Generalized transport coefficients for inelastic Maxwell mixtures under shear flow

The Boltzmann equation framework for inelastic Maxwell models is considered to determine the transport coefficients associated with the mass, momentum, and heat fluxes of a granular binary mixture in spatially inhomogeneous states close to the simple shear flow. The Boltzmann equation is solved by means of a Chapman-Enskog–type expansion around the (local) shear flow distributions $f_r^{\left(0\right)}$ for each species that retain all the hydrodynamic orders in the shear rate. Due to the anisotropy induced by the shear flow, tensorial quantities are required to describe the transport processes instead of the conventional scalar coefficients. These tensors are given in terms of the solutions of a set of coupled equations, which can be analytically solved as functions of the shear rate $a$, the coefficients of restitution ${\alpha }_{rs}$, and the parameters of the mixture (masses, diameters, and composition). Since the reference distribution functions $f_r^{\left(0\right)}$ apply for arbitrary values of the shear rate and are not restricted to weak dissipation, the corresponding generalized coefficients turn out to be nonlinear functions of both $a$ and ${\alpha }_{rs}$. The dependence of the relevant elements of the three diffusion tensors on both the shear rate and dissipation is illustrated in the tracer limit case, the results showing that the deviation of the generalized transport coefficients from their forms for vanishing shear rates is in general significant. A comparison with the previous results obtained analytically for inelastic hard spheres by using Grad's moment method is carried out, showing a good agreement over a wide range of values for the coefficients of restitution. Finally, as an application of the theoretical expressions derived here for the transport coefficients, thermal diffusion segregation of an intruder immersed in a granular gas is also studied.

Figure 1

(a) Shear-rate dependence of the (dimensionless) coefficients $D_{xx}^{*}/D_0^{*}$ and $D_{xy}^{*}/D_0^{*}$ for $d=3, {\sigma }_1/{\sigma }_2=1, m_1/m_2=2$ and three different values of the (common) coefficient of restitution $\alpha$: $\alpha =1$ (solid line), $\alpha =0.8$ (dashed red line), and $\alpha =0.6$ (dashed-dotted blue line). (b) Shear-rate dependence of the (dimensionless) coefficients $D_{yy}^{*}/D_0^{*}$ and $D_{yx}^{*}/D_0^{*}$ for the same parameter values. These results pertain to model A ($\beta =0$).

Figure 2

Same as Fig. 1 for $D_{p,yy}^{*}/D_{p,0}^{*}$ and $D_{p,xy}^{*}/D_{p,0}^{*}$ (a) $D_{T,yy}^{*}/D_{T,0}^{*}$ and $D_{T,xy}^{*}/D_{T,0}^{*}$ (b).

Figure 3

Plot of the ratio ${\eta }_{xyxy}^{*}/{\eta }^{*}$ as a function of the coefficient of restitution $\alpha$ in the steady state for a monodisperse granular gas (IMM with $\beta =0$, model A). The solid line is the result for a three-dimensional system ($d=3$) while the dashed line corresponds to a two-dimensional system ($d=2$).

Figure 4

Plot of the diagonal (dimensionless) coefficients $D_{xx}^{*}/D_0^{*}$ (a) and $D_{yy}^{*}/D_0^{*}$ (b) as functions of the (common) coefficient of restitution $\alpha$ in the steady USF state for $d=3$ in the cases ${\sigma }_1/{\sigma }_2=1$ and $m_1/m_2=2$ (C) and ${\sigma }_1/{\sigma }_2=2$ and $m_1/m_2=4$ (E). The solid lines correspond to the results derived here for IMM (models A and B) while the dashed lines are the results obtained for IHS [28, 29].

Figure 5

Same as Fig. 4 for $D_{xy}^{*}/D_0^{*}$ (a) and $D_{yx}^{*}/D_0^{*}$ (b).

Figure 6

Phase diagram for segregation for a three-dimensional system ($d=3$) and three different values of the (common) coefficient of restitution $\alpha \equiv {\alpha }_{22}={\alpha }_{12}$: $\alpha =0.9$ (solid line), $\alpha =0.8$ (dashed line), and $\alpha =0.7$ (dotted line).

Figure 7

Phase diagram for segregation for a three-dimensional system ($d=3$) with ${\alpha }_{22}=0.9$ and ${\alpha }_{12}=0.7$. The solid line corresponds to the theoretical prediction obtained from Eq. (92) while the symbols refer to computer simulations carried out in Ref. [48] for IHS in the so-called LTu flow (Couette flow with uniform heat flux).

Figure 8

Plot of the scalar diffusion coefficient $D^{*}\left(\alpha \right)$ (relative to its elastic value) and the zero shear-rate diffusion coefficient $D_0^{*}$ (relative to its elastic value) as functions of the (common) coefficient of restitution ${\alpha }_{12}={\alpha }_{22}\equiv \alpha$ in the steady state for $d=3$ and two different systems: ${\sigma }_1/{\sigma }_2=2$ and $m_1/m_2=2$ (solid lines) and ${\sigma }_1/{\sigma }_2=1$ and $m_1/m_2=3$ (dashed lines).