees of freedom are coupled. A normal solution to the Boltzmann equation is obtained by means of the Chapman-Enskog method for states near the homogeneous cooling state. The analysis is carried out to first order in the spatial gradients of the number density, the flow velocity, and the granular temperature. The constitutive equations for the momentum and heat fluxes and for the cooling rate are derived, and the associated transport coefficients are expressed in terms of the solutions of linear integral equations. For practical purposes, a first Sonine approximation is used to obtain explicit expressions of the transport coefficients as nonlinear functions of both coefficients of restitution and the moment of inertia. Known results for purely smooth inelastic spheres and perfectly elastic and rough spheres are recovered in the appropriate limits.

Figure 1

Plot of the characteristic relaxation times ${\ell }_1^{-1}$ and ${\ell }_2^{-1}$ as functions of $\beta$ for $\kappa =\frac{2}{5}$ and $\alpha =0.8$.

Figure 2

Plot of (a) $X=I{\Omega }^2/3T_r$ and (b) $\theta =T_r/T_t$ as functions of the number of collisions per particle ($s$) scaled by ${(1+\beta )}^{-2}$ for $\kappa =\frac{2}{5}$, $\alpha =0.8$, and $\beta =-0.5$, 0, $0.5$, and 1, starting from the initial condition $\theta \left(0\right)=1$, $X\left(0\right)=\frac{1}{2}$. The horizontal lines in panel (b) correspond to the respective stationary values ${\theta }_{\infty }$.

Figure 3

Plot of $X=I{\Omega }^2/3T_r$ (dashed line) and ${(1+\beta )}^2\theta ={(1+\beta )}^2{T}_r/T_t$ (solid line) as functions of the number of collisions per particle ($s$) scaled by ${(1+\beta )}^{-1}$ for $\kappa =\frac{2}{5}$, $\alpha =0.8$, and $\beta =-0.99$, starting from the initial condition $\theta \left(0\right)=1$, $X\left(0\right)=\frac{1}{2}$. The dash-dotted line represents ${(1+\beta )}^2\theta$ if $X\left(0\right)=0$. The horizontal line corresponds to the stationary value ${(1+\beta )}^2{\theta }_{\infty }$.

Figure 4

(a) Reduced cooling rate ${\zeta }^{*}$ as a function of $\beta$ for $\kappa =\frac{2}{5}$ and $\alpha =0.6$, $0.8$, and 1. (b) Reduced cooling rate ${\zeta }^{*}$ as a function of $\alpha$ for $\kappa =\frac{2}{5}$ and $\beta =-0.5$, 0, $0.5$, and 1, as well as in the quasismooth limit $\beta \to -1$ and for purely smooth spheres ($\beta =-1$). (c) Density plot of the reduced cooling rate ${\zeta }^{*}$ for $\kappa =\frac{2}{5}$. The contour lines correspond to ${\zeta }^{*}=0.025,0.05,...,0.35$.

Figure 5

(a) Reduced shear viscosity ${\eta }^{*}$ as a function of $\beta$ for $\kappa =\frac{2}{5}$ and $\alpha =0.6$, $0.8$, and 1. (b) Reduced shear viscosity ${\eta }^{*}$ as a function of $\alpha$ for $\kappa =\frac{2}{5}$ and $\beta =-0.5$, 0, $0.5$, and 1, as well as in the quasismooth limit $\beta \to -1$ and for purely smooth spheres ($\beta =-1$). (c) Density plot of the reduced shear viscosity ${\eta }^{*}$ for $\kappa =\frac{2}{5}$. The contour lines correspond to ${\eta }^{*}=1,1.05,...,1.4$.

Figure 6

Same as in Fig. 5 but for the reduced bulk viscosity ${\eta }_b^{*}$. The contour lines in panel (c) correspond to ${\eta }_b^{*}=0.5,0.75,...,3.5$.

Figure 7

Same as in Fig. 5 but for the reduced thermal conductivity ${\lambda }^{*}$. The contour lines in panel (c) correspond to ${\lambda }^{*}=1,1.25,...,3.75$.

Figure 8

Same as in Fig. 5 but for the reduced Dufour-like coefficient ${\mu }^{*}$. The contour lines in panel (c) correspond to ${\mu }^{*}=0.25,0.5,...,3.5$.

Figure 9

Same as in Fig. 5 but for the cooling rate transport coefficient $\xi$. The contour lines in panel (c) correspond to $\xi =0,0.05,...,0.4$.