Inelastic Maxwell Models for Dilute Granular Gases

Abstract

Inelastic Maxwell models for dilute granular gases are introduced in this chapter. As with ordinary gases, in these models the collision rate of two colliding particles is independent of their relative velocity. This simplification allows us to exactly evaluate the moments of the Boltzmann collision operator. Consequently, in contrast to the previous chapters where the analytic results for hard spheres have been approximate, the use of Maxwell models opens up the possibility of obtaining the exact forms of the Navier–Stokes transport coefficients for mono- and multicomponent granular gases as well as the rheological properties in sheared granular systems. The purpose of this chapter then is to offer a brief survey on hydrodynamic properties derived in the context of inelastic Maxwell models for systems close to the homogeneous cooling state and for far from equilibrium situations. The results obtained for inelastic Maxwell models will be compared with the theoretical results derived for inelastic hard spheres using analytic approximate methods and the DSMC method. Finally, a surprising “nonequilibrium phase transition” for a sheared binary mixture in the tracer limit will be identified.

Appendix A

Some of the collisional moments of the Boltzmann collision operator of IMM needed to achieve the Navier–Stokes transport coefficients for mono- and multicomponent granular gases are evaluated in this Appendix. Let us consider the general collisional integral

$$\begin{aligned} I[\psi (\mathbf {V})]=\int \mathrm{d}\mathbf {v}\; \psi (\mathbf {v}) J_\text {IMM}[\mathbf {v}|f,f], \end{aligned}$$

(9.94)

where $J_\text {IMM}[f,f]$ is defined by Eq. (9.1) for monocomponent gases. Therefore, Eq. (9.94) can be more explicitly written as

$$\begin{aligned} I[\psi (\mathbf {V}_1)]=\frac{\nu _\text {M}}{n S_d} \int \mathrm{d}\mathbf {v}_1 \int \mathrm{d}\mathbf{v}_{2}\int \mathrm{d}\widehat{\varvec{\sigma }} \left[ \alpha ^{-1}f(\mathbf{v}_{1}'')f(\mathbf{v}_{2}'')\psi (\mathbf {V}_1)- f(\mathbf{v}_{1})f(\mathbf{v}_{2})\psi (\mathbf {V}_1)\right] . \end{aligned}$$

(9.95)

As we did in Appendix A of Chap. 1, we again change variables to integrate over $\mathbf {v}_1''$ and $\mathbf {v}_2''$ instead of $\mathbf {v}_1$ and $\mathbf {v}_2$ in the first integral on the right-hand side of Eq. (9.95). Since $\mathrm{d}\mathbf {v}_1 \mathrm{d}\mathbf {v}_2=\alpha \mathrm{d}\mathbf {v}_1'' \mathrm{d}\mathbf {v}_2''$, Eq. (9.95) can be rewritten as

$$\begin{aligned} I[\psi (\mathbf {V}_1)]=\frac{\nu _\text {M}}{n S_d} \int \mathrm{d}\mathbf {v}_1 \int \mathrm{d}\mathbf{v}_{2}f(\mathbf{v}_{1})f(\mathbf{v}_{2})\int \mathrm{d}\widehat{\varvec{\sigma }}\left[ \psi (\mathbf {V}_1')-\psi (\mathbf {V}_1)\right] , \end{aligned}$$

(9.96)

where

$$\begin{aligned} \mathbf {V}_1'=\mathbf {V}_1-\frac{1}{2}(1+\alpha )(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})\widehat{\varvec{\sigma }}. \end{aligned}$$

(9.97)

Let us particularize Eq. (9.96) to $\psi (\mathbf {V})=m \mathbf {V}\mathbf {V}$. The scattering rule (9.97) yields

$$\begin{aligned} V_{1i}'V_{1j}'-V_{1i}V_{1j}=\frac{1+\alpha }{2}(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12}) \left[ \frac{1+\alpha }{2}(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})\widehat{\sigma }_i\widehat{\sigma }_j-\left( V_{1i}\widehat{\sigma }_j +V_{1j}\widehat{\sigma }_i\right) \right] . \end{aligned}$$

(9.98)

To perform the angular integrals, we require the results

$$\begin{aligned} \int \mathrm{d}\widehat{\varvec{\sigma }} (\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})^{2k+1}\widehat{\sigma }_i= \widetilde{B}_{k+1} g_{12}^{2k}g_{12,i}, \end{aligned}$$

(9.99)

$$\begin{aligned} \int \mathrm{d}\widehat{\varvec{\sigma }} (\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})^{2k}\widehat{\sigma }_i\widehat{\sigma }_j=\frac{\widetilde{B}_k}{2k+d}g_{12}^{2(k-1)}\left( 2k g_{12,i} g_{12,j}+g_{12}^2 \delta _{ij}\right) , \end{aligned}$$

(9.100)

where $\widetilde{B}_k$ is [12]

$$\begin{aligned} \widetilde{B}_k=\int \ \mathrm{d}\widehat{\varvec{\sigma }} (\widehat{\varvec{\sigma }}\cdot \widehat{\mathbf {g}}_{12})^{2k}= S_d \pi ^{-1/2}\frac{\varGamma \left( \frac{d}{2}\right) \varGamma \left( k+\frac{1}{2}\right) }{ \varGamma \left( k+\frac{d}{2}\right) }. \end{aligned}$$

(9.101)

Using Eqs. (9.98)–(9.100) in Eq. (9.96), we achieve [25]

$$\begin{aligned} I[m V_{1i}V_{1j}]= & {} \frac{\nu _\text {M}}{n d}m \frac{1+\alpha }{2}\int \mathrm{d}\mathbf {v}_1 \int \mathrm{d}\mathbf{v}_{2}f(\mathbf{v}_{1})f(\mathbf{v}_{2})\left[ \frac{1+\alpha }{2(d+2)}\left( 2g_{12,i}g_{12,j}+g_{12}^2 \delta _{ij}\right) \right. \nonumber \\&\left. -\left( V_{1i}g_{12,j} +V_{1j}g_{12,i}\right) \right] \nonumber \\= & {} -\nu _\text {M}\frac{1+\alpha }{2(d+2)}\left[ \frac{2}{d}\left( d+1-\alpha \right) P_{ij}-(1+\alpha )p\delta _{ij}\right] , \end{aligned}$$

(9.102)

where $p=n T$ and $P_{ij}$ is the pressure tensor. For the sake of convenience, Eq. (9.102) can be rewritten in terms of the traceless tensor $\varPi _{ij}=P_{ij}-p\delta _{ij}$ as

$$\begin{aligned} I[m V_{1i}V_{1j}]=-\nu _{0|2}\varPi _{ij}-\zeta p \delta _{ij}, \end{aligned}$$

(9.103)

where the cooling rate $\zeta$ is given by Eq. (9.3) and

$$\begin{aligned} \nu _{0|2}=\zeta +\frac{(1+\alpha )^2}{2(d+2)}\nu _\text {M}=\frac{(1+\alpha )(d+1-\alpha )}{d(d+2)}\nu _\text {M}. \end{aligned}$$

(9.104)

Next, the third-degree moments are considered. These moments can be obtained by following similar mathematical steps to those made before for the second-degree moments. Their expressions are [26]

$$\begin{aligned} \int \mathrm{d}\mathbf {v}\; \frac{m}{2}V^2 V_i\; J_\text {IMM}[\mathbf {v}|f,f]=-\nu _{2|1}q_i, \end{aligned}$$

(9.105)

$$\begin{aligned} \int \mathrm{d}\mathbf {v}\; \frac{m}{2} V_i V_j V_k\; J_\text {IMM}[\mathbf {v}|f,f]=-\nu _{0|3}Q_{ijk}-\frac{\nu _{2|1}-\nu _{0|3}}{d+2} \left( q_i \delta _{jk}+q_j \delta _{ik}+q_k \delta _{ij}\right) , \end{aligned}$$

(9.106)

where $\nu _{0|3}=\frac{3}{2}\nu _{0|2}$,

$$\begin{aligned} \nu _{2|1}=\frac{3}{2}\zeta +\frac{(d-1)(1+\alpha )^2}{2d(d+2)} \nu _\text {M}=\frac{(1+\alpha )\left[ 5d+4-(d+8)\alpha \right] }{4d(d+2)}\nu _\text {M}, \end{aligned}$$

(9.107)

and we have introduced the third-rank tensor

$$\begin{aligned} Q_{ijk}=\int \mathrm{d}\mathbf {v}\; \frac{m}{2}V_iV_j V_k f(\mathbf {V}). \end{aligned}$$

(9.108)

Note that the relation (9.105) can be easily obtained from Eq. (9.106) by taking its trace.

The expressions of the fourth-order collisional moments are larger and so will be omitted here. They can be found elsewhere [26]. As an illustration, only the collisional moment associated with the isotropic moment $V^4$ is considered. It is given by [24, 26]

$$\begin{aligned} \int \mathrm{d}\mathbf {v}\; m V^4\; J_\text {IMM}[\mathbf {v}|f,f]=-\nu _{4|0}M_4+\frac{\lambda _1}{m n}d^2 p^2-\frac{\lambda _2}{m n} \varPi _{k\ell }\varPi _{\ell k}, \end{aligned}$$

(9.109)

where the coefficients $\nu _{4|0}$, $\lambda _1$, and $\lambda _2$ are

$$\begin{aligned} \nu _{4|0}=2\zeta +\frac{(1+\alpha )^2(4d-7+6\alpha -3\alpha ^2)}{8d(d+2)}\nu _\text {M}, \end{aligned}$$

(9.110)

$$\begin{aligned} \lambda _1=\frac{(1+\alpha )^2\left( 4d-1-6\alpha +3\alpha ^2\right) }{8d^2}\nu _\text {M}, \end{aligned}$$

(9.111)

$$\begin{aligned} \lambda _2=\frac{(1+\alpha )^2\left( 1+6\alpha -3\alpha ^2\right) }{4d(d+2)}\nu _\text {M}. \end{aligned}$$

(9.112)

Now the case of multicomponent granular mixtures is considered. Let us consider the first-degree collisional moment. It is given by

$$\begin{aligned} \int \mathrm{d}\mathbf{v} m_i \mathbf{V}J_{\text {IMM},ij}[f_i,f_j]=m_i\frac{\nu _{\text {M},ij}}{n_j S_d} \int \mathrm{d}\mathbf {v}_1 \int \mathrm{d}\mathbf{v}_{2}\;f_i(\mathbf{v}_{1}) f_j(\mathbf{v}_{2})\int \mathrm{d}\widehat{\varvec{\sigma }}\left( \mathbf {V}_1'-\mathbf {V}_1\right) , \end{aligned}$$

(9.113)

where use has been made of the relation (9.96) and

$$\begin{aligned} \mathbf {V}_1'=\mathbf {V}_1-\frac{m_j}{m_i+m_j}(1+\alpha _{ij})(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})\widehat{\varvec{\sigma }}. \end{aligned}$$

(9.114)

The collision integral (9.113) can be easily evaluated after performing the angular integration. The final result is

$$\begin{aligned} \int \mathrm{d}\mathbf{v} m_i\mathbf{V}J_{\text {IMM},ij}[f_i,f_j]=-\frac{\nu _{\text {M},ij}}{d \rho _j}\mu _{ji}(1+\alpha _{ij}) \left( \rho _j\mathbf{j}_i-\rho _i\mathbf{j}_j\right) , \end{aligned}$$

(9.115)

where we recall that $\mu _{ij}=m_i/(m_i+m_j)$, $\rho _i=m_i n_i$ is the mass density of species i and

$$\begin{aligned} \mathbf {j}_i=\int \mathrm{d}\mathbf {v}\; m_i \mathbf {V} f_i(\mathbf {V}). \end{aligned}$$

(9.116)

The second-degree collisional moment can be obtained by following similar steps. The result is [6]

$$\begin{aligned} \int \mathrm{d}\mathbf{v} m_i\mathbf{V}\mathbf{V}J_{\text {IMM},ij}[f_i,f_j]= & {} -\frac{\nu _{\text {M},ij}}{d \rho _j}\mu _{ji}(1+\alpha _{ij}) \Bigg \{2\rho _j\mathsf{P}_i-\left( \mathbf{j}_i\mathbf{j}_j+\mathbf{j}_j\mathbf{j}_i\right) \nonumber \\&-\frac{2}{d+2}\mu _{ji}(1+\alpha _{ij})\Bigg [\rho _j\mathsf{P}_i+\rho _i\mathsf{P}_j- \left( \mathbf{j}_i\mathbf{j}_j+\mathbf{j}_j\mathbf{j}_i\right) \nonumber \\&+\left( \frac{d}{2}\left( \rho _ip_j+\rho _jp_i\right) -\mathbf{j}_i\cdot \mathbf{j}_j\right) \mathsf {I} \Bigg ]\Bigg \}, \end{aligned}$$

(9.117)

where $p_i=n_i T_i$ is the partial pressure of species i, $\mathsf {I}$ is the $d\times d$ unit tensor, and

$$\begin{aligned} \mathsf {P}_i=\int \mathrm{d}\mathbf {v}\; m_i \mathbf {V} \mathbf {V}\; f_i(\mathbf {V}). \end{aligned}$$

(9.118)

From Eq. (9.117), one can easily obtain the partial cooling rates $\zeta _i$:

$$\begin{aligned} \zeta _i= & {} -\sum _j\;\frac{1}{dn_iT_i}\int \mathrm{d}\mathbf {v}\; m_i V^2 J[f_i,f_j] \nonumber \\= & {} \sum _j\frac{2\nu _{\text {M},ij}}{d}\mu _{ji}(1+\alpha _{ij})\Bigg [1-\frac{\mu _{ji}}{2}(1+\alpha _{ij}) \frac{\theta _i+\theta _j}{\theta _j}\nonumber \\&+\frac{\mu _{ji}(1+\alpha _{ij})-1}{d\rho _j p_i}\mathbf {j}_i\cdot \mathbf {j}_j\Bigg ]. \end{aligned}$$

(9.119)

The third-degree collisional moment associated with heat flux has been explicitly evaluated elsewhere [39].