Navier–Stokes Transport Coefficients for Monocomponent Granular Gases. I. Theoretical Results

Abstract

A normal solution to the revised Enskog kinetic theory of smooth monocomponent granular gases is obtained via the Chapman–Enskog method for states close to the local homogeneous cooling state. The analysis is performed to first-order in spatial gradients, allowing the identification of Navier–Stokes transport coefficients associated with heat and momentum fluxes along with the first-order contribution to the cooling rate. The transport coefficients are determined from the solution to a set of coupled linear integral equations analogous to those for elastic collisions. These integral equations are solved by using different approximate methods that yield explicit expressions for the transport coefficients in terms of the coefficient of restitution and the solid volume fraction. Finally, the results obtained from the Chapman–Enskog method are compared against those derived from different approaches.

Appendix A

Some technical details concerning evaluation of some of the collision integrals appearing in the chapter are provided in this Appendix. First, we consider the integral

$$\begin{aligned} \int \; \mathrm {d}\mathbf{V}\; R_{ij}(\mathbf{V}) \mathcal {K}_i\left[ \frac{\partial f^{(0)}}{\partial V_j} \right] . \end{aligned}$$

(3.121)

This integral is involved in the determination of the kinetic contribution $\eta _\text {k}$ to the shear viscosity coefficient. According to the definition (3.27) of the operator $\mathcal {K}_i$, the integral (3.121) is

$$\begin{aligned} \int \; \mathrm {d}\mathbf {V}\; R_{ij}(\mathbf {V}) \mathcal {K}_i\left[ \frac{\partial f^{(0)}}{\partial V_j}\right]= & {} \sigma ^{d}\chi \int \mathrm {d}\mathbf {V}_{1}\int \mathrm {d}\mathbf {V}_{2}\;R_{ij}(\mathbf {V}_1)\; \int \mathrm {d}\widehat{\varvec{\sigma }}\Theta (\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})\widehat{\sigma }_i\nonumber \\&\times \left[ \alpha ^{-2}f^{(0)}(\mathbf {V} _{1}'')\frac{\partial f^{(0)}(\mathbf {V}_2'')}{\partial V_{2j}''} +f^{(0)}(\mathbf {V}_{1})\frac{\partial f^{(0)}(\mathbf {V}_2)}{\partial V_{2j}}\right] . \nonumber \\ \end{aligned}$$

(3.122)

As we saw in previous chapters, a simpler form of this integral is obtained by changing variables to integrate over $\mathbf {V}_1''$ and $\mathbf {V}_2''$ instead of $\mathbf {V}_1$ and $\mathbf {V}_2$ in the first term of Eq. (3.122). According to the relations (1.10) and (1.11), the Jacobian of the transformation is $\alpha$ and $\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12}=-\alpha \widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12}''$. Also, $\mathbf {V}_1(\mathbf {V}_1'',\mathbf {V}_2'')= \mathbf {V}_1'=\mathbf {V}_1-\frac{1}{2}(1+\alpha )(\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12})\widehat{\varvec{\sigma }}$. The integral (3.122) then becomes

$$\begin{aligned} \int \mathrm {d}\mathbf {V} R_{ij}(\mathbf {V}) \mathcal{K}_i \left[ \frac{\partial f^{(0)}}{\partial V_j}\right]= & {} -\chi \sigma ^d\int \mathrm {d}\mathbf {V}_{1}\int \mathrm {d}\mathbf {V}_{2}\;f^{(0)}(\mathbf {V}_1) \frac{\partial f^{(0)}(\mathbf {V}_2)}{\partial V_{2j}}\nonumber \\&\times \int \;\mathrm {d}\widehat{\varvec{\sigma }}\Theta (\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})\widehat{\sigma }_i\left[ R_{ij}(\mathbf {V}_1')-R_{ij}(\mathbf {V}_1)\right] . \nonumber \\ \end{aligned}$$

(3.123)

The scattering rules (1.4a) and (1.4b) give

$$\begin{aligned} R_{ij}(\mathbf {V}_1')-R_{ij}(\mathbf {V}_1)= & {} \frac{1}{4}m(1+\alpha )(\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12})\left[ (1+\alpha )(\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12})\left( \widehat{\sigma }_i\widehat{\sigma }_j-\frac{\delta _{ij}}{d}\right) \right. \nonumber \\&\left. -2\left( V_{1i}\widehat{\sigma }_j+V_{1j}\widehat{\sigma }_i \right) +\frac{4}{d}\delta _{ij}(\widehat{\varvec{\sigma }} \cdot \mathbf {V}_{1})\right] , \end{aligned}$$

(3.124)

and so Eq. (3.123) reads

$$\begin{aligned}&\int \mathrm {d}\mathbf {V} R_{ij}(\mathbf {V}) \mathcal{K}_i \left[ \frac{\partial f^{(0)}}{\partial V_j}\right] =m\chi \frac{1+\alpha }{2d} \sigma ^d \int \mathrm {d}\mathbf {V}_{1}\int \mathrm {d}\mathbf {V}_{2}\;f^{(0)}(\mathbf {V}_1) \frac{\partial f^{(0)}(\mathbf {V}_2)}{\partial V_{2j}}\nonumber \\&\; \; \times \, \int \;\mathrm {d}\widehat{\varvec{\sigma }}\Theta (\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12}) (\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})^2\Bigg \{d \; V_{1j}+\widehat{\sigma }_j\Big [(d-2)(\widehat{\varvec{\sigma }} \cdot \mathbf {V}_{1})\nonumber \\&\; \; -\,\frac{d-1}{2}(1+\alpha )(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})\Big ]\Bigg \}=m\chi \frac{d-1}{2d} \sigma ^d (1+\alpha )\int \mathrm {d}\mathbf {V}_{1}\int \mathrm {d}\mathbf {V}_{2}\;f^{(0)}(\mathbf {V}_1) \nonumber \\&\; \; \times \, f^{(0)}(\mathbf {V}_2) \int \;\mathrm {d}\widehat{\varvec{\sigma }}\Theta (\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12}) \left[ 4(\widehat{\varvec{\sigma }} \cdot \mathbf {V}_{1})-\frac{3}{2}(1+\alpha ) (\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12})\right] \nonumber \\= & {} m\chi \frac{d-1}{2d} B_2 \sigma ^d (1+\alpha )\int \mathrm {d}\mathbf {V}_{1}\int \mathrm {d}\mathbf {V}_{2}\;f^{(0)}(\mathbf {V}_1) f^{(0)}(\mathbf {V}_2)\nonumber \\&\times \, \left[ 4(\mathbf {V}_1\cdot \mathbf {g}_{12})-\frac{3}{2}(1+\alpha )g_{12}^2\right] \nonumber \\= & {} 2^{d-2}(d-1)\chi \phi (1+\alpha )(1-3\alpha )n T. \nonumber \\ \end{aligned}$$

(3.125)

In the integration over the angle $\widehat{\varvec{\sigma }}$ use has been made of the result (2.139).

The integrals involving the operator $\varvec{\mathcal {K}}$ in evaluation of the heat flux transport coefficients can be computed in a similar way. Thus, the integral (3.69) in particular is

$$\begin{aligned} \int \mathrm {d}\mathbf {V} \mathbf {S}(\mathbf {V}) \cdot \varvec{\mathcal {K}} \left[ f^{(0)}\right]= & {} -\chi \sigma ^{d-1}\int \mathrm {d}\mathbf {V}_{1}\int \mathrm {d}\mathbf {V}_{2}\;f^{(0)}(\mathbf {V}_1) f^{(0)}(\mathbf {V}_2)\nonumber \\&\times \,\int \;\mathrm {d}\widehat{\varvec{\sigma }}\Theta (\widehat{\varvec{\sigma }} \cdot \mathbf {g}_{12})(\widehat{\varvec{\sigma }}\cdot \mathbf {g}_{12})\varvec{\sigma }\cdot \left[ \mathbf {S}(\mathbf {V}_1')-\mathbf {S}(\mathbf {V}_1)\right] , \nonumber \\ \end{aligned}$$

(3.126)

where

$$\begin{aligned} \mathbf{S}(\mathbf{V}_{1}')-\mathbf{S}(\mathbf{V}_{1})= & {} \frac{m}{4}(1+\alpha )(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})\Bigg \{ \bigg [ \frac{1-\alpha ^{2}}{4}(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})^{2}-G_{12}^{2}-\frac{1}{4}g_{12}^{2}-(\mathbf{g}_{12}\cdot \mathbf{G}_{12})\nonumber \\&+\,(1+\alpha ) (\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12})(\widehat{\varvec{\sigma }} \cdot \mathbf{G}_{12})+(d+2)\frac{T}{m}\bigg ] \widehat{\varvec{\sigma }} -\bigg [\frac{1-\alpha }{2}(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}) \nonumber \\&+\,2(\widehat{\varvec{\sigma }}\cdot \mathbf{G}_{12})\bigg ] \mathbf{G}_{12} -\bigg [\frac{1-\alpha }{4}(\widehat{\varvec{\sigma }} \cdot \mathbf{g}_{12})+(\widehat{\varvec{\sigma }}\cdot \mathbf{G}_{12}) \bigg ] \mathbf{g}_{12}\Bigg \}. \end{aligned}$$

(3.127)

Here, $\mathbf {G}_{12}=(\mathbf {V}_1+\mathbf {V}_2)/2$. Substitution of Eq. (3.127) into Eq. (3.126) and integration over the solid angle leads to the expression (3.69). The result (3.68) may be derived by following similar steps.

Now the (reduced) collision frequency $\nu _\eta ^*$ is considered. According to the definition (3.60) and the scattering rule (3.124), we obtain

$$\begin{aligned} \nu _\eta ^*= & {} \frac{(2d+3-3\alpha )(1+\alpha )}{2(d+2)(d-1)(d+3)}\frac{m}{nT^2\nu _0}\sigma ^{d-1}B_3\int \mathrm {d}\mathbf {V}_1 \int \mathrm {d}\mathbf {V}_2\; f^{(0)}(\mathbf {V}_1) f_\text {M}(\mathbf {V}_2) \nonumber \\&\times \, R_{ij}(\mathbf {V}_2)g_{12} g_{12i}g_{12j}, \end{aligned}$$

(3.128)

where $\nu _0= n T/\eta _0$ and the standard first Sonine approximation (3.88) for $\mathcal{C}_{ij}$ has been considered. Additionally, use has been made of Eqs. (2.139) and (2.148). The leading Sonine approximation (3.76) to $f^{(0)}$ can be written as $f^{(0)}(\mathbf {V})=n \upsilon _\text {th}^{-d} \varphi (\mathbf {c})$, where $\varphi (\mathbf {c})=\pi ^{-d/2}\lim _{\epsilon _1\rightarrow 1}(1+\varXi _1)\mathrm {e}^{-\epsilon _1 c^2}$ and

$$\begin{aligned} \varXi _i = \frac{a_2}{2}\left[ \frac{\partial ^2}{\partial \epsilon _i^2}+(d+2)\frac{\partial }{\partial \epsilon _i}+ \frac{d(d+2)}{4}\right] . \end{aligned}$$

(3.129)

Substitution of the above approximation into Eq. (3.128) yields

$$\begin{aligned} \nu _\eta ^*=\frac{\varGamma \left( \frac{d}{2}\right) }{\varGamma \left( \frac{d+3}{2}\right) }\frac{(2d+3-3\alpha )(1+\alpha )}{2\sqrt{2}(d-1)(d+3)}\lim _{\epsilon _1 \rightarrow 1}\left( 1+\varXi _1\right) I_\eta (\epsilon _1), \end{aligned}$$

(3.130)

where

$$\begin{aligned} I_\eta (\epsilon _1)= \pi ^{-d}\int \mathrm {d}\mathbf {c}_1 \int \mathrm {d}\mathbf {c}_2 \mathrm {e}^{-(\epsilon _1 c_1^2+c_2^2)} \left( c_{2i}c_{2j}-\frac{1}{d}c_2^2 \delta _{ij} \right) g_{12}^* g_{12i}^*g_{12j}^* \end{aligned}$$

(3.131)

and $\mathbf {g}_{12}^*=\mathbf {c}_1-\mathbf {c}_2$. The integral $I_\eta$ can be performed by the change of variables (2.133), i.e., $\mathbf {x}=\mathbf {c}_1-\mathbf {c}_2$ and $\mathbf {y}=\epsilon _1 \mathbf {c}_1+\mathbf {c}_2$. With this change, we obtain

$$\begin{aligned} I_\eta (\epsilon _1)=\frac{d-1}{d}\frac{\varGamma \left( \frac{d+5}{2}\right) }{\varGamma \left( \frac{d}{2}\right) }\epsilon _1^{-d/2} \left( \frac{1+\epsilon _1}{\epsilon _1}\right) ^{1/2}. \end{aligned}$$

(3.132)

Use of (3.131) in Eq. (3.129) leads to the result displayed in Table 3.1. The expression of $\nu _{\kappa }^*$ can be obtained by following similar steps to those made for $\nu _{\eta }^*$.

In the modified first Sonine approximation, $\nu _\eta ^*$ is given by

$$\begin{aligned} \nu _\eta ^*= & {} \frac{(2d+3-3\alpha )(1+\alpha )}{2(d+2)(d-1)(d+3)}\frac{1}{1+a_2}\frac{m}{nT^2\nu _0}\sigma ^{d-1}B_3\int \mathrm {d}\mathbf {V}_1 \int \mathrm {d}\mathbf {V}_2\; f^{(0)}(\mathbf {V}_1) f^{(0)}(\mathbf {V}_2) \nonumber \\&\times \, R_{ij}(\mathbf {V}_2)g_{12} g_{12i}g_{12j}. \end{aligned}$$

(3.133)

To compute this integral, the leading Sonine approximation (3.85) for $f^{(0)}$ is employed, as before. By using the operators defined by Eq. (3.129), the integral (3.133) can be rewritten as

$$\begin{aligned} \nu _\eta ^*=\frac{\varGamma \left( \frac{d}{2}\right) }{\varGamma \left( \frac{d+3}{2}\right) }\frac{(2d+3-3\alpha )(1+\alpha )}{2\sqrt{2}(d-1)(d+3)}\frac{1}{1+a_2}\lim _{\epsilon _1 \rightarrow 1}\lim _{\epsilon _2 \rightarrow 1} (1+\varXi _1+\varXi _2)F_{\eta }(\epsilon _1,\epsilon _2), \end{aligned}$$

(3.134)

where

$$\begin{aligned} F_\eta (\epsilon _1,\epsilon _2)= \pi ^{-d}\int \mathrm {d}\mathbf {c}_1 \int \mathrm {d}\mathbf {c}_2 \mathrm {e}^{-(\epsilon _1 c_1^2+\epsilon _2 c_2^2)} \left( c_{2i}c_{2j}-\frac{1}{d}c_2^2 \delta _{ij} \right) g_{12}^* g_{12i}^*g_{12j}^*. \end{aligned}$$

(3.135)

In the numerator of Eq. (3.134) nonlinear terms in $a_2$ have, as usual, been neglected, so $(1+\varXi _1)(1+\varXi _2)\rightarrow 1+\varXi _1+\varXi _2$. Note that the standard first Sonine approximation to $\nu _\eta ^*$ is recovered when $\varXi _2=0$ and $a_2=0$ in the numerator and denominator, respectively, of Eq. (3.134). The integral $F_\eta (\epsilon _1,\epsilon _2)$ can easily be performed by introducing the variables $\mathbf {x}=\mathbf {c}_1-\mathbf {c}_2$ and $\mathbf {y}=\epsilon _1 \mathbf {c}_1+\epsilon _2 \mathbf {c}_2$, with the Jacobian $(\epsilon _1+\epsilon _2)^{-d}$. After some algebra, we obtain

$$\begin{aligned} F_\eta (\epsilon _1,\epsilon _2)=\frac{d-1}{d}\frac{\varGamma \left( \frac{d+5}{2}\right) }{\varGamma \left( \frac{d}{2}\right) } \left( \epsilon _1\epsilon _2\right) ^{-(d+5)/2}\epsilon _1^2 \left( \epsilon _1+\epsilon _2\right) ^{1/2}. \end{aligned}$$

(3.136)

As expected, $F_\eta (\epsilon _1,1)=I_\eta (\epsilon _1)$. The expression for $\nu _\eta ^*$ provided in Table 3.1 follows directly from Eqs. (3.134) and (3.136) after expanding the term $(1+a_2)^{-1}$ and neglecting nonlinear terms in $a_2$. Similarly, the expression for $\nu _\kappa ^*$ in the modified first Sonine approximation can be derived by using similar mathematical steps. Its explicit form is given in Table 3.1.