Transport coefficients for driven granular mixtures at low density

The transport coefficients of a granular binary mixture driven by a stochastic bath with friction are determined from the inelastic Boltzmann kinetic equation. A normal solution is obtained via the Chapman-Enskog method for states near homogeneous steady states. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. They are given in terms of the solutions of a set of coupled linear integral equations. As in the monocomponent case, since the collisional cooling cannot be compensated for locally by the heat produced by the external driving, the reference distributions (zeroth-order approximations) $f_i^{\left(0\right)}$ ($i=1,2$) for each species depend on time through their dependence on the pressure and the temperature. Explicit forms for the diffusion transport coefficients and the shear viscosity coefficient are obtained by assuming the steady-state conditions and by considering the leading terms in a Sonine polynomial expansion. A comparison with previous results obtained for granular Brownian motion and by using a (local) stochastic thermostat is also carried out. The present work extends previous theoretical results derived for monocomponent dense gases [Garzó, Chamorro, and Vega Reyes, Phys. Rev. E 87, 032201 (2013)] to granular mixtures at low density.

Figure 1

Plot of the (steady) reduced temperature $T_{1,\text{s}}/T_{\text{b}}$ of a Brownian particle as a function of the volume fraction $\varphi$ for hard disks ($d=2$). The parameters of the system (impurity particle plus granular gas) are $m_1=100m_2$, ${\sigma }_1={\sigma }_2$, and ${\alpha }_{11}={\alpha }_{12}={\alpha }_{22}=0.8$. The solid line refers to the results derived from Eq. (41), while the dashed line corresponds to the results obtained from Eq. (44) in the Brownian limit ($m_1/m_2\to \infty$). In both cases, $\beta =1$ and $\lambda =2$. Symbols are the simulation results obtained in Ref. [30] by means of the DSMC method (red diamonds) and MD simulations (black circles).

Figure 2

Temperature ratio $T_{1,\text{s}}/T_{2,\text{s}}$ versus the (common) coefficient of restitution $\alpha$ for hard disks (top panel) and hard spheres (bottom panel) for $x_1=\frac{2}{3}$, ${\sigma }_1={\sigma }_2$, and three different values of the mass ratio $m_1/m_2$: (a) $m_1/m_2=0.1$, (b) $m_1/m_2=2$, and (c) $m_1/m_2=10$. The parameters of the system are the same as those considered in Fig. 1.

Figure 3

Plot of the self-diffusion coefficient $\overline{D}$ as a function of the (common) coefficient of restitution $\alpha \equiv {\alpha }_{22}={\alpha }_{12}$ for a two-dimensional system ($d=2$). The parameters of the system are $m_1=100m_2$, ${\sigma }_1={\sigma }_2$, $\varphi =0.00785$, ${\xi }_b^2=0.2$, and ${\gamma }_{\text{b}}=0.1$. Symbols are the simulation results obtained in Ref. [30] by means of the DSMC method (red diamonds) and MD simulations (black circles). The solid line is the theoretical result obtained from Eq. (92) for $m_1=100m_2$, while the dashed line corresponds to the theoretical result obtained from Eq. (95) in the Brownian limit ($m_1/m_2\to \infty$).

Figure 4

Reduced diffusion coefficients $D^{*},D_p^{*}$, and $D_T^{*}$ as a function of the (common) coefficient of restitution ${\alpha }_{11}={\alpha }_{12}={\alpha }_{22}\equiv \alpha$ for an equimolar binary mixture ($x_1=\frac{1}{2}$) of hard disks ($d=2$) with ${\sigma }_1/{\sigma }_2=1$ and $m_1/m_2=2$. Different driven systems are plotted: (a) global stochastic thermostat (${\gamma }_{\text{b}}=0,\lambda =0$), (b) local stochastic thermostat (${\gamma }_{\text{b}}=0,\lambda =0$), (c) stochastic bath with friction (${\xi }_{\text{b}}^2=0.2,{\gamma }_{\text{b}}=0.1,\lambda =2,$ and $\beta =1$), and (d) undriven system (${\xi }_{\text{b}}^2={\gamma }_{\text{b}}=0$).

Figure 5

Plot of the (reduced) shear viscosity coefficient $\eta \left(\alpha \right)/\eta \left(1\right)$ versus the (common) coefficient of restitution ${\alpha }_{11}={\alpha }_{12}={\alpha }_{22}\equiv \alpha$ for an equimolar binary mixture ($x_1=\frac{1}{2}$) of hard disks (top panel) and hard spheres (bottom panel) with ${\sigma }_1/{\sigma }_2=1$ and three different values of the mass ratio: (a) $m_1/m_2=1$, (b) $m_1/m_2=2$, and (c) $m_1/m_2=4$. The lines correspond to the theoretical results derived for the stochastic thermostat (${\gamma }_{\text{b}}=0,\lambda =0$). The symbols are the DSMC results for a mixture of mechanically equivalent particles driven by the stochastic thermostat (Ref. [43]).

Figure 6

Plot of the (reduced) shear viscosity coefficient $\eta \left(\alpha \right)\eta /\eta \left(1\right)$ versus the (common) coefficient of restitution ${\alpha }_{11}={\alpha }_{12}={\alpha }_{22}\equiv \alpha$ for an equimolar binary mixture ($x_1=\frac{1}{2}$) of hard disks (top panel) and hard spheres (bottom panel) with ${\sigma }_1/{\sigma }_2=1$ and three different values of the mass ratio: (a) $m_1/m_2=1$, (b) $m_1/m_2=2$, and (c) $m_1/m_2=4$. The lines are the results derived for the general driven system with model parameters ${\xi }_{\text{b}}^2=0.2,{\gamma }_{\text{b}}=0.1,\lambda =2$, and $\beta =1$.

Figure 7

Plot of the (reduced) diffusion coefficient $D^{*}$ as a function of the (common) coefficient of restitution ${\alpha }_{11}={\alpha }_{12}={\alpha }_{22}\equiv \alpha$ for an equimolar binary mixture ($x_1=\frac{1}{2}$) of hard disks with ${\sigma }_1/{\sigma }_2=2$ and $m_1/m_2=8$. Here, the mixture is driven by a global stochastic thermostat (${\gamma }_{\text{b}}=0,\lambda =0$). The solid line is the result derived from Eq. (102), while the dashed line has been obtained by neglecting the derivatives of ${\chi }_1$ and ${\zeta }_0^{*}$ with respect to ${\xi }^{*}$ in Eq. (102).