2.1 Macroscopic quantities

The macroscopic properties such as mass density, momentum, energy and pressure are moments of the distribution function. Other moments that have physical interpretations are the pressure tensor and heat flux vector. General moment definition based on the trace free part of the central moment is

(2.4) $$\begin{eqnarray}u_{i_{1}\ldots i_{n}}^{\unicode[STIX]{x1D70D},A}=m\iint (I^{2/\unicode[STIX]{x1D6FF}})^{A}C^{2\unicode[STIX]{x1D70D}}C_{{<}i_{1}}C_{i_{2}}\ldots C_{i_{n}>}f\,\text{d}\boldsymbol{c}\,\text{d}I,\end{eqnarray}$$

where

(2.5) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}A=0,1,2,3,\ldots \\ \unicode[STIX]{x1D70D}=0,1,2,3,\ldots .\end{array}\right\}\end{eqnarray}$$

The basic and most important moments are:

(2.6a ) $$\begin{eqnarray}\displaystyle \text{Density }\unicode[STIX]{x1D70C} & = & \displaystyle m\iint f\,\text{d}\boldsymbol{c}\,\text{d}I=\int \unicode[STIX]{x1D70C}_{I}\,\text{d}I=u^{0,0},\end{eqnarray}$$

(2.6b ) $$\begin{eqnarray}\displaystyle \text{Velocity }\unicode[STIX]{x1D70C}v_{i} & = & \displaystyle m\iint c_{i}\,f\,\text{d}\boldsymbol{c}\,\text{d}I\quad \text{or}\quad 0=m\iint C_{i}\,f\,\text{d}\boldsymbol{c}\,\text{d}I=u_{i}^{0,0},\quad \qquad\end{eqnarray}$$

(2.6c ) $$\begin{eqnarray}\displaystyle \text{Stress }\unicode[STIX]{x1D748}_{ij} & = & \displaystyle m\iint C_{{<}i}C_{j>}f\,\text{d}\boldsymbol{c}\,\text{d}I=\boldsymbol{u}_{ij}^{0,0},\end{eqnarray}$$

(2.6d ) $$\begin{eqnarray}\displaystyle \text{Translational energy }\unicode[STIX]{x1D70C}u_{tr} & = & \displaystyle \frac{3}{2}p=m\iint \frac{C^{2}}{2}f\,\text{d}\boldsymbol{c}\,\text{d}I=\frac{1}{2}u^{1,0},\end{eqnarray}$$

(2.6e ) $$\begin{eqnarray}\displaystyle \text{Internal energy }\unicode[STIX]{x1D70C}u_{int} & = & \displaystyle m\iint I^{2/\unicode[STIX]{x1D6FF}}f\,\text{d}\boldsymbol{c}\,\text{d}I=\int I^{2/\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D70C}_{I}\,\text{d}I=u^{0,1},\end{eqnarray}$$

(2.6f ) $$\begin{eqnarray}\displaystyle \text{Translational heat flux }\boldsymbol{q}_{i,tr} & = & \displaystyle m\iint C_{i}\frac{C^{2}}{2}f\,\text{d}\boldsymbol{c}\,\text{d}I=\frac{1}{2}u_{i}^{1,0},\end{eqnarray}$$

(2.6g ) $$\begin{eqnarray}\displaystyle \text{Internal heat flux }\boldsymbol{q}_{i,int} & = & \displaystyle m\iint C_{i}I^{2/\unicode[STIX]{x1D6FF}}f\,\text{d}\boldsymbol{c}\,\text{d}I=u_{i}^{0,1}.\end{eqnarray}$$

$C_{i}=c_{i}-v_{i}$

$\unicode[STIX]{x1D70C}_{I}=m\int f\,\text{d}\boldsymbol{c}$

$u_{tr}$

$u_{int}$

$\boldsymbol{q}_{i,tr}$

$\boldsymbol{q}_{i,int}$

Regarding the energies, $u_{tr}$ and $u_{int}$ , the classical equipartition theorem (Reif Reference Reif2009) states that in full thermal equilibrium, and with $\unicode[STIX]{x1D6FF}$ fully excited internal DoF, each DoF contributes an energy of $(1/2)\unicode[STIX]{x1D703}$ to the energy of a particle, where $\unicode[STIX]{x1D703}=(k_{b}/m)T$ is temperature in specific energy units. Therefore the energy in equilibrium, for fully excited internal DoF, can be written as

(2.7) $$\begin{eqnarray}u=u_{tr|E}+u_{int|E}=\left(\frac{3}{2}+\frac{\unicode[STIX]{x1D6FF}}{2}\right)\unicode[STIX]{x1D703},\end{eqnarray}$$

where $u_{tr|E}=(3/2)\unicode[STIX]{x1D703}$ is the contribution from translational energy and $u_{int|E}=(\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D703}$ is the contribution from the internal DoF.

In non-equilibrium processes, translational energy and internal energies cannot be described by a single temperature, rather at least two distinct temperatures are required. Furthermore, the equipartition theorem as stated above is only valid when the internal DoF are either fully excited or frozen. Typically, for polyatomic gases, some internal DoF are only partially excited, with the degree of excitation determined by temperature. Therefore, $\unicode[STIX]{x1D6FF}$ is not an integer, but a continuous function of temperature, that has integer values only for fully excited internal DoF. For convenience we introduce the translational temperature $\unicode[STIX]{x1D703}_{tr}$ and the internal temperature $\unicode[STIX]{x1D703}_{int}$ which are defined through the energies as

(2.8a,b ) $$\begin{eqnarray}u_{tr}=\frac{3}{2}\unicode[STIX]{x1D703}_{tr}\quad \text{and}\quad u_{int}=\frac{\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703}_{int})}{2}\unicode[STIX]{x1D703}_{int}.\end{eqnarray}$$

We note that these temperatures are not measurable temperatures as in the equilibrium case, but rather convenient quantities to use instead of the energies $u_{tr}$ and $u_{int}$ . In thermal equilibrium, both temperatures agree, and are equal to the thermodynamic temperature, $\unicode[STIX]{x1D703}_{tr}=\unicode[STIX]{x1D703}_{int}=\unicode[STIX]{x1D703}$ . With these definitions, the ideal gas law in non-equilibrium reads $p=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}_{tr}$ .

Just as we defined temperatures for the energies $u_{tr}$ and $u_{int}$ , we can define an overall temperature $\unicode[STIX]{x1D703}$ for the total energy. Similar to $\unicode[STIX]{x1D703}_{tr}$ and $\unicode[STIX]{x1D703}_{int}$ , the temperature $\unicode[STIX]{x1D703}$ is a measure for energy, and not a measurable temperature. By definition, in equilibrium $\unicode[STIX]{x1D703}$ is the thermodynamic temperature of the gas

(2.9) $$\begin{eqnarray}u=\left(\frac{3}{2}+\frac{\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703})}{2}\right)\unicode[STIX]{x1D703}=\frac{3}{2}\unicode[STIX]{x1D703}_{tr}+\frac{\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703}_{int})}{2}\unicode[STIX]{x1D703}_{int}.\end{eqnarray}$$

The function $\unicode[STIX]{x1D6FF}$ can be found from measurements of specific heat in equilibrium as

(2.10) $$\begin{eqnarray}C_{v}=\frac{\text{d}u}{\text{d}\unicode[STIX]{x1D703}}=\frac{3+\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703})+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703})}{\displaystyle \text{d}\unicode[STIX]{x1D703}}}{2}.\end{eqnarray}$$

We will use third-order polynomial function for constant volume specific heat,

(2.11) $$\begin{eqnarray}C_{v}=C_{0}+C_{1}\unicode[STIX]{x1D703}+C_{2}\unicode[STIX]{x1D703}^{2}+C_{3}\unicode[STIX]{x1D703}^{3};\end{eqnarray}$$

the coefficients $C_{\unicode[STIX]{x1D6FC}}$ can be determined from tabulated data, and are available in databases (Borgnakke & Sonntag Reference Borgnakke and Sonntag2009). From (2.10) and (2.11) we find $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703})$ as a polynomial of the same order,

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703})=2C_{0}-3+C_{1}\unicode[STIX]{x1D703}+\frac{2}{3}C_{2}\unicode[STIX]{x1D703}^{2}+\frac{C_{3}}{2}\unicode[STIX]{x1D703}^{3}.\end{eqnarray}$$

From the definition of the overall temperature (2.9) we find a relation for $\unicode[STIX]{x1D6FF}$ evaluated at $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D703}_{int}$ , $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703})=\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703}_{int})(\unicode[STIX]{x1D703}_{int}/\unicode[STIX]{x1D703})-(3/2)((\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{tr})/\unicode[STIX]{x1D703})$ . From now on, we shall consider mainly states not far from equilibrium, so that $\unicode[STIX]{x1D703}_{int}/\unicode[STIX]{x1D703}\backsimeq 1\gg (\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{tr})/\unicode[STIX]{x1D703}$ holds. For convenience, we shall ignore the small difference between $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703}_{int})$ and $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703})$ , and will have only the latter appear in our equations.

In equilibrium the three temperatures agree, $\unicode[STIX]{x1D703}_{tr|E}=\unicode[STIX]{x1D703}_{int|E}=\unicode[STIX]{x1D703}$ , while in non-equilibrium they will differ. We define the non-equilibrium part of the temperature, named the dynamic temperature, as

(2.13) $$\begin{eqnarray}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{tr}.\end{eqnarray}$$

The first 36 raw moments are listed as,

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{36}=\{\unicode[STIX]{x1D70C},v_{i},\unicode[STIX]{x1D703},\unicode[STIX]{x0394}\unicode[STIX]{x1D703},\unicode[STIX]{x1D748}_{ij},\boldsymbol{q}_{i,tr},\boldsymbol{q}_{i,int},\boldsymbol{u}_{ij}^{1,0},u^{2,0},\boldsymbol{u}_{ij}^{0,1},u^{1,1},\boldsymbol{u}_{ijk}^{0,0}\}.\end{eqnarray}$$

2.2 Generalized S-model

The state of molecules and the distribution function will be changed by interaction between molecules (collisions). Different exchange processes occur on different characteristic time scales. In all collisions, the translational energy is exchanged between particles. However, only in some of the collisions is internal energy exchanged as well. These differences, and their relation to the macroscopic time scale, is a key feature for defining the state of a gas as being in non-equilibrium or equilibrium. The macroscopic time scale, which is the time needed for any changes to happen in the dynamics of the gas, is the reference time scale here. In cases when there are two different microscopic characteristic time scales, one smaller than and one comparable to, the macroscopic time scale, both rapid equilibrium and slow non-equilibrium processes exist in the gas. The case with a smaller time scale is associated with many collisions within the macroscopic time scale and results in a rapid equilibrium process. On the other hand, the case with a comparable time scale results in a slow non-equilibrium process. Also, all the processes with characteristics time much larger than macroscopic time scale appear to be frozen during the macroscopic time scale. While in every collision the translational energy is exchanged between molecules, internal energy is only exchanged in some of the collisions. Therefore, the translational microscopic time, $\unicode[STIX]{x1D70F}_{tr}$ , is smaller than the internal microscopic time, $\unicode[STIX]{x1D70F}_{int}$ .

To model the collision term, we use a two term BGK-type collision operator,

(2.15a ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}t}+c_{k}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}x_{k}} & = & \displaystyle S_{tr}+S_{int},\end{eqnarray}$$

(2.15b ) $$\begin{eqnarray}\displaystyle S_{tr} & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70F}_{tr}}(f-f_{tr}),\end{eqnarray}$$

(2.15c ) $$\begin{eqnarray}\displaystyle S_{int} & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70F}_{int}}(f-f_{int}).\end{eqnarray}$$

$S_{tr}$

$\unicode[STIX]{x1D70F}_{tr}$

$f_{tr}$

$S_{int}$

$\unicode[STIX]{x1D70F}_{int}$

$f_{int}$

In the standard BGK model (Bourgat et al. Reference Bourgat, Desvillettes, Le Tallec and Perthame1994; Andries et al. Reference Andries, Tallec, Perlat and Perthame2000; Rahimi & Struchtrup Reference Rahimi and Struchtrup2014a ), the targeted distributions, $f_{tr}$ and $f_{int}$ are chosen as the Maxwellian equilibrium distributions $f_{tr_{0}}$ and $f_{int_{0}}$ , respectively, which are given by

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle f_{tr_{0}}=\frac{\unicode[STIX]{x1D70C}_{I}}{m}\left(\frac{1}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D703}_{tr}}\right)^{3/2}\exp \left[-\frac{1}{2\unicode[STIX]{x1D703}_{tr}}C^{2}\right], & \displaystyle\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle f_{int_{0}}=\frac{\unicode[STIX]{x1D70C}}{m}\frac{1}{(2\unicode[STIX]{x03C0})^{3/2}\unicode[STIX]{x1D703}^{(\unicode[STIX]{x1D6FF}+3)/2}}\frac{1}{\unicode[STIX]{x1D6E4}\left(1+{\displaystyle \frac{\unicode[STIX]{x1D6FF}}{2}}\right)}\exp \left[-\frac{1}{\unicode[STIX]{x1D703}}\left(\frac{C^{2}}{2}+I^{2/\unicode[STIX]{x1D6FF}}\right)\right]. & \displaystyle\end{eqnarray}$$

Here, $f_{tr_{0}}$ is the equilibrium distribution for particles with frozen internal DoF. Also, $f_{int_{0}}$ is the full equilibrium distribution for the gas, which experienced full exchange of the translational and internal energies.

However, the equations for moments $\unicode[STIX]{x1D748}_{ij},\boldsymbol{q}_{i,tr},\boldsymbol{q}_{i,int},\boldsymbol{u}_{ij}^{1,0},u^{2,0},\boldsymbol{u}_{ij}^{0,1},u^{1,1}$ and $\boldsymbol{u}_{ijk}^{0,0}$ corresponding to the original BGK model, have the form of (Rahimi & Struchtrup Reference Rahimi and Struchtrup2014a )

(2.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}u_{i_{1}\ldots i_{n}}^{\unicode[STIX]{x1D70D},A}}{\unicode[STIX]{x2202}t}+\cdots =-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]u_{i_{1}\ldots i_{n}}^{\unicode[STIX]{x1D70D},A},\end{eqnarray}$$

with the same relaxation time for all of them. This could not predict the correct relaxation of the higher moments and the Prandtl number. Shakhov (Reference Shakhov1968) proposed a modified BGK model for a monatomic gases to obtain the correct $Pr$ number. Rykov (Reference Rykov1975) introduced a model to diatomic molecules with rotational movements and Wu et al. (Reference Wu, White, Scanlon, Reese and Zhang2015) extended this to polyatomic gases. Here, we introduce a generalized and modified S-model for polyatomic gases by obtaining $f_{tr}$ and $f_{int}$ which provide the correct $Pr$ number and distinct relaxation times for higher moments.

The relaxation times of the Boltzmann collision term for Maxwell molecules in the case of a monatomic gases for some higher moments are presented in table 1 (Truesdell & Muncaster Reference Truesdell and Muncaster1980; Torrilhon, Au & Struchtrup Reference Torrilhon, Au and Struchtrup2003). The relaxation time of $\boldsymbol{u}_{ij}^{1,0}$ is close to the relaxation time of $\unicode[STIX]{x1D748}_{ij}$ , but for other moments the differences are considerable and should not be ignored. Therefore, we obtained distribution functions, $f_{tr}$ and $f_{int}$ , in order to predict distinct relaxation of these higher moments and their internal moment counterparts $\{\boldsymbol{q}_{i,tr},\boldsymbol{q}_{i,int},u^{2,0},u^{1,1}\}$ by introducing four free relaxation parameters, $R_{q_{tr}}$ , $R_{q_{int}}$ , $R_{u^{2,0}}$ and $R_{u^{1,1}}$ . The resulting relaxation time for these higher moments from our proposed model are shown in table 2 and compared with the relaxation of the BGK model. Values of these relaxation parameters will be obtained using fitting to experimental and DSMC simulation data. It should be pointed out here that two relaxation parameters, $R_{q_{tr}}$ and $R_{q_{int}}$ , are analogies to the $Pr$ number (5.11). The microscopic relaxation in our model is independent of molecular velocity. In this sense the behaviour of our model is similar to the behaviour of Maxwell molecules. This basically influences the molecular velocity distribution function, but the physically meaningful moments are not influenced (Gallis & Torczynski Reference Gallis and Torczynski2011). The procedure and final form of the distribution functions are given in appendix A, along with some important features of the proposed model, e.g. recovering Maxwellian functions in equilibrium and the linear H-theorem.

Table 1.

Maxwell molecule relaxation times.

Table 2.

Relaxation times for higher moments based on four new free parameters and comparison with the BGK model.

3.1 General moment equation

The moment equations are obtained by taking weighted averages of the kinetic equation. Multiplying the kinetic equation (2.15a ) with $m(I^{2/\unicode[STIX]{x1D6FF}})^{A}C^{2\unicode[STIX]{x1D70D}}C_{{<}i_{1}}C_{i_{2}}\ldots C_{i_{n}>}$ and subsequent integration over velocity space and the internal energy parameter gives the general moment equation as

(3.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}u_{i_{1}\ldots i_{n}}^{\unicode[STIX]{x1D70D},A}}{\text{D}t}+2\unicode[STIX]{x1D70D}u_{i_{1}\ldots i_{n}k}^{\unicode[STIX]{x1D70D}-1,A}\frac{\text{D}v_{k}}{\text{D}t}+2\unicode[STIX]{x1D70D}u_{i_{1}\ldots i_{n}kj}^{\unicode[STIX]{x1D70D}-1,A}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{k}}+\frac{n}{2n+1}2\unicode[STIX]{x1D70D}u_{j<i_{1}\ldots i_{n-1}}^{\unicode[STIX]{x1D70D},A}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i_{n}>}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\unicode[STIX]{x2202}u_{i_{1}\ldots i_{n}k}^{\unicode[STIX]{x1D70D},A}}{\unicode[STIX]{x2202}x_{k}}+2\unicode[STIX]{x1D70D}\frac{n+1}{2n+3}u_{{<}i_{1}\ldots i_{n}}^{\unicode[STIX]{x1D70D},A}\frac{\unicode[STIX]{x2202}v_{k>}}{\unicode[STIX]{x2202}x_{k}}+\frac{n}{2n+1}\frac{\unicode[STIX]{x2202}u_{{<}i_{1}\ldots i_{n-1}}^{\unicode[STIX]{x1D70D}+1,A}}{\unicode[STIX]{x2202}x_{i_{n}>}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2\unicode[STIX]{x1D70D}\frac{n}{2n+1}u_{{<}i_{1}\ldots i_{n-1}}^{\unicode[STIX]{x1D70D},A}\frac{\text{D}v_{i_{n}>}}{\text{D}t}+\frac{n-1}{2n-1}nu_{{<}i_{1}\ldots i_{n-2}}^{\unicode[STIX]{x1D70D}+1,A}\frac{\unicode[STIX]{x2202}v_{i_{n-1}}}{\unicode[STIX]{x2202}x_{i_{n}>}}+nu_{{<}i_{1}\ldots i_{n-1}}^{\unicode[STIX]{x1D70D},A}\frac{\text{D}v_{i_{n}>}}{\text{D}t}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,u_{i_{1}\ldots i_{n}}^{\unicode[STIX]{x1D70D},A}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+\frac{n}{2n+1}\frac{n-1}{2n-1}2\unicode[STIX]{x1D70D}u_{{<}i_{1}\ldots i_{n-2}}^{\unicode[STIX]{x1D70D}+1,A}\frac{\unicode[STIX]{x2202}v_{i_{n-1}}}{\unicode[STIX]{x2202}x_{i_{n}>}}+nu_{k<i_{1}\ldots i_{n-1}}^{\unicode[STIX]{x1D70D},A}\frac{\unicode[STIX]{x2202}v_{i_{n}>}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{\unicode[STIX]{x1D70F}_{tr}}\left[u_{i_{1}\ldots i_{n}\mid E,tr}^{\unicode[STIX]{x1D70D},A}-u_{i_{1}\ldots i_{n}}^{\unicode[STIX]{x1D70D},A}\right]+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\left[u_{i_{1}\ldots i_{n}\mid E,int}^{\unicode[STIX]{x1D70D},A}-u_{i_{1}\ldots i_{n}}^{\unicode[STIX]{x1D70D},A}\right].\end{eqnarray}$$

Here, the relation $u_{\langle i_{1}\ldots i_{n}\rangle k}^{\unicode[STIX]{x1D70D},A}=u_{i_{1}\ldots i_{n}k}^{\unicode[STIX]{x1D70D},A}+(n/(2n+1))u_{{<}i_{1}\ldots i_{n-1}}^{\unicode[STIX]{x1D70D}+1,A}\unicode[STIX]{x1D6FF}_{i_{n}>k}$ is used (Struchtrup Reference Struchtrup2005b ).

3.2 Conservation laws

Conservation laws for mass ( $\unicode[STIX]{x1D70D}=A=n=0$ ), momentum ( $\unicode[STIX]{x1D70D}=A=0$ , $n=1$ ) and the balance laws for translational ( $\unicode[STIX]{x1D70D}=1$ , $A=n=0$ ) and internal ( $\unicode[STIX]{x1D70D}=0$ , $A=1$ , $n=0$ ) energies are obtained from the general moment equation (3.1) as

(3.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}\unicode[STIX]{x1D70C}}{\text{D}t}+\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=0, & \displaystyle\end{eqnarray}$$

(3.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}v_{i}}{\text{D}t}+\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D748}_{ij}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{tr}}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x1D703}_{tr}}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{i}}=0, & \displaystyle\end{eqnarray}$$

(3.2c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3}{2}\unicode[STIX]{x1D70C}\frac{\text{D}\unicode[STIX]{x1D703}_{tr}}{\text{D}t}+\frac{\unicode[STIX]{x2202}\boldsymbol{q}_{i,tr}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}_{tr}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=\frac{3\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70F}_{int}}\frac{(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{tr})}{2}, & \displaystyle\end{eqnarray}$$

(3.2d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\frac{\text{D}{\displaystyle \frac{\unicode[STIX]{x1D6FF}}{2}}\unicode[STIX]{x1D703}_{int}}{\text{D}t}+\frac{\unicode[STIX]{x2202}\boldsymbol{q}_{i,int}}{\unicode[STIX]{x2202}x_{i}}=-\frac{3\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70F}_{int}}\frac{(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{tr})}{2}. & \displaystyle\end{eqnarray}$$

The conservation of the total energy results from summation of the balance laws for translational and internal energies as

(3.2e ) $$\begin{eqnarray}\unicode[STIX]{x1D70C}\frac{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{2}\frac{\text{D}\unicode[STIX]{x1D703}}{\text{D}t}+\frac{\unicode[STIX]{x2202}\boldsymbol{q}_{i,int}}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}\boldsymbol{q}_{i,tr}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=0.\end{eqnarray}$$

Therefore, the balance law for dynamic temperature (2.13) is obtained as

(3.2f ) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D70C}\frac{\text{D}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\text{D}t}+\frac{2}{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\frac{\unicode[STIX]{x2202}\boldsymbol{q}_{i,int}}{\unicode[STIX]{x2202}x_{i}}-\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\frac{\unicode[STIX]{x2202}\boldsymbol{q}_{i,tr}}{\unicode[STIX]{x2202}x_{i}}-\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=-\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70F}_{int}}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}.\end{eqnarray}$$

3.3 Balance laws

Moment equations for stress tensor, $\unicode[STIX]{x1D748}_{ij}=\boldsymbol{u}_{ij}^{0,0}$ , translational heat flux, $\boldsymbol{q}_{i,tr}=(1/2)u_{i}^{1,0}$ and internal heat flux, $u_{i}^{0,1}=\boldsymbol{q}_{i,int}$ , which are present in the conservation laws, are obtained from the general moment equation (3.1) as

(3.3a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\unicode[STIX]{x1D748}_{ij}}{\text{D}t}+\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{ijk}^{0,0}}{\unicode[STIX]{x2202}x_{k}}+\frac{4}{5}\frac{\unicode[STIX]{x2202}q_{{<}i,tr}}{\unicode[STIX]{x2202}x_{j>}}+2\unicode[STIX]{x1D70E}_{k<i}\frac{\unicode[STIX]{x2202}v_{j>}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,2\unicode[STIX]{x1D70C}[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}]\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}=-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\unicode[STIX]{x1D70E}_{ij},\end{eqnarray}$$

(3.3b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\boldsymbol{q}_{i,tr}}{\text{D}t}-\frac{5}{2}\left[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right]\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D748}_{ij}}{\unicode[STIX]{x2202}x_{j}}+\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}-\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D70E}_{ik}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}-[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}]\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{kj}}{\unicode[STIX]{x2202}x_{j}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{1}{2}\frac{\unicode[STIX]{x2202}u_{ik}^{1,0}}{\unicode[STIX]{x2202}x_{k}}+\frac{1}{6}\frac{\unicode[STIX]{x2202}u^{2,0}}{\unicode[STIX]{x2202}x_{i}}+\boldsymbol{u}_{ijk}^{0,0}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{k}}+\frac{7}{5}\boldsymbol{q}_{i,tr}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+\frac{7}{5}q_{k,tr}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{2}{5}q_{j,tr}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}-\frac{5}{2}[\unicode[STIX]{x1D703}^{2}-2\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}]\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{i}}=-R_{q_{tr}}\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\boldsymbol{q}_{i,tr},\end{eqnarray}$$

(3.3c ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\boldsymbol{q}_{i,int}}{\text{D}t}-\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}+3\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{2}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D748}_{ij}}{\unicode[STIX]{x2202}x_{j}}+\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}-\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D70C}[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}]\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{i}}\right]+\frac{\unicode[STIX]{x2202}u_{ik}^{0,1}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{1}{3}\frac{\unicode[STIX]{x2202}u^{1,1}}{\unicode[STIX]{x2202}x_{i}}+q_{k,int}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{k}}+\boldsymbol{q}_{i,int}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}=-R_{q_{int}}\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\boldsymbol{q}_{i,int}.\end{eqnarray}$$

These equations contain the higher moments $\boldsymbol{u}_{ij}^{1,0}$ , $u^{2,0}$ , $\boldsymbol{u}_{ijk}^{0,0}$ , $\boldsymbol{u}_{ij}^{0,1}$ and $u^{1,1}$ , for which full moment equations are obtained from (3.1) with the appropriate choices for $\unicode[STIX]{x1D70D}$ and $A$ . Choosing all moments mentioned so far as variables will construct a set of equations for 36 moments listed in (2.14).

Set of 36 moments equations contain higher moments, $\{\boldsymbol{u}_{ijk}^{1,0},u_{i}^{2,0},u_{ijkl}^{0,0},\boldsymbol{u}_{ijk}^{0,1},u_{i}^{1,1}\}$ , under the space derivatives. We need constitutive equations for these higher moments to close the system of equations with only 36 variables. The generalized Grad’s distribution function (B 1) for polyatomic gases (see appendix B) is used to obtain constitutive equations as

(3.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\boldsymbol{u}_{ijk}^{1,0}=9\unicode[STIX]{x1D703}\boldsymbol{u}_{ijk}^{0,0},\quad u_{i}^{2,0}=28\unicode[STIX]{x1D703}\boldsymbol{q}_{i,tr},\quad u_{ijkl}^{0,0}=0,\\ \boldsymbol{u}_{ijk}^{0,1}={\displaystyle \frac{\unicode[STIX]{x1D6FF}}{2}}\unicode[STIX]{x1D703}\boldsymbol{u}_{ijk}^{0,0},\quad u_{i}^{1,1}=(5\boldsymbol{q}_{i,int}+\unicode[STIX]{x1D6FF}\boldsymbol{q}_{i,tr})\unicode[STIX]{x1D703}.\end{array}\right\}\end{eqnarray}$$

Substituting these equations into the balance laws for the moments $\boldsymbol{u}_{ij}^{1,0}$ , $u^{2,0}$ , $\boldsymbol{u}_{ijk}^{0,0}$ , $\boldsymbol{u}_{ij}^{0,1}$ and $u^{1,1}$ , results in

(3.5a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\boldsymbol{u}_{ij}^{1,0}}{\text{D}t}-\boldsymbol{u}_{ijk}^{0,0}2\left[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right]\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}-\frac{28}{5}\left[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right]q_{{<}i,tr}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,7\boldsymbol{u}_{ijk}^{0,0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}+2\boldsymbol{u}_{ijk}^{0,0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}+\frac{28}{5}q_{{<}i,tr}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{28}{5}q_{{<}i,tr}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}+9\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{ijk}^{0,0}}{\unicode[STIX]{x2202}x_{k}}+\frac{2}{5}28\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{{<}i,tr}}{\unicode[STIX]{x2202}x_{j>}}+\frac{6}{7}u_{{<}ij}^{1,0}\frac{\unicode[STIX]{x2202}v_{k>}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{4}{5}u_{j<i}^{1,0}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{j>}}+2u_{k<i}^{1,0}\frac{\unicode[STIX]{x2202}v_{j>}}{\unicode[STIX]{x2202}x_{k}}+\boldsymbol{u}_{ij}^{1,0}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+\frac{14}{15}u^{2,0}\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle \quad -\,2\frac{1}{\unicode[STIX]{x1D70C}}\boldsymbol{u}_{ijk}^{0,0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{kl}}{\unicode[STIX]{x2202}x_{l}}-\frac{28}{5}\frac{1}{\unicode[STIX]{x1D70C}}q_{{<}i,tr}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{j>l}}{\unicode[STIX]{x2202}x_{l}}=-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\boldsymbol{u}_{ij}^{1,0},\end{eqnarray}$$

(3.5b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}u^{2,0}}{\text{D}t}-8q_{k,tr}[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}]\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}+28\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{k,tr}}{\unicode[STIX]{x2202}x_{k}}-8\frac{q_{k,tr}}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{kj}}{\unicode[STIX]{x2202}x_{j}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,20q_{k,tr}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}+8q_{k,tr}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}+4u_{kj}^{1,0}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{k}}+\frac{7}{3}u^{2,0}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{R_{u^{2,0}}}{\unicode[STIX]{x1D70F}_{tr}}[(15\unicode[STIX]{x1D70C}[\unicode[STIX]{x1D703}^{2}-2\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}])-u^{2,0}]+\frac{R_{u^{2,0}}}{\unicode[STIX]{x1D70F}_{int}}[(15\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}^{2})-u^{2,0}],\end{eqnarray}$$

(3.5c ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\boldsymbol{u}_{ijk}^{0,0}}{\text{D}t}-3\frac{\unicode[STIX]{x1D70E}_{{<}ij}}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{k>l}}{\unicode[STIX]{x2202}x_{l}}+\frac{3}{7}\frac{\unicode[STIX]{x2202}u_{{<}ij}^{1,0}}{\unicode[STIX]{x2202}x_{k>}}-3\unicode[STIX]{x1D70E}_{{<}ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k>}}\nonumber\\ \displaystyle & & \displaystyle \quad -\,3[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}]\unicode[STIX]{x1D70E}_{{<}ij}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k>}}+3\unicode[STIX]{x1D70E}_{{<}ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k>}}+3u_{l<ij}^{0,0}\frac{\unicode[STIX]{x2202}v_{k>}}{\unicode[STIX]{x2202}x_{l}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\boldsymbol{u}_{ijk}^{0,0}\frac{\unicode[STIX]{x2202}v_{l}}{\unicode[STIX]{x2202}x_{l}}+\frac{12}{5}q_{{<}i,tr}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{k>}}=-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\boldsymbol{u}_{ijk}^{0,0},\end{eqnarray}$$

(3.5d ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\boldsymbol{u}_{ij}^{0,1}}{\text{D}t}-2q_{{<}i,int}\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{j>k}}{\unicode[STIX]{x2202}x_{k}}-2\frac{\left[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right]}{\unicode[STIX]{x1D70C}}q_{{<}i,int}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{ij>}}+\frac{\unicode[STIX]{x1D6FF}}{2}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{ijk}^{0,0}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\boldsymbol{u}_{ijk}^{0,0}\frac{\unicode[STIX]{x1D703}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}x_{k}}+\frac{\unicode[STIX]{x1D6FF}}{2}\boldsymbol{u}_{ijk}^{0,0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}+\frac{2\unicode[STIX]{x1D6FF}}{5}q_{{<}i,tr}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}+2q_{{<}i,int}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,2\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{{<}i,int}}{\unicode[STIX]{x2202}x_{j>}}+\frac{2\unicode[STIX]{x1D6FF}}{5}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{{<}i,tr}}{\unicode[STIX]{x2202}x_{j>}}+\frac{2}{5}\unicode[STIX]{x1D703}q_{{<}i,tr}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}x_{j>}}+2u_{k<i}^{0,1}\frac{\unicode[STIX]{x2202}v_{j>}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\boldsymbol{u}_{ij}^{0,1}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+\frac{2}{3}u^{1,1}\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}=-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\boldsymbol{u}_{ij}^{0,1},\end{eqnarray}$$

(3.5e ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}u^{1,1}}{\text{D}t}-2\frac{q_{k,int}}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{kj}}{\unicode[STIX]{x2202}x_{j}}-2q_{k,int}[\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}]\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}+2q_{k,int}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,(3q_{k,int}+\unicode[STIX]{x1D6FF}q_{k,tr})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}+5\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{k,int}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{k,tr}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D703}q_{k,tr}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}x_{k}}+2u_{kj}^{0,1}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{k}}+\frac{5}{3}u^{1,1}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{R_{u^{1,1}}}{\unicode[STIX]{x1D70F}_{tr}}\left[3\unicode[STIX]{x1D70C}\left[\frac{\unicode[STIX]{x1D6FF}}{2}\unicode[STIX]{x1D703}+\frac{3}{2}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right][\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}]-u^{1,1}\right]+\frac{R_{u^{1,1}}}{\unicode[STIX]{x1D70F}_{int}}\left[\left(3\frac{\unicode[STIX]{x1D6FF}}{2}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}^{2}\right)-u^{1,1}\right].\qquad\end{eqnarray}$$

These balance laws, along with the conservation laws (3.2f ), form a closed set of 36 equations associated with the variables listed in (2.14).

Applying the order of magnitude method to this set of 36 moment equations will ensure that the minimum number of moments with optimized definitions are used for any wanted order of accuracy in terms of the powers of the Knudsen numbers. This method, in the first step, applies the CE expansion on the moment equations to find their leading-order terms. Then, new moment definitions are constructed such that only those which are linearly independent have the same order of accuracy. This will give the minimum number of moments at a certain order of accuracy. The CE expansion is carried out and new optimized moment definitions are obtained in the next section.

5.1 A recipe for choosing the set of equations

The relaxation of the internal DoF leads to various ordering sequences for different values of $\unicode[STIX]{x1D6FC}$ , which differ in particular in the terms associated with the dynamic temperature $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ . The accounting of these terms, which depends on the value of $\unicode[STIX]{x1D6FC}$ and the accuracy under consideration, needs great care. In order to decide which set of equations we need to consider for a particular problem, the relaxation times, their ratios and characteristic time or length scale must be known. Therefore, the particular problem under consideration determines which set of equations should be used. This choice depends on the values of both Knudsen numbers: if the value of $Kn_{tr}$ is rather small while $Kn_{int}$ is relatively large, one will choose a model with high power in $\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}}$ and low power in $\unicode[STIX]{x1D716}$ ; these are models with internal corrections to the NSF equations. On the other hand, if both Knudsen numbers are small, one can use a lower accuracy model, like the refined NSF equations. In problems when both Knudsen numbers are large, particularly for order-unity values of $Kn_{tr}$ , a higher order of accuracy is an essential choice, e.g. one would choose the third-order R19 equations.

The following sections will discuss different sets of equations based on the desired order of accuracy in the powers of $\unicode[STIX]{x1D716}$ . For this, we will consider the increasing orders as laid out in (5.2) up to third order, but only present four main cases, $\unicode[STIX]{x1D716}^{0}$ , $\unicode[STIX]{x1D716}^{1}$ , $\unicode[STIX]{x1D716}^{2}$  and  $\unicode[STIX]{x1D716}^{3}$ .

5.2 Zeroth order, $\unicode[STIX]{x1D716}^{0}$ : Euler equations

We begin the reduction process with considering only the zeroth-order terms in the conservation laws,

(5.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}\unicode[STIX]{x1D70C}}{\text{D}t}+\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=0, & \displaystyle\end{eqnarray}$$

(5.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}v_{i}}{\text{D}t}+\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}=0, & \displaystyle\end{eqnarray}$$

(5.3c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{2}\unicode[STIX]{x1D70C}\frac{\text{D}\unicode[STIX]{x1D703}}{\text{D}t}+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=0. & \displaystyle\end{eqnarray}$$

These equations form a closed set of equations for the variables $\{\unicode[STIX]{x1D70C},v_{i},\unicode[STIX]{x1D703}\}$ , the Euler equations for polyatomic gases, with specific heat given in (2.10).

5.3 Order $\unicode[STIX]{x1D716}^{1}$ : Refined Navier–Stokes–Fourier equations

For the first order, terms up to $\unicode[STIX]{x1D716}^{1}$ order must be considered in the conservation laws. Hence, now all terms are relevant in the conservation laws,

(5.4) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \frac{\text{D}\unicode[STIX]{x1D70C}}{\text{D}t}+\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=0,\\ \displaystyle \unicode[STIX]{x1D70C}\frac{\text{D}v_{i}}{\text{D}t}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703})}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D748}_{ij}}{\unicode[STIX]{x2202}x_{j}}=0,\\ \displaystyle \frac{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{2}\unicode[STIX]{x1D70C}\frac{\text{D}\unicode[STIX]{x1D703}}{\text{D}t}+\frac{\unicode[STIX]{x2202}q_{i}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}=0.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

These equations must be furnished with equations for $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ , $\unicode[STIX]{x1D748}_{ij}$ , $q_{i}$ at the required order. To first order, the balance law for $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ , which is at order $\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}}$ , with terms up to order $\unicode[STIX]{x1D716}^{1-\unicode[STIX]{x1D6FC}}$ must be considered as

(5.5) $$\begin{eqnarray}\unicode[STIX]{x1D70C}\frac{\text{D}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\text{D}t}+\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x0394}\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}\right)\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=-\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70F}_{int}}\unicode[STIX]{x0394}\unicode[STIX]{x1D703},\end{eqnarray}$$

while the leading terms for stress tensor and total heat flux are obtained from (C 1b ) and (C 1c ) as

(5.6) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D748}_{ij}=-\unicode[STIX]{x1D70F}_{tr}2\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}},\\ \displaystyle q_{i}=-\unicode[STIX]{x1D70F}_{tr}\frac{5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}{2R_{q_{int}}R_{q_{tr}}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

From the conservation laws, we recognize that in a moving gas the pressure is not just the equilibrium ideal gas pressure $\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}$ , but $p=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}-\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ . For this reason, one often denotes the second term as the dynamic pressure, (Kremer Reference Kremer2010; Arima et al. Reference Arima, Taniguchi, Ruggeri and Sugiyama2012), $\unicode[STIX]{x1D6F1}=-\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ .

These first-order equations for $\unicode[STIX]{x1D748}_{ij}$ and $q_{i}$ are the classical NSF equations, which relate the stress deviator and heat flux to the gradients of velocity and temperature. The factors between them are the shear viscosity $\unicode[STIX]{x1D707}$ and the heat conductivity $\unicode[STIX]{x1D705}$ which we identify as

(5.7a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D707}=\unicode[STIX]{x1D70F}_{tr}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\quad \text{and}\quad \unicode[STIX]{x1D705}=\unicode[STIX]{x1D70F}_{tr}\frac{5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}{2R_{q_{int}}R_{q_{tr}}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}.\end{eqnarray}$$

We have internal DoF corrections to the (5.6) by considering order- $\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}}$ terms from (C 1b ) and (C 1c ), which results in corrected shear viscosity and heat conductivity as,

(5.8a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D707}=\unicode[STIX]{x1D70F}_{tr}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703})\quad \text{and}\quad \unicode[STIX]{x1D705}=\unicode[STIX]{x1D70F}_{tr}\frac{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}{2R_{q_{int}}R_{q_{tr}}}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}).\end{eqnarray}$$

The obtained relation between relaxation time and the shear viscosity is identical to that for a monatomic gas. Internal DoF affect the heat conductivity, which differs from the monatomic gas as extra means of energy transport (internal DoF) are present in polyatomic gases. Therefore, the heat conductivity (5.8) consists of two parts based on (4.3d ) and (4.3h ) as,

(5.9a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{tr} & = & \displaystyle \unicode[STIX]{x1D70F}_{tr}\frac{5}{2R_{q_{tr}}}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}),\end{eqnarray}$$

(5.9b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{int} & = & \displaystyle \unicode[STIX]{x1D70F}_{tr}\frac{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{2R_{q_{int}}}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}).\end{eqnarray}$$

In the classical Navier–Stokes equations, the dynamic pressure has the form $\unicode[STIX]{x1D6F1}=-\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}v_{i}/\unicode[STIX]{x2202}x_{i})$ where $\unicode[STIX]{x1D708}$ is the bulk viscosity. Comparing with the above, we identify a relation between relaxation time $\unicode[STIX]{x1D70F}_{int}$ and the bulk viscosity,

(5.10) $$\begin{eqnarray}\unicode[STIX]{x1D710}=\unicode[STIX]{x1D70F}_{int}\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}).\end{eqnarray}$$

The bulk viscosity is a function of the internal relaxation time, hence it will vanish in a monatomic gas where no internal energy exchange occurs ( $\unicode[STIX]{x1D6FF}=0$ ).

Experimental data of viscosities and specific heat (Poling et al. Reference Poling, Prausnitz and O’connell2001; Borgnakke & Sonntag Reference Borgnakke and Sonntag2009; Cramer Reference Cramer2012) will be used to obtain the temperature-dependent relaxation times, $\unicode[STIX]{x1D70F}_{tr}$ and $\unicode[STIX]{x1D70F}_{int}$ , using (5.10) and (5.8).

The Prandtl number is defined as the dimensionless ratio of specific heat and shear viscosity over heat conductivity (Struchtrup Reference Struchtrup2005b ),

(5.11) $$\begin{eqnarray}Pr=\frac{5+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{2}\frac{\unicode[STIX]{x1D707}}{k}.\end{eqnarray}$$

This is a measure of the importance of momentum over thermal diffusivity. Based on the obtained shear viscosity and heat conductivity definitions (5.8), the Prandtl number is

(5.12) $$\begin{eqnarray}Pr=\frac{\left(5+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{int}}R_{q_{tr}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}.\end{eqnarray}$$

The values of the modelling parameters $R_{q_{int}}$ and $R_{q_{tr}}$ are restricted by the Prandtl number and one of them depends on the other one through the $Pr$ number. Viscosity and heat conductivity values could be used to determine the heat fluxes relaxation parameters. Therefore, the model provides the freedom to fit two parameters ( $R_{u^{2,0}}$ and $R_{u^{1,1}}$ ). These values can be found from fitting to experimental or DSMC simulation data for rarefied flows, such as heat conduction, damping of ultrasound, light scattering experiments or shockwave structure.

What we have obtained here at first order are not the classical NSF equations, since we have to use the full balance law (5.5) for $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ (or dynamic pressure). The classical NSF equations constitute a five variables model for $\{\unicode[STIX]{x1D70C},v_{i},\unicode[STIX]{x1D703}\}$ , while, the refined NSF (RNSF) equations obtained have six independent field variables, $\{\unicode[STIX]{x1D70C},v_{i},\unicode[STIX]{x1D703},\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\}$ .

5.4 Order $\unicode[STIX]{x1D716}^{2}$ : refined Grad’s 14 moment equations

The relevant equations at the second order of accuracy are the conservation laws (5.4) and the dynamic temperature equation with additional terms up to order $\unicode[STIX]{x1D716}^{2-\unicode[STIX]{x1D6FC}}$ ,

(5.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D70C}\frac{\text{D}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\text{D}t}+\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}+\frac{2}{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\frac{\unicode[STIX]{x2202}q_{i}}{\unicode[STIX]{x2202}x_{i}}-\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}-\frac{10R_{q_{int}}}{3\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\left[\frac{\unicode[STIX]{x2202}q_{i}}{\unicode[STIX]{x2202}x_{i}}+\frac{R_{q_{tr}}\left(2\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)}{3\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\left(\unicode[STIX]{x0394}q_{i}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}-q_{i}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}{3\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}q_{i}}{\unicode[STIX]{x2202}x_{i}}=-\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70F}_{int}}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}.\quad \qquad\end{eqnarray}$$

For the second order of accuracy, the balance laws for stress and heat flux must be considered up to first order, because the variables, $\unicode[STIX]{x1D748}_{ij}$ and $q_{i}$ , are at first order, see (4.8). From (C 1b ) and (C 1c ) with terms up to order $\unicode[STIX]{x1D716}^{1}$ , we have second-order equations as

(5.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\unicode[STIX]{x1D748}_{ij}}{\text{D}t}+\frac{4R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\frac{\unicode[STIX]{x2202}q_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}-\frac{4R_{q_{int}}R_{q_{tr}}\left(2{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)}{\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}q_{{<}i}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,2\unicode[STIX]{x1D70E}_{k<i}\frac{\unicode[STIX]{x2202}v_{j>}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+2\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}=-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\unicode[STIX]{x1D748}_{ij},\end{eqnarray}$$

(5.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}q_{i}}{\text{D}t}+\unicode[STIX]{x1D70E}_{ik}\left[\frac{5+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}-\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}\right]-\frac{2}{39}\frac{\unicode[STIX]{x2202}B^{+}}{\unicode[STIX]{x2202}x_{i}}+\frac{5}{13}\frac{\unicode[STIX]{x2202}B^{-}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(1+\frac{2R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\left[q_{k}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{k}}+q_{i}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{2R_{q_{int}}q_{k}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{i}}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{ik}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}+\frac{5+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{2}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\left(\frac{R_{q_{int}}R_{q_{tr}}\left(5+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}q_{i}+\frac{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\left(R_{q_{tr}}-R_{q_{int}}\right)}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\unicode[STIX]{x1D6E5}q_{i}\right).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

For closing the set of equations we need constitutive equations for $B^{+}$ and $\unicode[STIX]{x0394}q_{i}$ up to order $\unicode[STIX]{x1D716}^{1-2\unicode[STIX]{x1D6FC}}$ from (C 2b ) and (C 1d ), as

(5.16) $$\begin{eqnarray}B^{+}=-\frac{39}{2}\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}+\frac{3\unicode[STIX]{x1D70F}_{tr}}{\unicode[STIX]{x1D70F}_{int}}\left[\frac{10}{R_{u^{2,0}}}+\frac{3-\unicode[STIX]{x1D6FF}}{2R_{u^{1,1}}}-\frac{23-\unicode[STIX]{x1D6FF}}{2}\right]\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703},\end{eqnarray}$$

(5.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}q_{i} & = & \displaystyle \frac{5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}{5R_{q_{int}}^{2}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}^{2}}\frac{5}{2}\unicode[STIX]{x1D70F}_{tr}\left[\left(1+\frac{3R_{q_{int}}}{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \times \,\left(\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{i}}+\frac{2}{39}\frac{\unicode[STIX]{x2202}B^{+}}{\unicode[STIX]{x2202}x_{i}}\right)\nonumber\\ \displaystyle & & \displaystyle \left.+\,\frac{\left(R_{q_{int}}-R_{q_{tr}}\right)}{R_{q_{tr}}}\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}\vphantom{\left(1+\frac{3R_{q_{int}}}{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)}\right]-\frac{\left[1+\displaystyle \frac{\unicode[STIX]{x1D70F}_{tr}}{\unicode[STIX]{x1D70F}_{int}}\right]5R_{q_{int}}\left(R_{q_{tr}}-R_{q_{int}}\right)}{5R_{q_{int}}^{2}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}^{2}}q_{i},\qquad\end{eqnarray}$$

and for $B^{-}$  at leading order from (C 2a ) as,

(5.18) $$\begin{eqnarray}B^{-}=-\frac{3\unicode[STIX]{x1D70F}_{tr}}{\unicode[STIX]{x1D70F}_{int}}\left[\frac{3}{R_{u^{2,0}}}-\frac{3-\unicode[STIX]{x1D6FF}}{2R_{u^{1,1}}}-\frac{3+\unicode[STIX]{x1D6FF}}{2}\right]\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}.\end{eqnarray}$$

With balance laws for stress and heat flux, the second-order equations form a set of PDEs for the 14 variables $\{\unicode[STIX]{x1D70C},v_{i},\unicode[STIX]{x1D703},\unicode[STIX]{x0394}\unicode[STIX]{x1D703},\unicode[STIX]{x1D748}_{ij},q_{i}\}$ . Other authors discuss a 14 moment set for polyatomic gases, (Ruggeri & Sugiyama Reference Ruggeri and Sugiyama2015) where the equations differ from what we find. Indeed, our refined Grad’s 14 moment (RG14) equations contain additional terms of order $\unicode[STIX]{x1D716}^{1+2\unicode[STIX]{x1D6FC}}$ , which are the terms in the equations for overall heat flux (C 1c ) and the dynamic temperature (5.13) containing $B^{+}$ , $\unicode[STIX]{x0394}q_{i}$  and $B^{-}$ along with their constitutive equations, (5.16)–(5.18). Therefore, the proposed model is not a second-order accurate model. Also, the mentioned 14 field theory (Ruggeri & Sugiyama Reference Ruggeri and Sugiyama2015) contains three nonlinear terms from (C 1c ), which according to our analysis are of orders $\unicode[STIX]{x1D716}^{2+\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D716}^{3}$ , respectively. As will be seen below, if one wishes to have a theory at these orders, there will be additional terms that must be included.

The leading-order term of $\unicode[STIX]{x0394}q_{i}$ at order $\unicode[STIX]{x1D716}^{1+\unicode[STIX]{x1D6FC}}$ from (C 1d ) and (C 1c ) is obtained as

(5.19) $$\begin{eqnarray}\unicode[STIX]{x0394}q_{i}=\unicode[STIX]{x1D70F}_{tr}\frac{5\left(\displaystyle 3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)}{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}.\end{eqnarray}$$

We name the factor relating the heat flux difference and the gradient of dynamic temperature the dynamic heat conductivity,

(5.20) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6E5}}=\unicode[STIX]{x1D70F}_{tr}\frac{5\left(\displaystyle 3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)}{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}.\end{eqnarray}$$

5.5 Order $\unicode[STIX]{x1D716}^{3}$ : regularized 19 (R19) equations

Finally, we present the equations at third order of accuracy, which are: the conservation laws (5.4); the full equation for the dynamic temperature (5.13); the equation for heat flux difference with terms up to order $\unicode[STIX]{x1D716}^{2-2\unicode[STIX]{x1D6FC}}$ , due to the fact that $\unicode[STIX]{x0394}q_{i}$ is at order $\unicode[STIX]{x1D716}^{1+\unicode[STIX]{x1D6FC}}$ and first appears in the $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ equation (5.13) which itself is at order $\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}}$ ,

(5.21) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\unicode[STIX]{x0394}q_{i}}{\text{D}t}+\unicode[STIX]{x1D70E}_{ik}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{10R_{q_{int}}\left(2\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}^{2}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}^{2}}\right)\left(\unicode[STIX]{x0394}\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}\right)\left(\unicode[STIX]{x0394}q_{i}-q_{i}\right)}{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\unicode[STIX]{x0394}q_{j}}{5\left(5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}\right)}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x1D6FF}}{(42+25\unicode[STIX]{x1D6FF})}\left(7+\frac{15R_{q_{int}}}{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\frac{\unicode[STIX]{x2202}B_{ij}^{+}}{\unicode[STIX]{x2202}x_{j}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(\frac{25R_{q_{int}}+7\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}{5\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\right)\left(\unicode[STIX]{x0394}q_{i}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x0394}q_{k}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{k}}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{42}{(42+25\unicode[STIX]{x1D6FF})^{2}}\left(7+\frac{15R_{q_{int}}}{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}B_{ij}^{+}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j}}+\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{ik}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{5}{39}\left(1-\frac{10R_{q_{int}}}{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\frac{\unicode[STIX]{x2202}B^{-}}{\unicode[STIX]{x2202}x_{i}}-\frac{5}{39}\left(1+\frac{3R_{q_{int}}}{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\frac{\unicode[STIX]{x2202}B^{+}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{5}{2}\left[1-\frac{R_{q_{int}}}{R_{q_{tr}}}\right]\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D70E}_{ik}\left(\frac{5}{2}\left[1-\frac{R_{q_{int}}}{R_{q_{tr}}}\right]\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}-\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{2R_{q_{int}}}{5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}\left[q_{j}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}+q_{i}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+q_{k}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{k}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{5}{2}\left(1+\frac{3R_{q_{int}}}{R_{q_{tr}}\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\right)\left[\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D748}_{ij}}{\unicode[STIX]{x2202}x_{j}}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\left(\frac{5R_{q_{int}}\left(R_{q_{tr}}-R_{q_{int}}\right)}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}q_{i}+\frac{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}^{2}+5R_{q_{int}}^{2}}{5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}\unicode[STIX]{x0394}q_{i}\right),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

the equations for stress tensor and total heat flux with terms up to order $\unicode[STIX]{x1D716}^{2}$ , because the variables, $\unicode[STIX]{x1D748}_{ij}$ and $q_{i}$ , are at first order,

(5.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\unicode[STIX]{x1D748}_{ij}}{\text{D}t}+\frac{4R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\frac{\unicode[STIX]{x2202}q_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}-\frac{4R_{q_{int}}R_{q_{tr}}\left(\displaystyle 2{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)}{\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}q_{{<}i}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{4\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}{5\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}q_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}+2\unicode[STIX]{x1D70E}_{k<i}\frac{\unicode[STIX]{x2202}v_{j>}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{4R_{q_{int}}R_{q_{tr}}}{\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}\left(2\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)\unicode[STIX]{x0394}q_{{<}i}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}+\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{ijk}^{0,0}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,2\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}=-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\unicode[STIX]{x1D70E}_{ij},\end{eqnarray}$$

(5.23) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}q_{i}}{\text{D}t}+\unicode[STIX]{x1D70E}_{ik}\left[\frac{5+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}-\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}\right]-\frac{\unicode[STIX]{x1D70E}_{ik}}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{kj}}{\unicode[STIX]{x2202}x_{j}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(1+\frac{2R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\left[q_{k}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{k}}+q_{i}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\right]-\frac{2}{39}\frac{\unicode[STIX]{x2202}B^{+}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{5}{13}\frac{\unicode[STIX]{x2202}B^{-}}{\unicode[STIX]{x2202}x_{i}}+\frac{2R_{q_{int}}q_{k}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{i}}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{ik}}{\unicode[STIX]{x2202}x_{k}}-\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}+\frac{5+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{2}\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}{5\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\left[\unicode[STIX]{x0394}q_{k}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x0394}q_{i}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x0394}q_{k}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{i}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{168}{(42+25\unicode[STIX]{x1D6FF})^{2}}B_{ij}^{+}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j}}+\frac{4\unicode[STIX]{x1D6FF}}{\left(42+25\unicode[STIX]{x1D6FF}\right)}\frac{\unicode[STIX]{x2202}B_{ij}^{+}}{\unicode[STIX]{x2202}x_{j}}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D748}_{ij}}{\unicode[STIX]{x2202}x_{j}}+\unicode[STIX]{x1D70E}_{ik}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,7\left(\frac{1}{(14+\unicode[STIX]{x1D6FF})^{2}}-\frac{24}{\left(42+25\unicode[STIX]{x1D6FF}\right)^{2}}\right)\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}B_{ij}^{-}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j}}+\frac{7\left(3+\unicode[STIX]{x1D6FF}\right)\left(14+3\unicode[STIX]{x1D6FF}\right)}{(14+\unicode[STIX]{x1D6FF})(42+25\unicode[STIX]{x1D6FF})}\frac{\unicode[STIX]{x2202}B_{ij}^{-}}{\unicode[STIX]{x2202}x_{j}}+\boldsymbol{u}_{ijk}^{0,0}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\!\left(\frac{R_{q_{int}}R_{q_{tr}}\left(\displaystyle 5+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}q_{i}+\frac{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\left(R_{q_{tr}}-R_{q_{int}}\right)}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\unicode[STIX]{x0394}q_{i}\!\right)\!,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

equation for $B^{+}$ with terms up to order $\unicode[STIX]{x1D716}^{2-2\unicode[STIX]{x1D6FC}}$ , due to the fact that $B^{+}$ is at order $\unicode[STIX]{x1D716}^{2\unicode[STIX]{x1D6FC}}$ and first appears in the $\unicode[STIX]{x1D748}_{ij}$ and $q_{i}$ equations, which themselves are at order $\unicode[STIX]{x1D716}^{1}$ ,

(5.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}B^{+}}{\text{D}t}-8\unicode[STIX]{x1D703}\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}+\frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}-40R_{q_{int}}}{5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(\frac{50R_{q_{int}}R_{q_{tr}}\unicode[STIX]{x1D703}\left(\displaystyle 2{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)}{\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}+\frac{\left(5R_{q_{int}}+3R_{q_{tr}}\right)\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)-100R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,q_{k}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}-\left(2-\frac{50R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\unicode[STIX]{x1D703}q_{k}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}-\frac{20}{39}B^{-}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{85}{39}B^{+}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}+\frac{3\left(23-\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}}\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}+\left(26-\frac{78}{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{\unicode[STIX]{x1D70F}_{tr}}\!\left(\frac{9}{2}R_{u^{1,1}}+15R_{u^{2,0}}\!\right)\!\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}-\frac{1}{\unicode[STIX]{x1D70F}_{int}}\!\left[\frac{3}{2}\left(\left(3-\unicode[STIX]{x1D6FF}\right)R_{u^{1,1}}+20R_{u^{2,0}}-\left(23-\unicode[STIX]{x1D6FF}\right)\right)\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\left(\frac{3R_{u^{1,1}}+10R_{u^{2,0}}}{13}B^{+}+\frac{10}{13}\left(R_{u^{1,1}}-R_{u^{2,0}}\right)B^{-}\right),\end{eqnarray}$$

and the balance law for $B^{-}$ with terms up to order $\unicode[STIX]{x1D716}^{1}$ , because $B^{+}$ is at order $\unicode[STIX]{x1D716}^{1}$ and first appears in the $\unicode[STIX]{x1D748}_{ij}$ and $q_{i}$ equations, which themselves are at order $\unicode[STIX]{x1D716}^{1}$ ,

(5.25) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}B^{-}}{\text{D}t}+\frac{12}{5}\unicode[STIX]{x1D703}\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{i}}+\frac{12R_{q_{int}}+2\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}{5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{k}}{\unicode[STIX]{x2202}x_{k}}-\frac{2}{13}B^{+}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(\frac{\left(5R_{q_{int}}+3R_{q_{tr}}\right)\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)+30R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}-\frac{2R_{q_{tr}}R_{q_{int}}\unicode[STIX]{x1D703}\left(2\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}^{2}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}^{2}}\right)}{\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}\right)q_{k}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,2\left(1+\frac{R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right)\unicode[STIX]{x1D703}q_{k}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k}}-3\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}+\frac{71}{39}B^{-}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{\unicode[STIX]{x1D70F}_{tr}}\frac{9\left(R_{u^{1,1}}-R_{u^{2,0}}\right)}{2}\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}-\frac{1}{\unicode[STIX]{x1D70F}_{int}}\left(\frac{3}{2}\left(3+\unicode[STIX]{x1D6FF}+\left[3-\unicode[STIX]{x1D6FF}\right]R_{u^{1,1}}-6R_{u^{2,0}}\right)\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\left(\frac{10R_{u^{1,1}}+3R_{u^{2,0}}}{13}B^{-}+\left[\frac{3\left(R_{u^{1,1}}-R_{u^{2,0}}\right)}{13}B^{+}\right]\right).\end{eqnarray}$$

These 19 PDEs are closed with the constitutive equations for $B_{ij}^{+}$ up to $1-\unicode[STIX]{x1D6FC}$ order, because $B_{ij}^{+}$ is at order $\unicode[STIX]{x1D716}^{1+\unicode[STIX]{x1D6FC}}$ and first appears in the $\unicode[STIX]{x1D748}_{ij}$ and $q_{i}$ equations, which themselves are at order $\unicode[STIX]{x1D716}^{1}$ ,

(5.26) $$\begin{eqnarray}\displaystyle B_{ij}^{+} & = & \displaystyle -\frac{1}{\left[\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]}\left[2\left(\frac{\left(70+23\unicode[STIX]{x1D6FF}\right)R_{q_{int}}}{\unicode[STIX]{x1D6FF}\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}-\frac{14-\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D6FF}}\right)\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}\right.\nonumber\\ \displaystyle & & \displaystyle -\,\frac{4\left(70-19\unicode[STIX]{x1D6FF}\right)}{39\unicode[STIX]{x1D6FF}}B^{-}\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}+\frac{(14-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}-\unicode[STIX]{x1D6FF}(14+\unicode[STIX]{x1D6FF})}{\unicode[STIX]{x1D6FF}\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)}\unicode[STIX]{x1D703}\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle -\,2\left(\frac{\left(70+23\unicode[STIX]{x1D6FF}\right)R_{q_{int}}}{\unicode[STIX]{x1D6FF}\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}-\frac{14-\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D6FF}}\right)\unicode[STIX]{x1D703}q_{{<}i}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{2(42+25\unicode[STIX]{x1D6FF})}{39\unicode[STIX]{x1D6FF}}B^{+}\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}+2R_{q_{int}}\left(\frac{14+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.-\,\unicode[STIX]{x1D703}\frac{10\left(7R_{q_{int}}+2\left(7+3\unicode[STIX]{x1D6FF}\right)R_{q_{tr}}\right)\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}+14R_{q_{tr}}\unicode[STIX]{x1D703}\left(\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)^{2}+\left(70+23\unicode[STIX]{x1D6FF}\right)R_{q_{tr}}\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}^{2}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x1D6FF}\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \times \,\left.q_{{<}i}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}+4\unicode[STIX]{x1D703}\left(2\unicode[STIX]{x1D70E}_{k<i}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{j>}}+2\unicode[STIX]{x1D70E}_{k<i}\frac{\unicode[STIX]{x2202}v_{j>}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\right)-\frac{\left(42+25\unicode[STIX]{x1D6FF}\right)}{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}\right],\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and for $\boldsymbol{u}_{ijk}^{0,0}$ and $B_{ij}^{-}$ at their leading orders, due to the fact that they are at order $\unicode[STIX]{x1D716}^{2}$ and first appear in the $\unicode[STIX]{x1D748}_{ij}$ and $q_{i}$ equations, which themselves are at order $\unicode[STIX]{x1D716}^{1}$ ,

(5.27) $$\begin{eqnarray}\boldsymbol{u}_{ijk}^{0,0}=-\unicode[STIX]{x1D70F}_{tr}\left[3\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{{<}ij}}{\unicode[STIX]{x2202}x_{k>}}-3\unicode[STIX]{x1D703}\unicode[STIX]{x1D70E}_{{<}ij}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{k>}}+\frac{12R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}q_{{<}i}\frac{\unicode[STIX]{x2202}v_{j}}{\unicode[STIX]{x2202}x_{k>}}\right],\end{eqnarray}$$

(5.28) $$\begin{eqnarray}\displaystyle B_{ij}^{-} & = & \displaystyle -\unicode[STIX]{x1D70F}_{tr}\left[\frac{6(14+\unicode[STIX]{x1D6FF})}{7\left(3+\unicode[STIX]{x1D6FF}\right)}\unicode[STIX]{x1D703}\left(\unicode[STIX]{x1D70E}_{k<i}\frac{\unicode[STIX]{x2202}v_{j>}}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x1D70E}_{k<i}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{j>}}-\frac{2}{3}\unicode[STIX]{x1D748}_{ij}\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2(14+\unicode[STIX]{x1D6FF})R_{q_{int}}}{\left(3+\unicode[STIX]{x1D6FF}\right)\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}\left(\vphantom{\left[\left(\left({\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)^{2}+2{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)\right]}5R_{q_{int}}\left[3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right]\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,R_{q_{tr}}\left[\left(3+\unicode[STIX]{x1D6FF}\right)\unicode[STIX]{x1D6FF}+\left(7+2\unicode[STIX]{x1D6FF}\right)\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}+\left(\left({\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)^{2}+2{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)\unicode[STIX]{x1D703}^{2}\right]\right)q_{{<}i}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2(14+\unicode[STIX]{x1D6FF})}{3\left(3+\unicode[STIX]{x1D6FF}\right)}B^{-}\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}-\frac{2(14+\unicode[STIX]{x1D6FF})\left(3R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}{\left(3+\unicode[STIX]{x1D6FF}\right)\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\unicode[STIX]{x1D703}q_{{<}i}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{j>}}\nonumber\\ \displaystyle & & \displaystyle \left.+\,2\frac{(14+\unicode[STIX]{x1D6FF})\left(3R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}{(3+\unicode[STIX]{x1D6FF})\left(5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}\right)}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x2202}q_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}\right].\end{eqnarray}$$

This is the set of original regularized 19 (R19) equations corresponding to the third order of accuracy.

Next, we will introduce new forms of the balance and constitutive equations to replace (5.24)–(5.28) in the set of R19 equations. New forms eliminate some derivatives by substitution with stress tensor, heat flux and dynamic temperature while keeping the current order of accuracy. The new form requires the same number of boundary conditions for both linear and nonlinear cases, while this is not the case for the original R19 equations. Also, the boundary conditions needed for the new form of R19 equations are clear; and they are derived and presented in the next section. However, the exact type of boundary conditions needed for the original nonlinear R19 equations are not clear. The boundary conditions will be discussed further in the coming section.

5.5.1 Transformation of equations

The balance laws for $B^{+}$ and $B^{-}$ and the constitutive equations in the set of R19 equations can be rewritten using the leading-order heat flux difference, NSF stress and viscosities as

(5.29a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}_{ij}^{\mathit{NSF}} & = & \displaystyle -2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}v_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}=\unicode[STIX]{x1D748}_{ij}+O(\unicode[STIX]{x1D716}^{1+\unicode[STIX]{x1D6FC}})+\cdots \,,\end{eqnarray}$$

(5.29b ) $$\begin{eqnarray}\displaystyle q_{i}^{\mathit{NSF}} & = & \displaystyle -\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}}=q_{i}+O(\unicode[STIX]{x1D716}^{1+\unicode[STIX]{x1D6FC}})+\cdots \,,\end{eqnarray}$$

(5.29c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6E5}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x_{i}} & = & \displaystyle \unicode[STIX]{x0394}q_{i}+O(\unicode[STIX]{x1D716}^{1+2\unicode[STIX]{x1D6FC}})+\cdots \,,\end{eqnarray}$$

(5.29d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\mathit{NSF}} & = & \displaystyle \frac{\unicode[STIX]{x1D710}}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}v_{i}}{\unicode[STIX]{x2202}x_{i}}=\unicode[STIX]{x0394}\unicode[STIX]{x1D703}+O(\unicode[STIX]{x1D716}^{2\unicode[STIX]{x1D6FC}})+\cdots \,,\end{eqnarray}$$

in a way that we still keep the proper order of accuracy. The balance laws take the form of

(5.30) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}B^{+}}{\text{D}t}-2\frac{20R_{q_{int}}-\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}\unicode[STIX]{x1D703}\left[\frac{\unicode[STIX]{x2202}q_{k}}{\unicode[STIX]{x2202}x_{k}}-q_{k}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}{\unicode[STIX]{x2202}x_{k}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left[\left(26-\frac{78}{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}+\frac{3\left(\displaystyle 23-\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)}{3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}+\frac{85}{39}B^{+}-\frac{20}{39}B^{-}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}-\!\left(\!\frac{50R_{q_{int}}R_{q_{tr}}\unicode[STIX]{x1D703}\!\left(\displaystyle 2{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)}{\left(5R_{q_{int}}+\!\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)\!R_{q_{tr}}\right)^{2}}+\frac{(5R_{q_{int}}+3R_{q_{tr}})\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)-100R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\!\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\frac{q_{k}q_{k}}{\unicode[STIX]{x1D705}}+2\frac{20R_{q_{int}}-\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\frac{\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}}\left[\frac{q_{k}q_{k}}{\unicode[STIX]{x1D705}}+\frac{q_{k}\unicode[STIX]{x0394}q_{k}}{\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6E5}}}\right]+4\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x1D70E}_{ij}\unicode[STIX]{x1D748}_{ij}}{\unicode[STIX]{x1D707}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{\unicode[STIX]{x1D70F}_{tr}}\!\left(\frac{9}{2}R_{u^{1,1}}+15R_{u^{2,0}}\!\right)\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}-\frac{1}{\unicode[STIX]{x1D70F}_{int}}\!\left[\frac{3}{2}((3-\unicode[STIX]{x1D6FF})R_{u^{1,1}}+20R_{u^{2,0}}-(23-\unicode[STIX]{x1D6FF}))\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\left(\frac{3R_{u^{1,1}}+10R_{u^{2,0}}}{13}B^{+}+\frac{10}{13}\left(R_{u^{1,1}}-R_{u^{2,0}}\right)B^{-}\right),\end{eqnarray}$$

(5.31) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}B^{-}}{\text{D}t}-2\frac{6R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}}\frac{\unicode[STIX]{x1D703}}{\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}\left[\frac{q_{k}q_{k}}{\unicode[STIX]{x1D705}}+\frac{q_{k}\unicode[STIX]{x0394}q_{k}}{\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6E5}}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad -\left(\frac{(5R_{q_{int}}+3R_{q_{tr}})\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)+30R_{q_{int}}}{5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}-\frac{2R_{q_{tr}}R_{q_{int}}\unicode[STIX]{x1D703}\left(2\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}^{2}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}^{2}}\right)}{\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}\right)\frac{q_{k}q_{k}}{\unicode[STIX]{x1D705}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{6}{5}\unicode[STIX]{x1D703}\frac{\unicode[STIX]{x1D748}_{ij}\unicode[STIX]{x1D748}_{ij}}{\unicode[STIX]{x1D707}}-\left(3\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}-\frac{71}{39}B^{-}+\frac{2}{13}B^{+}\right)\frac{\unicode[STIX]{x2202}v_{k}}{\unicode[STIX]{x2202}x_{k}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2\frac{6R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\unicode[STIX]{x1D703}\left[\frac{\unicode[STIX]{x2202}q_{k}}{\unicode[STIX]{x2202}x_{k}}-q_{k}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}{\unicode[STIX]{x2202}x_{k}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{\unicode[STIX]{x1D70F}_{tr}}\frac{9\left(R_{u^{1,1}}-R_{u^{2,0}}\right)}{2}\unicode[STIX]{x1D70C}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{2}-\frac{1}{\unicode[STIX]{x1D70F}_{int}}\left(\frac{3}{2}\left(3+\unicode[STIX]{x1D6FF}+\left[3-\unicode[STIX]{x1D6FF}\right]R_{u^{1,1}}-6R_{u^{2,0}}\right)\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad -\left[\frac{1}{\unicode[STIX]{x1D70F}_{tr}}+\frac{1}{\unicode[STIX]{x1D70F}_{int}}\right]\left(\frac{10R_{u^{1,1}}+3R_{u^{2,0}}}{13}B^{-}+\left[\frac{3\left(R_{u^{1,1}}-R_{u^{2,0}}\right)}{13}B^{+}\right]\right).\end{eqnarray}$$

The new constitutive equations for $B_{ij}^{+}$ , $B_{ij}^{-}$ and $\boldsymbol{u}_{ijk}^{0,0}$ in the set of R19 equations after some manipulation become

(5.32) $$\begin{eqnarray}\displaystyle B_{ij}^{+} & = & \displaystyle -\frac{1}{\left[\displaystyle 1+\frac{2\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)\unicode[STIX]{x1D707}}{3\left(\displaystyle 3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)v}\right]\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}}\left(\frac{2\left(70-19\unicode[STIX]{x1D6FF}\right)B^{-}+(42+25\unicode[STIX]{x1D6FF})B^{+}}{39\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D748}_{ij}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\frac{(42+25\unicode[STIX]{x1D6FF})}{2\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\unicode[STIX]{x1D70E}_{ij}-2R_{q_{int}}\left[\frac{14+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.-\,\unicode[STIX]{x1D703}\frac{10\left(7R_{q_{int}}+2\left(7+3\unicode[STIX]{x1D6FF}\right)R_{q_{tr}}\right)\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}+14R_{q_{tr}}\unicode[STIX]{x1D703}\left(\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)^{2}+\left(70+23\unicode[STIX]{x1D6FF}\right)R_{q_{tr}}\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x1D6FF}\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}\right]\nonumber\\ \displaystyle & & \displaystyle \times \,\left.\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D705}}q_{{<}i}q_{j>}+\left[\frac{(14-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}-\unicode[STIX]{x1D6FF}(14+\unicode[STIX]{x1D6FF})}{\unicode[STIX]{x1D6FF}\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\frac{\displaystyle \text{d}\unicode[STIX]{x1D6FF}}{\displaystyle \text{d}\unicode[STIX]{x1D703}}\right)}+\frac{28}{3}\right]\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D710}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\unicode[STIX]{x1D748}_{ij}-8\unicode[STIX]{x1D703}\unicode[STIX]{x1D70E}_{k<i}\unicode[STIX]{x1D70E}_{j>k}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2\unicode[STIX]{x1D707}}{1+{\displaystyle \frac{2\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)\unicode[STIX]{x1D707}}{3\left(3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)v}}}\left(\frac{\left(70+23\unicode[STIX]{x1D6FF}\right)R_{q_{int}}}{\unicode[STIX]{x1D6FF}\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}-\frac{14-\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D6FF}}\right)\frac{q_{{<}i}}{\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}\nonumber\\ \displaystyle & & \displaystyle \times \,\left[\frac{q_{j>}}{\unicode[STIX]{x1D705}}+\frac{\unicode[STIX]{x0394}q_{j>}}{\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6E5}}}\right]-\frac{2}{{\displaystyle \frac{1}{\unicode[STIX]{x1D707}}}+{\displaystyle \frac{2\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)}{3\left(\displaystyle 3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)v}}}\left(\frac{\left(70+23\unicode[STIX]{x1D6FF}\right)R_{q_{int}}}{\unicode[STIX]{x1D6FF}\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}\right.\nonumber\\ \displaystyle & & \displaystyle \left.-\,\frac{14-\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D6FF}}\right)\frac{1}{\unicode[STIX]{x1D70C}}\left[\frac{\unicode[STIX]{x2202}q_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}-q_{{<}i}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}{\unicode[STIX]{x2202}x_{j>}}\right],\end{eqnarray}$$

(5.33) $$\begin{eqnarray}\displaystyle B_{ij}^{-} & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}}\left[-\frac{6(14+\unicode[STIX]{x1D6FF})}{7\left(3+\unicode[STIX]{x1D6FF}\right)}\unicode[STIX]{x1D703}\unicode[STIX]{x1D70E}_{k<i}\unicode[STIX]{x1D70E}_{j>k}-\frac{\left(14+\unicode[STIX]{x1D6FF}\right)}{3\left(3+\unicode[STIX]{x1D6FF}\right)}B^{-}\unicode[STIX]{x1D748}_{ij}\right.\nonumber\\ \displaystyle & & \displaystyle -\,\frac{2(14+\unicode[STIX]{x1D6FF})R_{q_{int}}}{\left(3+\unicode[STIX]{x1D6FF}\right)\left(5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)^{2}}\left(5R_{q_{int}}\left[3+\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right]\right.\nonumber\\ \displaystyle & & \displaystyle \left.\left.+\,R_{q_{tr}}\left[\left(3+\unicode[STIX]{x1D6FF}\right)\unicode[STIX]{x1D6FF}+\left(7+2\unicode[STIX]{x1D6FF}\right)\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}+\left(\left({\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)^{2}+2{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}^{2}}}\right)\unicode[STIX]{x1D703}^{2}\right]\right)\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D705}}q_{{<}i}q_{j>}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D707}\frac{2(14+\unicode[STIX]{x1D6FF})\left(3R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}{(3+\unicode[STIX]{x1D6FF})\left(5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}\right)}\frac{q_{{<}i}}{\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}\left[\frac{q_{j>}}{\unicode[STIX]{x1D705}}+\frac{\unicode[STIX]{x0394}q_{j>}}{\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6E5}}}\right]\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D707}\frac{2(14+\unicode[STIX]{x1D6FF})\left(3R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}\right)}{(3+\unicode[STIX]{x1D6FF})\left(5R_{q_{int}}+\left(\displaystyle \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}\right)R_{q_{tr}}\right)}\frac{1}{\unicode[STIX]{x1D70C}}\left(\frac{\unicode[STIX]{x2202}q_{{<}i}}{\unicode[STIX]{x2202}x_{j>}}-q_{{<}i}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}{\unicode[STIX]{x2202}x_{j>}}\right),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(5.34) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{ijk}^{0,0} & = & \displaystyle \frac{6R_{q_{int}}}{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}\frac{q_{{<}i}\unicode[STIX]{x1D70E}_{jk>}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}}+3\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x1D70E}_{{<}ij}}{\unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}\left[\frac{q_{k>}}{\unicode[STIX]{x1D705}}+\frac{\unicode[STIX]{x0394}q_{k>}}{\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6E5}}}\right]\nonumber\\ \displaystyle & & \displaystyle -\,3\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D70C}}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{{<}ij}}{\unicode[STIX]{x2202}x_{k>}}-\unicode[STIX]{x1D70E}_{{<}ij}\frac{\unicode[STIX]{x2202}\ln \unicode[STIX]{x1D70C}\left(\unicode[STIX]{x1D703}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right)}{\unicode[STIX]{x2202}x_{k>}}\right),\end{eqnarray}$$

where the microscopic time scales are substituted by viscosities and specific heat using (5.8).

8.1 Refined NSF equations

For this case, the RNSF equations (5.4)–(5.6), reduce to

(8.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}=0, & \displaystyle\end{eqnarray}$$

(8.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3+\unicode[STIX]{x1D6FF}+\left(1+\unicode[STIX]{x1D703}\right)\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}}{2}\left(1+\unicode[STIX]{x1D70C}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}q}{\unicode[STIX]{x2202}x}=0, & \displaystyle\end{eqnarray}$$

(8.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D703}=0, & \displaystyle\end{eqnarray}$$

(8.4d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}_{11}=0, & \displaystyle\end{eqnarray}$$

(8.4e ) $$\begin{eqnarray}\displaystyle & \displaystyle q=-\unicode[STIX]{x1D70F}_{tr}\frac{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\left(1+\unicode[STIX]{x1D703}\right)\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}{2R_{q_{int}}R_{q_{tr}}}\left(1+\unicode[STIX]{x1D70C}\right)\left(1+\unicode[STIX]{x1D703}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}. & \displaystyle\end{eqnarray}$$

8.2 Boundary conditions

For obtaining the boundary conditions, we consider the steady state condition with 11 variables,

(8.5) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}=\left\{\unicode[STIX]{x1D703},\unicode[STIX]{x0394}\unicode[STIX]{x1D703},q,\unicode[STIX]{x0394}q,B^{+},B^{-},B_{11}^{+},B_{11}^{-},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D70E}_{11},u_{111}^{0,0}\right\},\end{eqnarray}$$

and write the system of equations as

(8.6) $$\begin{eqnarray}B(\unicode[STIX]{x1D6F7})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}y}=P(\unicode[STIX]{x1D6F7})\unicode[STIX]{x1D6F7}.\end{eqnarray}$$

The number of boundary conditions which must be described is the number of variables of the system (11) minus the number of multiplicity of the zero eigenvalues of the matrix $\unicode[STIX]{x1D63C}(\unicode[STIX]{x1D6F7})$ (Torrilhon & Struchtrup Reference Torrilhon and Struchtrup2008). Calculation of the eigenvalues shows that the matrix $\unicode[STIX]{x1D63C}(\unicode[STIX]{x1D6F7})$ possesses a zero eigenvalue with a multiplicity of 4. Therefore, we need to prescribe a total number of 7 boundary conditions for regularized 19 equations. Four associated null spaces of the matrix $\unicode[STIX]{x1D63C}(\unicode[STIX]{x1D6F7})$ give relations which describe $B^{+}$ , $B^{-}$ , $B_{11}^{+}$ and $B_{11}^{-}$ as functions of the rest of the variables of (8.5). Therefore, only 7 independent variables remain. Based on this reduced 7 field of variables of $\unicode[STIX]{x1D6F7}$ , we have the velocity and internal energy parameter functions corresponding to the fluxes with odd powers in the normal velocity component ( $q$ , $\unicode[STIX]{x0394}q$ and $u_{111}^{0,0}$ ) as

(8.7) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}=\left\{C_{1}\left(\frac{C^{2}}{2}+I^{2/\unicode[STIX]{x1D6FF}}\right),C_{1}\left(\frac{C^{2}}{2}-\frac{5R_{q_{int}}}{\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}I^{2/\unicode[STIX]{x1D6FF}}\right),C_{1}\left(C_{1}C_{1}-\frac{3}{5}C^{2}\right)\right\}.\end{eqnarray}$$

The microscopic boundary condition along with the $\unicode[STIX]{x1D6F9}$ function (7.1) are used to obtain 6 macroscopic boundary conditions which correspond to the boundary conditions for $q$ , $\unicode[STIX]{x0394}q$ and $u_{111}^{0,0}$ (7.9), (7.10) and (7.12), on each wall, and are given in appendix D.

Here, the fluid is considered to be stationary and the flow which exists due to the density changes in the unsteady state is ignored. We apply the prescribed mass condition as the seventh boundary condition to update the density during the unsteady processes as

(8.8) $$\begin{eqnarray}\int _{-L/2}^{L/2}\unicode[STIX]{x1D70C}\,\text{d}x=\unicode[STIX]{x1D70C}_{0}L.\end{eqnarray}$$

If we were to allow gas movements and convection to be a part of the problem, the conservation of mass and normal velocity of gas on the wall would replace the prescribed mass condition. Here we are interested in only stationary heat conduction, therefore all the velocities are set to zero; and the prescribed mass condition (8.2) is solved by the trapezoidal rule along with conservation of momentum to gain the distribution of density at each time step in the unsteady processes.

Also, the boundary condition for the RNSF equations along with the prescribed mass condition is the temperature jump condition obtained from (D 10) at order $\unicode[STIX]{x1D700}^{1}$  as,

(8.9) $$\begin{eqnarray}\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{w}=n_{y}\unicode[STIX]{x1D70F}_{tr}\frac{(2-\unicode[STIX]{x1D712})}{\unicode[STIX]{x1D712}}\sqrt{\frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D703}}{2}}\frac{5R_{q_{int}}+\left(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D703}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}\unicode[STIX]{x1D703}}\right)R_{q_{tr}}}{(4+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D701})R_{q_{int}}R_{q_{tr}}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}.\end{eqnarray}$$

8.3 Numerical scheme

The finite difference method is used to discretize our system of equations with central difference scheme in spatial discretization and first-order explicit discretization in time. The steady state condition is

(8.10) $$\begin{eqnarray}\frac{|\unicode[STIX]{x1D703}_{m/2}^{n+1}-\unicode[STIX]{x1D703}_{m/2}^{n}|}{|\unicode[STIX]{x1D703}_{m/2}^{n}|}\leqslant 10^{-6}.\end{eqnarray}$$

For RNSFs temperature jump boundary condition (8.9), second-order backward and forward finite difference discretizations are used.

8.4 Linear and steady case

First, we study the steady linearized set of equations with small disturbances from an equilibrium ground state $\{\unicode[STIX]{x1D70C}_{0},v_{i}^{0}=0,\unicode[STIX]{x1D703}_{0}\}$ . The set of linear steady equations are reduced to 5 coupled equations for $\unicode[STIX]{x1D6F7}=\{\unicode[STIX]{x0394}q,q,u_{111}^{0,0},\unicode[STIX]{x1D70E}_{11},\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\}$ . Equations for the rest of the variables $\{\unicode[STIX]{x1D70C},\unicode[STIX]{x1D703},B^{+},B^{-}\}$ are functions of coupled variables. The solution of the set of coupled equations, $A_{5\times 5}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}/\unicode[STIX]{x2202}x)=B_{5\times 5}\unicode[STIX]{x1D6F7}$ , is obtained using the eigenvalue method as

(8.11) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(x)=\mathop{\sum }_{n=1}^{5}C_{n}\unicode[STIX]{x1D751}_{n}\text{e}^{\unicode[STIX]{x1D706}_{n}x},\end{eqnarray}$$

where, $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D751}$ are eigenvalues and eigenvectors of the coefficient matrix, $\unicode[STIX]{x1D63C}^{-1}\unicode[STIX]{x1D63D}$ . The solution for the remaining variables are obtained by integrating of their equations.

Figure 2.

Comparison of temperature and density profiles for $Kn$ numbers equal to 0.071 and 0.71 with $t_{int}/t_{tr}=1$ . Results shown are obtained from: R19 equations (blue dashed line); RNSF equations (black line); DSMC method (red triangles).