## Appendix A High Order of Moments and Source Terms in Equation (12)  
(1). Algebraic expression for high order of moments \(f_{ijkl},\ \psi_{ij},\ \Omega_{i}\) in terms of derivatives of lower moments:  
\[\left\{\begin{array}{ll}\phi_{ijkl}&=-\frac{4\mu}{A_{I}\rho}\frac{\partial m_{ cijk}}{\partial x_{l_{0}}}-\frac{4\mu}{A_{I}\rho}\bigg{[}\frac{3}{7}R_{< ij}\frac{\partial u_{k}}{\partial x_{l_{0}}}-3 RT\sigma_{<ij}\frac{\partial u_{k}}{ \partial x_{l_{0}}}+m_{<ijk}\frac{\partial RT}{\partial x_{l_{0}}}-\frac{m_{< ijk}}{\rho}\bigg{(}\frac{\partial c_{l_{0}m}}{\partial x_{m}}+\frac{\partial p}{ \partial x_{l_{0}}}\bigg{)}\\ &-\frac{A_{2}}{\mathcal{A}_{I}}\frac{\partial c_{i}\phi_{jkl}}{\rho},\\ \psi_{ijk}&=-\frac{2\mu}{A_{I}\rho}\frac{\partial R_{<ij}}{\partial x_{l_{0}}} -\frac{2\mu}{A_{I}\rho}\bigg{(}\frac{28}{9}q_{<ij}\frac{\partial u_{j}}{ \partial x_{l_{0}}}-R_{<ij}\frac{\partial\ln\rho}{\partial x_{l_{0}}}+\frac{R _{<ij}+7 RT\sigma_{<ij}}{RT}\frac{\partial RT}{\partial x_{l_{0}}}\bigg{)}\\ &-\frac{A_{0}q_{<ij}\partial x_{l_{0}}+A_{0}\bar{\partial}_{<}\ln\mu_{<}}{A_{ \Delta}\rho},\\ \Omega_{i}&=-\frac{7}{3}\frac{\mu}{\alpha A_{I}\rho}\frac{\partial A}{\partial x _{i}}+\frac{28\mu}{A_{\Delta}\rho}\bigg{[}\frac{3\phi}{\partial x_{m}}\big{(} \frac{\partial u_{m}}{\partial x_{m}}+\sigma_{ml}\frac{\partial u_{m}}{ \partial x_{l}}\bigg{)}-\Big{(}\frac{4}{9}\frac{\partial u_{<ij}}{\partial x_ {l_{0}}}+\sigma_{ik}\frac{\partial RT}{\partial x_{k}}\Big{)}\bigg{]}\\ &-\frac{A_{0}q_{<ik}+A_{\Delta}\eta_{ij}\alpha_{I}}{A_{\Omega}\rho}.\end{array}\right. \tag{12}\]  
(2). Expression for the source terms \(\mathfrak{M}_{ijk},\ \mathfrak{R}_{ij},\ \mathfrak{N}\):  
\[\left\{\begin{array}{ll}\mathfrak{M}_{ijk}&=-3\sigma_{<ij}\frac{\partial RT }{\partial x_{l_{0}}}+\frac{3\sigma_{<ij}}{\rho}\bigg{(}\frac{\partial p}{ \partial x_{l_{0}}}+\frac{\partial x_{l_{0}}}{\partial x_{m}}\bigg{)}-\frac{1 }{5}q_{<i}\frac{\partial u_{j}}{\partial x_{l_{0}}}-3m_{<ij}\frac{\partial u_{ l_{0}}}{\partial x_{m}},\\ \mathfrak{R}_{ij}&=-A_{R}\frac{p}{\mu}\frac{\partial_{<i}\rho\lambda_{<k}}{ \mu}+\frac{8}{\partial RT\sigma_{i}-\frac{9}{2}R_{j}}\frac{\partial u_{k}}{ \partial x_{k}}-\frac{4}{9}(RT\sigma_{<ki}+R_{ki})\bigg{(}\frac{\partial u_{j}} {\partial x_{k}}+\frac{\partial u_{k}}{\partial x_{j}}\bigg{)}\\ &-2R_{ki}\frac{\partial u_{j}}{\partial x_{k}}+\frac{28}{5}\frac{\partial}{ \rho}\frac{\partial\partial p_{<k}}{\partial x_{k}}+\frac{28}{5}RTq_{<i}\bigg{(} \frac{\partial p\lambda_{>}}{\partial x_{l_{0}}}-2\frac{\partial T}{\partial x _{j_{0}}}\bigg{)}+2\frac{m_{i}\mu}{\rho}\frac{\partial u_{k}}{\partial x_{l}} \\ &+m_{ij}\bigg{(}\frac{2}{\rho}\frac{\partial p}{\partial x_{k}}-9\frac{\partial RT }{\partial x_{k}}\bigg{)}+\frac{14}{3}\frac{\sigma_{ij}}{\rho}\bigg{(}\frac{ \partial q_{m}}{\partial x_{m}}+\sigma_{ml}\frac{\partial u_{m}}{\partial x_{ l}}\bigg{)}-\frac{14}{15}\Delta\frac{\partial u_{ci}}{\partial x_{j_{0}}}-2f_{ ijkl}\frac{\partial u_{k}}{\partial x_{l}},\\ \mathbf{N}&=-A_{\Delta}\frac{p}{\mu}\frac{\partial_{ki}\rho}{\rho}-\frac{4}{3} \Delta\frac{\partial u_{l_{0}}}{\partial x_{k}}-4(2 RT\sigma_{kl}+R_{kl})\frac{ \partial u_{l_{0}}}{\partial x_{l}}+8\frac{4\bar{\alpha}_{l}}{\rho}\Big{(}\frac {\partial\phi_{l}}{\partial x_{l_{0}}}\Big{)}+RTq_{k}\Big{(}8\frac{\partial p }{\partial x_{k}}-28\frac{\partial T}{T\partial x_{k}}\bigg{)}.\end{array}\right. \tag{13}\]