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Granular Gaseous Flows pp 1–55Cite as

Kinetic Theory of Inelastic Hard Spheres

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Part of the book series: Soft and Biological Matter ((SOBIMA))

Abstract

The present chapter provides a concise introduction to the kinetic theory of granular gases (namely, gases of hard spheres with inelastic collisions) at low and moderate densities. We briefly review first the dynamics of binary collisions for some of the models most widely used in the literature, and then we outline heuristically the derivation of the Boltzmann and Enskog kinetic equations for monocomponent granular gases. A connection with hydrodynamics is established where the corresponding macroscopic balance equations for the densities of mass, momentum and energy are exactly derived from the above kinetic equations with expressions for the momentum and heat fluxes and the cooling rate as functionals of the one-particle velocity distribution function. These kinetic equations are then extended to the interesting case of multicomponent granular mixtures. The complexity of the kernel of the Enskog-Boltzmann collision operator, however, prevents the possibility of obtaining exact results, and for this reason there is often a preference for kinetic models that are mathematically simpler than the original equations but capture their most relevant physical properties. Thus, the chapter ends with the construction of some kinetic models proposed in the literature of granular gases based on the popular BGK model for ordinary (elastic) gases.

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Notes

  1. 1.

    In what follows we will take units for the granular temperature so that \(k_\text {B}=1\).

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  1. Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, Badajoz, Spain

    Vicente Garzó

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Appendices

Appendix A

In this appendix the production term \(\sigma _\psi ^{(c)}\) due to collisions is expressed in the form (1.84). Taking into account Eq. (1.77), the quantity \(\sigma _\psi ^{(c)}\) is

$$\begin{aligned} \sigma _\psi ^{(c)}= & {} \sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}\int {\mathrm {d}}\mathbf{v}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \alpha ^{-2}\chi (\mathbf {r},\mathbf {r}-\varvec{\sigma }) f(\mathbf {r},\mathbf {v}_1'';t) \nonumber \\&\times \, f(\mathbf {r}-\varvec{\sigma },\mathbf {v}_2'';t)\psi (\mathbf {v}_1)-\sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}\int {\mathrm {d}}\mathbf{v}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \nonumber \\&\times \,\chi (\mathbf {r},\mathbf {r}+\varvec{\sigma }) f(\mathbf {r},\mathbf {v}_1;t) f(\mathbf {r}+\varvec{\sigma },\mathbf {v}_2;t)\psi (\mathbf {v}_1). \end{aligned}$$
(1.187)

We 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 of the right-hand side of Eq. (1.187). According to the relations (1.58), \({\mathrm {d}}\mathbf {v}_1 {\mathrm {d}}\mathbf {v}_2=\alpha {\mathrm {d}}\mathbf {v}_1'' {\mathrm {d}}\mathbf {v}_2''\) and \((\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})=-\alpha (\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}'')\), so the above integral can be recast into the form

$$\begin{aligned}&\sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}''\int {\mathrm {d}}\mathbf{v}_{2}''\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (-\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12}'')(-\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}'') \chi (\mathbf {r},\mathbf {r}-\varvec{\sigma }) f(\mathbf {r},\mathbf {v}_1'';t)\nonumber \\&\times \,f(\mathbf {r}-\varvec{\sigma },\mathbf {v}_2'';t) \psi (\mathbf {v}_1). \end{aligned}$$
(1.188)

Equation (1.188) contains the pre-collisional velocities \((\mathbf {v}_1'',\mathbf {v}_2'')\) and the post-collisional velocities \((\mathbf {v}_1,\mathbf {v}_2)\). Since the transformation \((\mathbf {v}_1'',\mathbf {v}_2'')\rightarrow (\mathbf {v}_1,\mathbf {v}_2)\) is equivalent to \((\mathbf {v}_1,\mathbf {v}_2)\rightarrow (\mathbf {v}_1',\mathbf {v}_2')\), we can change the notation in (1.188) by using the (dummy) variables \((\mathbf {v}_1,\mathbf {v}_2)\) and making the change of integration from \(\widehat{\varvec{\sigma }}\rightarrow -\widehat{\varvec{\sigma }}\). Accordingly, \(\mathbf {v}_1(\mathbf {v}_1'',\mathbf {v}_2'')\) must be relabeled to \(\mathbf {v}_1'(\mathbf {v}_1,\mathbf {v}_2)\) to obtain

$$\begin{aligned} \sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}\int {\mathrm {d}}\mathbf{v}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})&\chi (\mathbf {r},\mathbf {r}+\varvec{\sigma }) f(\mathbf {r},\mathbf {v}_1;t)\nonumber \\&\times f(\mathbf {r}+\varvec{\sigma },\mathbf {v}_2;t) \psi (\mathbf {v}_1'), \end{aligned}$$
(1.189)

where \(\mathbf {v}_1'=\mathbf {v}_1-\frac{1}{2}\left( 1+\alpha \right) (\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})\widehat{\varvec{\sigma }}\). With this result, Eq. (1.187) can be written in the form (1.84). The property of the collision integral (1.83) will be used in several calculations throughout the present monograph.

Appendix B

Some details of the derivation of cooling rate and collisional transfer contributions to pressure tensor and heat flux are provided in this Appendix. First, in accordance with the property of the Enskog collision integral, we have the identity

$$\begin{aligned} I_\psi= & {} \int {\mathrm {d}}\mathbf {v}_1 \psi (\mathbf {v}_1) J_{\text {E}}[\mathbf {r},\mathbf {v}_1|f,f]\nonumber \\= & {} \sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}\int {\mathrm {d}}\mathbf{v}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \left[ \psi (\mathbf {v}_1')-\psi (\mathbf {v}_1)\right] \nonumber \\&\times \, f_2(\mathbf {r},\mathbf {v}_1,\mathbf {r}+\varvec{\sigma },\mathbf {v}_2;t), \end{aligned}$$
(1.190)

where \(\psi (\mathbf {v})\) is an arbitrary function of velocity and

$$\begin{aligned} f_2(\mathbf {r}_1,\mathbf {v}_1,\mathbf {r}_2,\mathbf {v}_2;t)= \chi (\mathbf {r}_1,\mathbf {r}_2) f(\mathbf {r}_1, \mathbf {v}_1;t)f(\mathbf {r}_2, \mathbf {v}_2;t). \end{aligned}$$
(1.191)

Equation (1.190) can be written in an equivalent form by interchanging \(\mathbf {v}_1\) and \(\mathbf {v}_2\) and changing \(\widehat{\varvec{\sigma }}\rightarrow -\widehat{\varvec{\sigma }}\). The result is

$$\begin{aligned} I_\psi =&\; \sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}\int {\mathrm {d}}\mathbf{v}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \left[ \psi (\mathbf {v}_2')-\psi (\mathbf {v}_2)\right] \nonumber \\&\times f_2(\mathbf {r},\mathbf {v}_2,\mathbf {r}-\varvec{\sigma },\mathbf {v}_1;t). \end{aligned}$$
(1.192)

The combination of Eqs. (1.190) and (1.192) yields

$$\begin{aligned} I_\psi= & {} \frac{1}{2}\sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}\int {\mathrm {d}}\mathbf{v}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \left\{ \left[ \psi (\mathbf {v}_1')-\psi (\mathbf {v}_1)\right] \right. \nonumber \\&\times \, \left. f_2(\mathbf {r},\mathbf {v}_1,\mathbf {r}+\varvec{\sigma },\mathbf {v}_2;t)+\left[ \psi (\mathbf {v}_2')-\psi (\mathbf {v}_2)\right] f_2(\mathbf {r},\mathbf {v}_2,\mathbf {r}-\varvec{\sigma },\mathbf {v}_1;t)\right\} .\qquad \qquad \end{aligned}$$
(1.193)

To simplify Eq. (1.193), note first the relation

$$\begin{aligned} f_2(\mathbf {r},\mathbf {v}_2,\mathbf {r}-\varvec{\sigma },\mathbf {v}_1;t)= f_2(\mathbf {r}-\varvec{\sigma },\mathbf {v}_1,\mathbf {r},\mathbf {v}_2;t) \end{aligned}$$
(1.194)

and then arrange the terms to arrive at

$$\begin{aligned} I_\psi= & {} \frac{1}{2}\sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}\int {\mathrm {d}}\mathbf{v}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \left\{ \left[ \psi (\mathbf {v}_1')+\psi (\mathbf {v}_2')\right. \right. \nonumber \\&-\,\left. \psi (\mathbf {v}_1)-\psi (\mathbf {v}_2)\right] f_2(\mathbf {r},\mathbf {v}_1,\mathbf {r}+\varvec{\sigma },\mathbf {v}_2;t)+ \left[ \psi (\mathbf {v}_1')-\psi (\mathbf {v}_1)\right] \nonumber \\&\times \, \left. \left[ f_2(\mathbf {r},\mathbf {v}_1, \mathbf {r}+\varvec{\sigma },\mathbf {v}_2;t)- f_2(\mathbf {r}-\varvec{\sigma },\mathbf {v}_1,\mathbf {r},\mathbf {v}_2;t)\right] \right\} . \end{aligned}$$
(1.195)

The first term on the integrand represents a collisional effect due to scattering with a change in velocities. This term is analogous to the one obtained for a dilute gas, see Eq. (1.86). The second term is a pure collisional effect (it vanishes for dilute gases) due to the spatial difference of the colliding pair. This second effect provides collisional transfer contributions to momentum and heat fluxes. It can be written as a divergence through the identity

$$\begin{aligned} F(\mathbf {r},\mathbf {r}+\varvec{\sigma })- F(\mathbf {r}-\varvec{\sigma },\mathbf {r})= & {} -\int _{0}^{1}\; {\mathrm {d}}\lambda \; \frac{\partial }{\partial \lambda }F\big (\mathbf {r}-\lambda \varvec{\sigma },\mathbf {r}+ (1-\lambda ) \varvec{\sigma }\big ) \nonumber \\= & {} \frac{\partial }{\partial \mathbf {r}}\cdot \varvec{\sigma } \int _{0}^{1}\; {\mathrm {d}}\lambda \;F\big (\mathbf {r}-\lambda \varvec{\sigma },\mathbf {r}+ (1-\lambda ) \varvec{\sigma }\big ), \nonumber \\ \end{aligned}$$
(1.196)

where \(F(\mathbf {r}_1,\mathbf {r}_2)= f_2(\mathbf {r}_1, \mathbf {v}_1, \mathbf {r}_2, \mathbf {v}_2;t)\). Using this identity, Eq. (1.195) can be recast into the form

$$\begin{aligned} I_\psi= & {} \frac{1}{2}\sigma ^{d-1}\int {\mathrm {d}}\mathbf{v}_{1}\int {\mathrm {d}}\mathbf{v}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \nonumber \\&\times \, \bigg \{\left[ \psi (\mathbf {v}_1')+\psi (\mathbf {v}_2')-\psi (\mathbf {v}_1)-\psi (\mathbf {v}_2)\right] f_2(\mathbf {r},\mathbf {v}_1,\mathbf {r}+\varvec{\sigma },\mathbf {v}_2;t)\nonumber \\&+\, \nabla \cdot \varvec{\sigma } \left[ \psi (\mathbf {v}_1')-\psi (\mathbf {v}_1)\right] \int _0^{1} {\mathrm {d}}\lambda f_2\big (\mathbf {r}-\lambda \varvec{\sigma },\mathbf {v}_1, \mathbf {r}+(1-\lambda )\varvec{\sigma },\mathbf {v}_2;t\big )\bigg \}. \nonumber \\ \end{aligned}$$
(1.197)

In the case \(\psi (\mathbf {v})=m\mathbf {v}\), the first term in the integrand (1.197) disappears since the momentum is conserved in all pair collisions, i.e., \(\mathbf {v}_1'+\mathbf {v}_2'=\mathbf {v}_1+\mathbf {v}_2\). Thus, Eq. (1.197) for \(\psi (\mathbf {v})=m\mathbf {v}\) reduces to

$$\begin{aligned} I_p\equiv & {} \int {\mathrm {d}}\mathbf {v} \; m \mathbf {v}\; J_{\text {E}}[\mathbf {r},\mathbf {v}|f,f] \nonumber \\= & {} -\nabla \cdot \frac{1+\alpha }{4}m\sigma ^{d} \int {\mathrm {d}}\mathbf {v}_1\int {\mathrm {d}}\mathbf {v}_2 \int {\mathrm {d}}\widehat{\varvec{\sigma }}\,\varTheta (\widehat{{\varvec{\sigma }}} \cdot \mathbf {g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})^2\widehat{\varvec{\sigma }} \widehat{\varvec{\sigma }} \nonumber \\&\times \, \int _{0}^{1}\; {\mathrm {d}}\lambda \; f_2\left( \mathbf {r}-\lambda \varvec{\sigma },\mathbf {v}_1, \mathbf {r}+(1-\lambda )\varvec{\sigma },\mathbf {v}_2;t\right) , \end{aligned}$$
(1.198)

where use has been made of the scattering law (1.52). According to the momentum balance equation (1.89), the divergence of the collisional transfer part \(\mathsf {P}^{\text {c}}\) is defined by

$$\begin{aligned} I_p=-\nabla \cdot \mathsf {P}^{\text {c}}. \end{aligned}$$
(1.199)

The explicit form (1.86) for \(\mathsf {P}^{\text {c}}\) may be easily identified after comparing Eqs. (1.198) and (1.199).

The case of kinetic energy \(\psi =\frac{1}{2}mv^2\) can be analyzed in a similar way except that energy is not conserved in collisions. This means that the first term on the right side of Eq. (1.197) does not disappear. As before, the second term on the right side of Eq. (1.198) gives the collisional transfer contribution to the heat flux. After some simple algebra, we see that

$$\begin{aligned} I_e&\equiv \int {\mathrm {d}}\mathbf {v} \frac{1}{2}m \mathbf {v}^2\; J_{\text {E}}[\mathbf {r},\mathbf {v}|f,f]\nonumber \\&=\;-\frac{(1-\alpha ^2)}{8}m\sigma ^{d-1}\int {\mathrm {d}}\mathbf {v}_1\int {\mathrm {d}}\mathbf {v}_2 \int d\widehat{\varvec{\sigma }}\,\varTheta (\widehat{{\varvec{\sigma }}} \cdot \mathbf {g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})^3 \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \times f_2(\mathbf {r},\mathbf {v}_1,\mathbf {r}+\varvec{\sigma },\mathbf {v}_2;t) \nonumber \\&-\,\nabla \cdot m\sigma ^{d}\frac{1+\alpha }{4}\int {\mathrm {d}}\mathbf {v}_1\int {\mathrm {d}}\mathbf {v}_2 \int {\mathrm {d}}\widehat{\varvec{\sigma }}\,\varTheta (\widehat{{\varvec{\sigma }}} \cdot \mathbf {g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})^2 \widehat{\varvec{\sigma }} \bigg [\frac{1-\alpha }{4}(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \nonumber \\&+\, \widehat{\varvec{\sigma }}\cdot \mathbf{G}_{12} +\widehat{\varvec{\sigma }}\cdot \mathbf{U}\bigg ] \int _{0}^{1}\; {\mathrm {d}}\lambda \; f_2\big (\mathbf {r}-\lambda \varvec{\sigma },\mathbf {v}_1, \mathbf {r}+(1-\lambda )\varvec{\sigma },\mathbf {v}_2;t\big ), \end{aligned}$$
(1.200)

bearing in mind that \(\mathbf {G}_{12}=(\mathbf {V}_1+\mathbf {V}_2)/2\) and \(\mathbf {V}=\mathbf {v}-\mathbf {U}\). Upon deriving Eq. (1.200), use has been made of Eq. (1.8) and the relation

$$\begin{aligned} v_1^2-v_1^{'2}=\frac{1-\alpha ^2}{4}(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})^2+(1+\alpha ) (\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})\left[ (\widehat{\varvec{\sigma }}\cdot \mathbf{G}_{12}) +(\widehat{\varvec{\sigma }}\cdot \mathbf{U})\right] . \end{aligned}$$
(1.201)

Moreover, note that the second term on the right-hand side of Eq. (1.200) (the one that is proportional to \(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}\)) vanishes by symmetry. The balance energy equation (1.91) yields

$$\begin{aligned} \int {\mathrm {d}}\mathbf{v} \frac{m}{2} (\mathbf{v}-\mathbf{U})^2 J_{\text {E}}[\mathbf {r},\mathbf{v}|f,f]=-\nabla \cdot \mathbf{q}^{\text {c}} -\mathsf {P}^{\text {c}}:\nabla \mathbf{U}-\frac{d}{2}nT \zeta , \end{aligned}$$
(1.202)

where \(\mathbf{q}^{\text {c}}\) is the collisional contribution to the heat flux and \(\zeta \) is the cooling rate. Comparing Eqs. (1.200) and (1.202) and taking into account Eq. (1.199), we can finally obtain the expressions (1.98) and (1.93) for \(\mathbf{q}^{\text {c}}\) and \(\zeta \), respectively.

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Garzó, V. (2019). Kinetic Theory of Inelastic Hard Spheres. In: Granular Gaseous Flows. Soft and Biological Matter. Springer, Cham. https://doi.org/10.1007/978-3-030-04444-2_1

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