Hegedus Aerodynamics

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Starting Equations (Ref. 1)

$${\widehat{F}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\xi }_x{\tau }_{xx}+{\xi }_y{\tau }_{xy}+{\xi }_z{\tau }_{xz}\\ {\xi }_x{\tau }_{xy}+{\xi }_y{\tau }_{yy}+{\xi }_z{\tau }_{yz}\\ {\xi }_x{\tau }_{xz}+{\xi }_y{\tau }_{yz}+{\xi }_z{\tau }_{zz}\\ {\xi }_x{b}_x+{\xi }_y{b}_y+{\xi }_z{b}_z\end{array}\right]\text{,}{\widehat{G}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\eta }_x{\tau }_{xx}+{\eta }_y{\tau }_{xy}+{\eta }_z{\tau }_{xz}\\ {\eta }_x{\tau }_{xy}+{\eta }_y{\tau }_{yy}+{\eta }_z{\tau }_{yz}\\ {\eta }_x{\tau }_{xz}+{\eta }_y{\tau }_{yz}+{\eta }_z{\tau }_{zz}\\ {\eta }_x{b}_x+{\eta }_y{b}_y+{\eta }_z{b}_z\end{array}\right]\text{,}{\widehat{H}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\zeta }_x{\tau }_{xx}+{\zeta }_y{\tau }_{xy}+{\zeta }_z{\tau }_{xz}\\ {\zeta }_x{\tau }_{xy}+{\zeta }_y{\tau }_{yy}+{\zeta }_z{\tau }_{yz}\\ {\zeta }_x{\tau }_{xz}+{\zeta }_y{\tau }_{yz}+{\zeta }_z{\tau }_{zz}\\ {\zeta }_x{b}_x+{\zeta }_y{b}_y+{\zeta }_z{b}_z\end{array}\right]$$ $$b_x=u{\tau }_{xx}+v{\tau }_{xy}+w{\tau }_{xz}-q_x$$ $$b_y=u{\tau }_{xy}+v{\tau }_{yy}+w{\tau }_{yz}-q_y$$ $$b_z=u{\tau }_{xz}+v{\tau }_{yz}+w{\tau }_{zz}-q_z$$ $${\tau }_{xx}=\frac{2}{3}\mu \left(2\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}-\frac{\partial w}{\partial z}\right)$$ $${\tau }_{yy}=\frac{2}{3}\mu \left(2\frac{\partial v}{\partial y}-\frac{\partial u}{\partial x}-\frac{\partial w}{\partial z}\right)$$ $${\tau }_{zz}=\frac{2}{3}\mu \left(2\frac{\partial w}{\partial z}-\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right)$$ $${\tau }_{xy}=\mu \left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)$$ $${\tau }_{xz}=\mu \left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)$$ $${\tau }_{xz}=\mu \left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)$$ $${\tau }_{yz}=\mu \left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)$$ $$q_x=-\kappa \frac{\partial a^2}{\partial x}\text{,}{q}_y=-\kappa \frac{\partial a^2}{\partial y}\text{,}{q}_z=-\kappa \frac{\partial a^2}{\partial z}$$ $$\frac{\partial }{\partial x}={\xi }_x\frac{\partial }{\partial \xi }+{\eta }_x\frac{\partial }{\partial \eta }+{\varsigma }_x\frac{\partial }{\partial \zeta }$$ $$\frac{\partial }{\partial y}={\xi }_y\frac{\partial }{\partial \xi }+{\eta }_y\frac{\partial }{\partial \eta }+{\varsigma }_y\frac{\partial }{\partial \zeta }$$ $$\frac{\partial }{\partial z}={\xi }_z\frac{\partial }{\partial \xi }+{\eta }_z\frac{\partial }{\partial \eta }+{\varsigma }_z\frac{\partial }{\partial \zeta }$$

Derive Vector Equations

$${\widehat{F}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\widehat{f}}_{v\text{,}\mathrm{\rho u}}\\ {\widehat{f}}_{v\text{,}\mathrm{\rho v}}\\ {\widehat{f}}_{v\text{,}\mathrm{\rho w}}\\ {\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}\end{array}\right]\text{,}{\widehat{G}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\widehat{g}}_{v\text{,}\mathrm{\rho u}}\\ {\widehat{g}}_{v\text{,}\mathrm{\rho v}}\\ {\widehat{g}}_{v\text{,}\mathrm{\rho w}}\\ {\widehat{g}}_{v\text{,}{\mathrm{\rho e}}_0}\end{array}\right]\text{,}{\widehat{H}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\widehat{h}}_{v\text{,}\mathrm{\rho u}}\\ {\widehat{h}}_{v\text{,}\mathrm{\rho v}}\\ {\widehat{h}}_{v\text{,}\mathrm{\rho w}}\\ {\widehat{h}}_{v\text{,}{\mathrm{\rho e}}_0}\end{array}\right]$$ $$\begin{array}{ccccccc}{{\widehat{f}}_{v\text{,}\mathrm{\rho u}}=\xi }_x{\tau }_{xx}+{\xi }_y{\tau }_{xy}+{\xi }_z{\tau }_{xz}& \text{,}& & {{\widehat{g}}_{v\text{,}\mathrm{\rho u}}=\eta }_x{\tau }_{xx}+{\eta }_y{\tau }_{xy}+{\eta }_z{\tau }_{xz}& \text{,}& & {{\widehat{h}}_{v\text{,}\mathrm{\rho u}}=\zeta }_x{\tau }_{xx}+{\zeta }_y{\tau }_{xy}+{\zeta }_z{\tau }_{xz}\\ {{\widehat{f}}_{v\text{,}\mathrm{\rho v}}=\xi }_x{\tau }_{xy}+{\xi }_y{\tau }_{yy}+{\xi }_z{\tau }_{yz}& \text{,}& & {{\widehat{g}}_{v\text{,}\mathrm{\rho v}}=\eta }_x{\tau }_{xy}+{\eta }_y{\tau }_{yy}+{\eta }_z{\tau }_{yz}& \text{,}& & {{\widehat{h}}_{v\text{,}\mathrm{\rho v}}=\zeta }_x{\tau }_{xy}+{\zeta }_y{\tau }_{yy}+{\zeta }_z{\tau }_{yz}\\ {{\widehat{f}}_{v\mathrm{,}\mathrm{\rho w}}=\xi }_x{\tau }_{xz}+{\xi }_y{\tau }_{yz}+{\xi }_z{\tau }_{zz}& \text{,}& & {{\widehat{g}}_{v\text{,}\mathrm{\rho w}}=\eta }_x{\tau }_{xz}+{\eta }_y{\tau }_{yz}+{\eta }_z{\tau }_{zz}& \text{,}& & {{\widehat{h}}_{v\text{,}\mathrm{\rho w}}=\zeta }_x{\tau }_{xz}+{\zeta }_y{\tau }_{yz}+{\zeta }_z{\tau }_{zz}\\ {{\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}=\xi }_x{b}_x+{\xi }_y{b}_y+{\xi }_z{b}_z& \text{,}& & {{\overrightarrow{g}}_{v\text{,}{\mathrm{\rho e}}_0}=\eta }_x{b}_x+{\eta }_y{b}_y+{\eta }_z{b}_z& \text{,}& & {{\widehat{h}}_{v\text{,}{\mathrm{\rho e}}_0}=\zeta }_x{b}_x+{\zeta }_y{b}_y+{\zeta }_z{b}_z\end{array}$$

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Momentum equation

${\widehat{f}}_{v\text{,}\mathrm{\rho u}}$ equation.

$${{\widehat{f}}_{v\text{,}\mathrm{\rho u}}=\xi }_x{\tau }_{xx}+{\xi }_y{\tau }_{xy}+{\xi }_z{\tau }_{xz}$$ $$=\mu \left[{\xi }_x\left(\frac{2}{3}\left(2{\xi }_x\frac{\partial u}{\partial \xi }+2{\eta }_x\frac{\partial u}{\partial \eta }+2{\zeta }_x\frac{\partial u}{\partial \zeta }-{\xi }_y\frac{\partial v}{\partial \xi }-{\eta }_y\frac{\partial v}{\partial \eta }-{\zeta }_y\frac{\partial v}{\partial \zeta }-{\xi }_z\frac{\partial w}{\partial \xi }-{\eta }_z\frac{\partial w}{\partial \eta }-{\zeta }_z\frac{\partial w}{\partial \zeta }\right)\right)+{\xi }_y\left({\xi }_y\frac{\partial u}{\partial \xi }+{\eta }_y\frac{\partial u}{\partial \eta }+{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_x\frac{\partial v}{\partial \xi }+{\eta }_x\frac{\partial v}{\partial \eta }+{\zeta }_x\frac{\partial v}{\partial \zeta }\right)+{\xi }_z\left({\xi }_x\frac{\partial w}{\partial \xi }+{\eta }_x\frac{\partial w}{\partial \eta }+{\zeta }_x\frac{\partial w}{\partial \zeta }+{\xi }_z\frac{\partial u}{\partial \xi }+{\eta }_z\frac{\partial u}{\partial \eta }+{\zeta }_z\frac{\partial u}{\partial \zeta }\right)\right]$$ $$=\mu \left[{\xi }_x\left(\frac{2}{3}\left(2{\xi }_x\frac{\partial u}{\partial \xi }+2{\eta }_x\frac{\partial u}{\partial \eta }+2{\zeta }_x\frac{\partial u}{\partial \zeta }-{\xi }_y\frac{\partial v}{\partial \xi }-{\eta }_y\frac{\partial v}{\partial \eta }-{\zeta }_y\frac{\partial v}{\partial \zeta }-{\xi }_z\frac{\partial w}{\partial \xi }-{\eta }_z\frac{\partial w}{\partial \eta }-{\zeta }_z\frac{\partial w}{\partial \zeta }\right)\right)+{\xi }_y\left({\xi }_y\frac{\partial u}{\partial \xi }+{\eta }_y\frac{\partial u}{\partial \eta }+{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_x\frac{\partial v}{\partial \xi }+{\eta }_x\frac{\partial v}{\partial \eta }+{\zeta }_x\frac{\partial v}{\partial \zeta }\right)+{\xi }_z\left({\xi }_x\frac{\partial w}{\partial \xi }+{\eta }_x\frac{\partial w}{\partial \eta }+{\zeta }_x\frac{\partial w}{\partial \zeta }+{\xi }_z\frac{\partial u}{\partial \xi }+{\eta }_z\frac{\partial u}{\partial \eta }+{\zeta }_z\frac{\partial u}{\partial \zeta }\right)\right]$$ $$=\mu \left[{\xi }_x\left(\frac{2}{3}\left(3{\xi }_x\frac{\partial u}{\partial \xi }+3{\eta }_x\frac{\partial u}{\partial \eta }+3{\zeta }_x\frac{\partial u}{\partial \zeta }-{\xi }_x\frac{\partial u}{\partial \xi }-{\xi }_y\frac{\partial v}{\partial \xi }-{\xi }_z\frac{\partial w}{\partial \xi }-{\eta }_x\frac{\partial u}{\partial \eta }-{\eta }_y\frac{\partial v}{\partial \eta }-{\eta }_z\frac{\partial w}{\partial \eta }-{\zeta }_x\frac{\partial u}{\partial \zeta }-{\zeta }_y\frac{\partial v}{\partial \zeta }-{\zeta }_z\frac{\partial w}{\partial \zeta }\right)\right)+{\xi }_y\left({\xi }_y\frac{\partial u}{\partial \xi }+{\eta }_y\frac{\partial u}{\partial \eta }+{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_x\frac{\partial v}{\partial \xi }+{\eta }_x\frac{\partial v}{\partial \eta }+{\zeta }_x\frac{\partial v}{\partial \zeta }\right)+{\xi }_z\left({\xi }_x\frac{\partial w}{\partial \xi }+{\eta }_x\frac{\partial w}{\partial \eta }+{\zeta }_x\frac{\partial w}{\partial \zeta }+{\xi }_z\frac{\partial u}{\partial \xi }+{\eta }_z\frac{\partial u}{\partial \eta }+{\zeta }_z\frac{\partial u}{\partial \zeta }\right)\right]$$ $$=\mu \left[{\xi }_x\left(2{\xi }_x\frac{\partial u}{\partial \xi }+2{\eta }_x\frac{\partial u}{\partial \eta }+2{\zeta }_x\frac{\partial u}{\partial \zeta }-\frac{2}{3}\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }-\frac{2}{3}\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\frac{2}{3}\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+{\xi }_y{\xi }_y\frac{\partial u}{\partial \xi }+{\xi }_y{\eta }_y\frac{\partial u}{\partial \eta }+{\xi }_y{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_y{\xi }_x\frac{\partial v}{\partial \xi }+{\xi }_y{\eta }_x\frac{\partial v}{\partial \eta }+{\xi }_y{\zeta }_x\frac{\partial v}{\partial \zeta }+{\xi }_z{\xi }_x\frac{\partial w}{\partial \xi }+{\xi }_z{\eta }_x\frac{\partial w}{\partial \eta }+{\xi }_z{\zeta }_x\frac{\partial w}{\partial \zeta }+{\xi }_z{\xi }_z\frac{\partial u}{\partial \xi }+{\xi }_z{\eta }_z\frac{\partial u}{\partial \eta }+{\xi }_z{\zeta }_z\frac{\partial u}{\partial \zeta }\right]$$ $$=\mu \left[{\xi }_x\left(2{\xi }_x\frac{\partial u}{\partial \xi }+2{\eta }_x\frac{\partial u}{\partial \eta }+2{\zeta }_x\frac{\partial u}{\partial \zeta }-\frac{2}{3}\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }-\frac{2}{3}\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\frac{2}{3}\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+{\xi }_y{\xi }_y\frac{\partial u}{\partial \xi }+{\xi }_z{\xi }_z\frac{\partial u}{\partial \xi }+{\xi }_y{\xi }_x\frac{\partial v}{\partial \xi }+{\xi }_z{\xi }_x\frac{\partial w}{\partial \xi }+{\xi }_y{\eta }_y\frac{\partial u}{\partial \eta }+{\xi }_z{\eta }_z\frac{\partial u}{\partial \eta }+{\xi }_y{\eta }_x\frac{\partial v}{\partial \eta }+{\xi }_z{\eta }_x\frac{\partial w}{\partial \eta }+{\xi }_y{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_z{\zeta }_z\frac{\partial u}{\partial \zeta }+{\xi }_y{\zeta }_x\frac{\partial v}{\partial \zeta }+{\xi }_z{\zeta }_x\frac{\partial w}{\partial \zeta }\right]$$ $$=\mu \left[{\xi }_x\left(-\frac{2}{3}\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }-\frac{2}{3}\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\frac{2}{3}\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+{\xi }_x{\xi }_x\frac{\partial u}{\partial \xi }+{\xi }_y{\xi }_y\frac{\partial u}{\partial \xi }+{\xi }_z{\xi }_z\frac{\partial u}{\partial \xi }+{\xi }_x{\xi }_x\frac{\partial u}{\partial \xi }+{\xi }_x{\xi }_y\frac{\partial v}{\partial \xi }+{\xi }_x{\xi }_z\frac{\partial w}{\partial \xi }+{\xi }_x{\eta }_x\frac{\partial u}{\partial \eta }+{\xi }_y{\eta }_y\frac{\partial u}{\partial \eta }+{\xi }_z{\eta }_z\frac{\partial u}{\partial \eta }+{\eta }_x{\xi }_x\frac{\partial u}{\partial \eta }+{\eta }_x{\xi }_y\frac{\partial v}{\partial \eta }+{\eta }_x{\xi }_z\frac{\partial w}{\partial \eta }+{\xi }_x{\zeta }_x\frac{\partial u}{\partial \zeta }+{\xi }_y{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_z{\zeta }_z\frac{\partial u}{\partial \zeta }+{\zeta }_x{\xi }_x\frac{\partial u}{\partial \zeta }+{\zeta }_x{\xi }_y\frac{\partial v}{\partial \zeta }+{\zeta }_x{\xi }_z\frac{\partial w}{\partial \zeta }\right]$$ $$=\mu \left[{\xi }_x\frac{2}{3}\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }-\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial u}{\partial \xi }+{\xi }_x\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial u}{\partial \eta }+{\eta }_x\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\eta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial u}{\partial \zeta }+{\zeta }_x\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\zeta }\right)\right]$$ $$=\mu \left[{\xi }_x\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial u}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial u}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial u}{\partial \zeta }-{\xi }_x\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)+{\eta }_x\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\eta }\right)-{\xi }_x\left(\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+{\zeta }_x\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\zeta }\right)\right]$$ $$=\mu \left[{\xi }_x\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial u}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial u}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial u}{\partial \zeta }+\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\xi }\right){\eta }_x-\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\eta }\right){\xi }_x+\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\xi }\right){\zeta }_x-\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\zeta }\right){\xi }_x\right]$$ ${\widehat{f}}_{v\text{,}\mathrm{\rho v}}$ equation.
$${{\widehat{f}}_{v\text{,}\mathrm{\rho v}}=\xi }_x{\tau }_{xy}+{\xi }_y{\tau }_{yy}+{\xi }_z{\tau }_{yz}$$ $$=\mu \left[{\xi }_x\left({\xi }_y\frac{\partial u}{\partial \xi }+{\eta }_y\frac{\partial u}{\partial \eta }+{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_x\frac{\partial v}{\partial \xi }+{\eta }_x\frac{\partial v}{\partial \eta }+{\zeta }_x\frac{\partial v}{\partial \zeta }\right)+{\xi }_y\left(\frac{2}{3}\left(2{\xi }_y\frac{\partial v}{\partial \xi }+2{\eta }_y\frac{\partial v}{\partial \eta }+2{\zeta }_y\frac{\partial v}{\partial \zeta }-{\xi }_x\frac{\partial u}{\partial \xi }-{\eta }_x\frac{\partial u}{\partial \eta }-{\zeta }_x\frac{\partial u}{\partial \zeta }-{\xi }_z\frac{\partial w}{\partial \xi }-{\eta }_z\frac{\partial w}{\partial \eta }-{\zeta }_z\frac{\partial w}{\partial \zeta }\right)\right)+{\xi }_z\left({\xi }_z\frac{\partial v}{\partial \xi }+{\eta }_z\frac{\partial v}{\partial \eta }+{\zeta }_z\frac{\partial v}{\partial \zeta }+{\xi }_y\frac{\partial w}{\partial \xi }+{\eta }_y\frac{\partial w}{\partial \eta }+{\zeta }_y\frac{\partial w}{\partial \zeta }\right)\right]$$ $$=\mu \left[{\xi }_x\left({\xi }_y\frac{\partial u}{\partial \xi }+{\eta }_y\frac{\partial u}{\partial \eta }+{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_x\frac{\partial v}{\partial \xi }+{\eta }_x\frac{\partial v}{\partial \eta }+{\zeta }_x\frac{\partial v}{\partial \zeta }\right)+{\xi }_y\left(\frac{2}{3}\left(2{\xi }_y\frac{\partial v}{\partial \xi }+2{\eta }_y\frac{\partial v}{\partial \eta }+2{\zeta }_y\frac{\partial v}{\partial \zeta }-{\xi }_x\frac{\partial u}{\partial \xi }-{\eta }_x\frac{\partial u}{\partial \eta }-{\zeta }_x\frac{\partial u}{\partial \zeta }-{\xi }_z\frac{\partial w}{\partial \xi }-{\eta }_z\frac{\partial w}{\partial \eta }-{\zeta }_z\frac{\partial w}{\partial \zeta }\right)\right)+{\xi }_z\left({\xi }_z\frac{\partial v}{\partial \xi }+{\eta }_z\frac{\partial v}{\partial \eta }+{\zeta }_z\frac{\partial v}{\partial \zeta }+{\xi }_y\frac{\partial w}{\partial \xi }+{\eta }_y\frac{\partial w}{\partial \eta }+{\zeta }_y\frac{\partial w}{\partial \zeta }\right)\right]$$ $$=\mu \left[{\xi }_x\left({\xi }_y\frac{\partial u}{\partial \xi }+{\eta }_y\frac{\partial u}{\partial \eta }+{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_x\frac{\partial v}{\partial \xi }+{\eta }_x\frac{\partial v}{\partial \eta }+{\zeta }_x\frac{\partial v}{\partial \zeta }\right)+{\xi }_y\left(\frac{2}{3}\left(3{\xi }_y\frac{\partial v}{\partial \xi }+3{\eta }_y\frac{\partial v}{\partial \eta }+3{\zeta }_y\frac{\partial v}{\partial \zeta }-{\xi }_x\frac{\partial u}{\partial \xi }-{\xi }_y\frac{\partial v}{\partial \xi }-{\xi }_z\frac{\partial w}{\partial \xi }-{\eta }_x\frac{\partial u}{\partial \eta }-{\eta }_y\frac{\partial v}{\partial \eta }-{\eta }_z\frac{\partial w}{\partial \eta }-{\zeta }_x\frac{\partial u}{\partial \zeta }-{\zeta }_y\frac{\partial v}{\partial \zeta }-{\zeta }_z\frac{\partial w}{\partial \zeta }\right)\right)+{\xi }_z\left({\xi }_z\frac{\partial v}{\partial \xi }+{\eta }_z\frac{\partial v}{\partial \eta }+{\zeta }_z\frac{\partial v}{\partial \zeta }+{\xi }_y\frac{\partial w}{\partial \xi }+{\eta }_y\frac{\partial w}{\partial \eta }+{\zeta }_y\frac{\partial w}{\partial \zeta }\right)\right]$$ $$=\mu \left[{\xi }_x{\xi }_y\frac{\partial u}{\partial \xi }+{\xi }_x{\eta }_y\frac{\partial u}{\partial \eta }+{\xi }_x{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_x{\xi }_x\frac{\partial v}{\partial \xi }+{\xi }_x{\eta }_x\frac{\partial v}{\partial \eta }+{\xi }_x{\zeta }_x\frac{\partial v}{\partial \zeta }+{\xi }_y\left(2{\xi }_y\frac{\partial v}{\partial \xi }+2{\eta }_y\frac{\partial v}{\partial \eta }+2{\zeta }_y\frac{\partial v}{\partial \zeta }-\frac{2}{3}\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }-\frac{2}{3}\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\frac{2}{3}\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+{\xi }_z{\xi }_z\frac{\partial v}{\partial \xi }+{\xi }_z{\eta }_z\frac{\partial v}{\partial \eta }+{\xi }_z{\zeta }_z\frac{\partial v}{\partial \zeta }+{\xi }_z{\xi }_y\frac{\partial w}{\partial \xi }+{\xi }_z{\eta }_y\frac{\partial w}{\partial \eta }+{\xi }_z{\zeta }_y\frac{\partial w}{\partial \zeta }\right]$$ $$=\mu \left[{\xi }_y\left(2{\xi }_y\frac{\partial v}{\partial \xi }+2{\eta }_y\frac{\partial v}{\partial \eta }+2{\zeta }_y\frac{\partial v}{\partial \zeta }-\frac{2}{3}\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }-\frac{2}{3}\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\frac{2}{3}\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+{\xi }_x{\xi }_x\frac{\partial v}{\partial \xi }+{\xi }_z{\xi }_z\frac{\partial v}{\partial \xi }+{\xi }_x{\xi }_y\frac{\partial u}{\partial \xi }+{\xi }_z{\xi }_y\frac{\partial w}{\partial \xi }+{\xi }_x{\eta }_x\frac{\partial v}{\partial \eta }+{\xi }_z{\eta }_z\frac{\partial v}{\partial \eta }+{\xi }_x{\eta }_y\frac{\partial u}{\partial \eta }+{\xi }_z{\eta }_y\frac{\partial w}{\partial \eta }+{\xi }_x{\zeta }_x\frac{\partial v}{\partial \zeta }+{\xi }_z{\zeta }_z\frac{\partial v}{\partial \zeta }+{\xi }_x{\zeta }_y\frac{\partial u}{\partial \zeta }+{\xi }_z{\zeta }_y\frac{\partial w}{\partial \zeta }\right]$$ $$=\mu \left[{\xi }_y\left(-\frac{2}{3}\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }-\frac{2}{3}\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\frac{2}{3}\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+{\xi }_x{\xi }_x\frac{\partial v}{\partial \xi }+{\xi }_y{\xi }_y\frac{\partial v}{\partial \xi }+{\xi }_z{\xi }_z\frac{\partial v}{\partial \xi }+{\xi }_y{\xi }_x\frac{\partial u}{\partial \xi }+{\xi }_y{\xi }_y\frac{\partial v}{\partial \xi }+{\xi }_y{\xi }_z\frac{\partial w}{\partial \xi }+{\xi }_x{\eta }_x\frac{\partial v}{\partial \eta }+{\xi }_y{\eta }_y\frac{\partial v}{\partial \eta }+{\xi }_z{\eta }_z\frac{\partial v}{\partial \eta }+{\eta }_y{\xi }_x\frac{\partial u}{\partial \eta }+{\eta }_y{\xi }_y\frac{\partial v}{\partial \eta }+{\eta }_y{\xi }_z\frac{\partial w}{\partial \eta }+{\xi }_x{\zeta }_x\frac{\partial v}{\partial \zeta }+{\xi }_y{\zeta }_y\frac{\partial v}{\partial \zeta }+{\xi }_z{\zeta }_z\frac{\partial v}{\partial \zeta }+{\zeta }_y{\xi }_x\frac{\partial u}{\partial \zeta }+{\zeta }_y{\xi }_y\frac{\partial v}{\partial \zeta }+{\zeta }_y{\xi }_z\frac{\partial w}{\partial \zeta }\right]$$ $$=\mu \left[{\xi }_y\frac{2}{3}\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }-\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial v}{\partial \xi }+{\xi }_y\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial v}{\partial \eta }+{\eta }_y\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\eta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial v}{\partial \zeta }+{\zeta }_y\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\zeta }\right)\right]$$ $$=\mu \left[{\xi }_y\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial v}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial v}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial v}{\partial \zeta }-{\xi }_y\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)+{\eta }_y\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\eta }\right)-{\xi }_y\left(\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+{\zeta }_y\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\zeta }\right)\right]$$ $$=\mu \left[{\xi }_y\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial v}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial v}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial v}{\partial \zeta }+\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\xi }\right){\eta }_y-\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\eta }\right){\xi }_y+\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\xi }\right){\zeta }_y-\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\zeta }\right){\xi }_y\right]$$ Similarly, ${\widehat{f}}_{v\text{,}\mathrm{\rho w}}$ is found.

$${{\widehat{f}}_{v\text{,}\mathrm{\rho w}}=\xi }_x{\tau }_{xz}+{\xi }_y{\tau }_{yz}+{\xi }_z{\tau }_{zz}$$ $$=\mu \left[{\xi }_z\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial w}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial w}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial w}{\partial \zeta }+\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\xi }\right){\eta }_z-\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\eta }\right){\xi }_z+\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\xi }\right){\zeta }_z-\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\zeta }\right){\xi }_z\right]$$ Combining the above, the vector form of the viscous portion of the momentum equation, ${\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}$ , is found.
$$\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\xi }\right)\overrightarrow{\eta }-\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\eta }\right)\overrightarrow{\xi }+\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\xi }\right)\overrightarrow{\zeta }-\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\zeta }\right)\overrightarrow{\xi }\right]$$ or
$$\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\eta }\times \left(\overrightarrow{\eta }\times \overrightarrow{\xi }\right)+\Delta {\overrightarrow{U}}_{\zeta }\times \left(\overrightarrow{\zeta }\times \overrightarrow{\xi }\right)\right]$$ Similarly, ${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}$ is found.
$$\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\eta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\eta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\eta }\right)\overrightarrow{\xi }-\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\xi }\right)\overrightarrow{\eta }+\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\eta }\right)\overrightarrow{\zeta }-\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\zeta }\right)\overrightarrow{\eta }\right]$$ or
$$\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\eta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\eta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\xi }\times \left(\overrightarrow{\xi }\times \overrightarrow{\eta }\right)+\Delta {\overrightarrow{U}}_{\zeta }\times \left(\overrightarrow{\zeta }\times \overrightarrow{\eta }\right)\right]$$ Similarly, ${\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}$ is found.
$$\mu \left[\overrightarrow{\zeta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\zeta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\zeta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\zeta }\right)\overrightarrow{\xi }-\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\xi }\right)\overrightarrow{\zeta }+\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\zeta }\right)\overrightarrow{\eta }-\left(\Delta {\overrightarrow{U}}_{\eta }·\overrightarrow{\eta }\right)\overrightarrow{\zeta }\right]$$ or
$$\mu \left[\overrightarrow{\zeta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\zeta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\zeta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\xi }\times \left(\overrightarrow{\xi }\times \overrightarrow{\zeta }\right)+\Delta {\overrightarrow{U}}_{\eta }\times \left(\overrightarrow{\eta }\times \overrightarrow{\zeta }\right)\right]$$

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Energy equation

${\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}$ equation.

$${\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}={\xi }_x{b}_x+{\xi }_y{b}_y+{\xi }_z{b}_z$$ $$={\xi }_x\left(u{\tau }_{xx}+v{\tau }_{xy}+w{\tau }_{xz}\right)+{\xi }_y\left(u{\tau }_{xy}+v{\tau }_{yy}+w{\tau }_{yz}\right)+{\xi }_z\left(u{\tau }_{xz}+v{\tau }_{yz}+w{\tau }_{zz}\right)-{\xi }_x{q}_x-{\xi }_y{q}_y-{\xi }_z{q}_z$$ $$=u\left({\xi }_x{\tau }_{xx}+{\xi }_y{\tau }_{xy}+{\xi }_y{\tau }_{xz}\right)+v\left({\xi }_x{\tau }_{xy}+{\xi }_y{\tau }_{yy}+{\xi }_z{\tau }_{yz}\right)+w\left({\xi }_x{\tau }_{xz}+{\xi }_y{\tau }_{yz}+{\xi }_z{\tau }_{zz}\right)-{{\xi }_{x}q}_x-{{\xi }_{y}q}_y-{{\xi }_{z}q}_z$$ Focusing on the last three terms.

$$-{\xi }_x{q}_x-{\xi }_y{q}_y-{\xi }_z{q}_z=\kappa \left({\xi }_x\frac{\partial a^2}{\partial x}+{\xi }_y\frac{\partial a^2}{\partial y}+{\xi }_z\frac{\partial a^2}{\partial z}\right)$$ $$=\kappa \left({\xi }_x\left({\xi }_x\frac{\partial a^2}{\partial \xi }+{\eta }_x\frac{\partial a^2}{\partial \eta }+{\zeta }_x\frac{\partial a^2}{\partial \zeta }\right)+{\xi }_y\left({\xi }_y\frac{\partial a^2}{\partial \xi }+{\eta }_y\frac{\partial a^2}{\partial \eta }+{\zeta }_y\frac{\partial a^2}{\partial \zeta }\right)+{\xi }_z\left({\xi }_z\frac{\partial a^2}{\partial \xi }+{\eta }_z\frac{\partial a^2}{\partial \eta }+{\zeta }_z\frac{\partial a^2}{\partial \zeta }\right)\right)$$ $$=\kappa \left(\left({\xi }_x{\xi }_x+{\xi }_y{\xi }_y+{\xi }_z{\xi }_z\right)\frac{\partial a^2}{\partial \xi }+\left({\xi }_x{\eta }_x+{\xi }_y{\eta }_y+{\xi }_z{\eta }_z\right)\frac{\partial a^2}{\partial \eta }+\left({\xi }_x{\zeta }_x+{\xi }_y{\zeta }_y+{\xi }_z{\zeta }_z\right)\frac{\partial a^2}{\partial \zeta }\right)$$ $$=\kappa \left(\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$ Putting it back together.

$${\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}=u\left({\xi }_x{\tau }_{xx}+{\xi }_y{\tau }_{xy}+{\xi }_y{\tau }_{xz}\right)+v\left({\xi }_x{\tau }_{xy}+{\xi }_y{\tau }_{yy}+{\xi }_z{\tau }_{yz}\right)+w\left({\xi }_x{\tau }_{xz}+{\xi }_y{\tau }_{yz}+{\xi }_z{\tau }_{zz}\right)+\kappa \left(\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$ $${\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$ Similarily:

$${\widehat{g}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\left(\overrightarrow{\eta }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }+\left(\overrightarrow{\eta }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$ $${\widehat{h}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\left(\overrightarrow{\zeta }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\zeta }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$

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Axisymmetric

The following geometry relation will be assumed.

$$\overrightarrow{\eta }·\overrightarrow{\xi }=0$$ $$\overrightarrow{\eta }·\overrightarrow{\zeta }=0$$ ${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}$ equation.
$${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}=\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\eta }\right)\overrightarrow{\xi }-\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\xi }\right)\overrightarrow{\eta }+\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\eta }\right)\overrightarrow{\zeta }-\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\zeta }\right)\overrightarrow{\eta }\right]$$ Next, taking the vector triple product
$$\overrightarrow{\eta }\times \left(\overrightarrow{\eta }\times \Delta {\overrightarrow{U}}_{\eta }\right)=\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)\overrightarrow{\eta }-\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }$$ and rearranging it
$$\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }=\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)\overrightarrow{\eta }-\overrightarrow{\eta }\times \left(\overrightarrow{\eta }\times \Delta {\overrightarrow{U}}_{\eta }\right)$$ and then inserting it in the ${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}$ equation leads to
$${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}=\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)\overrightarrow{\eta }-\overrightarrow{\eta }\times \left(\overrightarrow{\eta }\times \Delta {\overrightarrow{U}}_{\eta }\right)+\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\eta }\right)\overrightarrow{\xi }-\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\xi }\right)\overrightarrow{\eta }+\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\eta }\right)\overrightarrow{\zeta }-\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\zeta }\right)\overrightarrow{\eta }\right]$$ Next, the velocity difference in the η direction is given by
$$\Delta {\overrightarrow{U}}_{\eta }=\overrightarrow{U}\times \Delta \overrightarrow{\Theta }$$ where
$$\Delta \overrightarrow{\Theta }$$ is along the axis.

Assuming U is in the η plane, then
$$\overrightarrow{\eta }\times \Delta {\overrightarrow{U}}_{\eta }=0$$ $$\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\xi }=0$$ $$\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\zeta }=0$$ $$\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\eta }=0$$ $$\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\eta }=0$$ Therefore
$${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}=\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)\overrightarrow{\eta }-\left(\Delta {\overrightarrow{U}}_{\xi }·\overrightarrow{\xi }\right)\overrightarrow{\eta }-\left(\Delta {\overrightarrow{U}}_{\zeta }·\overrightarrow{\zeta }\right)\overrightarrow{\eta }\right]$$ $$=\mu \left[\overrightarrow{\eta }\frac{2}{3}\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+2\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)\right]$$ $$=\mu \left[\overrightarrow{\eta }\frac{2}{3}\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+2\overrightarrow{\eta }·\left(\overrightarrow{U}\times \Delta \overrightarrow{\Theta }\right)-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)\right]$$ $$=\mu \left[\overrightarrow{\eta }\frac{2}{3}\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+2\left(\Delta \overrightarrow{\Theta }\times \overrightarrow{\eta }\right)·\overrightarrow{U}-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)\right]$$ Dot products are independent of the η index, therefore the following results when differencing in η direction.
$${\Delta }_{\eta }{\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}={\mu }_{i\text{,}j\text{,}k}\left[\left({\overrightarrow{\eta }}_{i\text{,}\text{j}+\mathrm{1/2}\text{,}k}-{\overrightarrow{\eta }}_{i\text{,}j-\mathrm{1/2}\text{,}k}\right)\frac{2}{3}{\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+2\left(\Delta \overrightarrow{\Theta }\times \overrightarrow{\eta }\right)·\overrightarrow{U}-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)}_{i\text{,}j\text{,}k}\right]$$ $${\Delta }_{\eta }{\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}={\mu }_{i\text{,}j\text{,}k}\left[\frac{1}{2}\left(\frac{{\overrightarrow{S}}_{i\text{,}j+1\text{,}k}^{\eta }}{{\mathrm{vol}}_{i\text{,}j+1\text{,}j}}-\frac{{\overrightarrow{S}}_{i\text{,}j-1\text{,}k}^{\eta }}{{\mathrm{vol}}_{i\text{,}j-1\text{,}k}}\right)\frac{2}{3}{\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+2\left(\Delta \overrightarrow{\Theta }\times \overrightarrow{\eta }\right)·\overrightarrow{U}-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)}_{i\text{,}j\text{,}k}\right]$$ Where $\overrightarrow{S}$ is the area vector for a given cell side (η in this case) and vol is the volume of the cell.

Next, the relationship that the area vectors of the cell walls must add to zero will be used.
$$\left({\overrightarrow{S}}_{i+1\text{,}j\text{,}k}^{\xi }-{\overrightarrow{S}}_{i-1\text{,}j\text{,}k}^{\xi }\right)+\left({\overrightarrow{S}}_{i\text{,}j+1\text{,}k}^{\eta }-{\overrightarrow{S}}_{i\text{,}j-1\text{,}k}^{\eta }\right)+\left({\overrightarrow{S}}_{i\text{,}j\text{,}k+1}^{\zeta }-{\overrightarrow{S}}_{i\text{,}j\text{,}k-1}^{\zeta }\right)=0$$ Rearranging,
$$\left({\overrightarrow{S}}_{i\text{,}j+1\text{,}k}^{\eta }-{\overrightarrow{S}}_{i\text{,}j-1\text{,}k}^{\eta }\right)=-\left({\overrightarrow{S}}_{i+1\text{,}j\text{,}k}^{\xi }-{\overrightarrow{S}}_{i-1\text{,}j\text{,}k}^{\xi }\right)-\left({\overrightarrow{S}}_{i\text{,}j\text{,}k+1}^{\zeta }-{\overrightarrow{S}}_{i\text{,}j\text{,}k-1}^{\zeta }\right)$$ Substituting this into ${\Delta }_{\eta }{\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}$ gives
$${\Delta }_{\eta }{\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}=-{\mu }_{i\text{,}j\text{,}k}\left[\frac{1}{2}\frac{\left({\overrightarrow{S}}_{i+1\text{,}j\text{,}k}^{\xi }-{\overrightarrow{S}}_{i-1\text{,}j\text{,}k}^{\xi }\right)+\left({\overrightarrow{S}}_{i\text{,}jk+1}^{\zeta }-{\overrightarrow{S}}_{i\text{,}j\text{,}k-1}^{\zeta }\right)}{{\mathrm{vol}}_{i\text{,}j\text{,}k}}\frac{2}{3}{\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+2\left(\Delta \overrightarrow{\Theta }\times \overrightarrow{\eta }\right)·\overrightarrow{U}-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)}_{i\text{,}j\text{,}k}\right]$$

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3D Thin Layer Set

Rearrange the 3D viscous momentum equations.

$${\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }\right]+\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\eta }\times \left(\overrightarrow{\eta }\times \overrightarrow{\xi }\right)+\Delta {\overrightarrow{U}}_{\zeta }\times \left(\overrightarrow{\zeta }\times \overrightarrow{\xi }\right)\right]$$ $${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }\right]+\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\eta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\eta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\xi }\times \left(\overrightarrow{\xi }\times \overrightarrow{\eta }\right)+\Delta {\overrightarrow{U}}_{\zeta }\times \left(\overrightarrow{\zeta }\times \overrightarrow{\eta }\right)\right]$$ $${\overrightarrow{\widehat{h}}}_{v\text{,}\rho U}=\mu \left[\overrightarrow{\zeta }\frac{1}{3}\left(\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }\right]+\mu \left[\overrightarrow{\zeta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)+\left(\overrightarrow{\zeta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\zeta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\Delta {\overrightarrow{U}}_{\xi }\times \left(\overrightarrow{\xi }\times \overrightarrow{\zeta }\right)+\Delta {\overrightarrow{U}}_{\eta }\times \left(\overrightarrow{\eta }\times \overrightarrow{\zeta }\right)\right]$$ Next, the cross product and dot terms are dropped, leaving only the thin-layer set.

$${\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }\right]$$ $${\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }$$ $${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }\right]$$ $${\widehat{g}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }$$ $${\overrightarrow{\widehat{h}}}_{v\text{,}\rho U}=\mu \left[\overrightarrow{\zeta }\frac{1}{3}\left(\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }\right]$$ $${\widehat{h}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }$$

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Viscous Jacobian

The viscous Jacobians are derived from the 3D thin layer set.

To help derive the viscous Jacobian the following viscous solution vector is defined.

$$Q_v=\left[\begin{array}{c}\rho \\ \frac{\left(\rho u\right)}{\rho }\\ \frac{\left(\rho v\right)}{\rho }\\ \frac{\left(\rho w\right)}{\rho }\\ \rho e_0\end{array}\right]$$ The viscous Jacobian in the η direction is then defined as follows.

$$\frac{\partial {\widehat{F}}_v}{\partial Q}=\frac{\partial {\widehat{F}}_v}{\partial Q_v}\frac{\partial Q_v}{\partial Q}$$ The derivative of the viscous Flux is then split into a part that explicitly contains the velocity and explicitly contains the speed of sound.

$$\frac{\partial {\widehat{F}}_v}{\partial Q_v}=\frac{\partial {\widehat{F}}_v}{\partial U}+\frac{\partial {\widehat{F}}_v}{\partial a^2}\frac{\partial \left(a^2\right)}{\partial Q_v}$$ Starting with, $\frac{\partial {\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}}{\partial U}$

$${\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}=\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }\right]=\mu \left[\begin{array}{ccc}\frac{1}{3}{\xi }_x{\xi }_x+\overrightarrow{\xi }·\overrightarrow{\xi }& \frac{1}{3}{\xi }_x{\xi }_y& \frac{1}{3}{\xi }_x{\xi }_z\\ \frac{1}{3}{\xi }_y{\xi }_x& \frac{1}{3}{\xi }_y{\xi }_y+\overrightarrow{\xi }·\overrightarrow{\xi }& \frac{1}{3}{\xi }_y{\xi }_z\\ \frac{1}{3}{\xi }_z{\xi }_x& \frac{1}{3}{\xi }_z{\xi }_y& \frac{1}{3}{\xi }_z{\xi }_z+\overrightarrow{\xi }·\overrightarrow{\xi }\end{array}\right]\left[\begin{array}{c}\Delta u\\ \Delta v\\ \Delta w\end{array}\right]$$ $${\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]\left[\begin{array}{c}\Delta u\\ \Delta v\\ \Delta w\end{array}\right]$$ $$\frac{\partial {\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}}{\partial U}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]$$ Next, $\frac{\partial \left({\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}·\overrightarrow{U}\right)}{\partial U}$

$${\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}·\overrightarrow{U}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{2}\left[\begin{array}{ccc}\left(u_{i+1}+u_i\right)& \left(v_{i+1}+v_i\right)& \left(w_{i+1}+w_i\right)\end{array}\right]\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]\left[\begin{array}{c}\Delta u\\ \Delta v\\ \Delta w\end{array}\right]$$ $$\frac{\partial \left({\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}·\overrightarrow{U}\right)}{\partial u}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{2}\left[\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]\left[\begin{array}{c}\Delta u_{\xi }\\ \Delta v_{\xi }\\ \Delta w_{\xi }\end{array}\right]+\left[\begin{array}{ccc}\left(u_{i+1}+u_i\right)& \left(v_{i+1}+v_i\right)& \left(w_{i+1}+w_i\right)\end{array}\right]\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\right]$$ Since the transpose of a scaler is equal to the scaler, the second matrix multiplication set can be transposed.

$$\frac{\partial \left({\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}·\overrightarrow{U}\right)}{\partial u}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{2}\left[\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]\left[\begin{array}{c}\Delta u_{\xi }\\ \Delta v_{\xi }\\ \Delta w_{\xi }\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]\left[\begin{array}{c}{u}_{i+1}+u_i\\ v_{i+1}+v_i\\ w_{i+1}+w_i\end{array}\right]\right]$$ $$\frac{\partial \left({\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}·\overrightarrow{U}\right)}{\partial u}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]\left[\begin{array}{c}\Delta u_{\xi }+\left(u_{i+1}+u_i\right)\\ \Delta v_{\xi }+\left(v_{i+1}+v_i\right)\\ \Delta w_{\xi }+\left(w_{i+1}+w_i\right)\end{array}\right]$$ $$\frac{\partial \left({\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}·\overrightarrow{U}\right)}{\partial u}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{ccc}\frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z\\ \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1\end{array}\right]\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$$ $$\frac{\partial \left({\overrightarrow{\widehat{f}}}_{v\text{,}\rho U}·\overrightarrow{U}\right)}{\partial u}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)$$ Finally the derivative of the viscous flux due to the explicit velocity terms is,

$$\frac{\partial {\widehat{F}}_v}{\partial U}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z& 0\\ 0& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z& 0\\ 0& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1& 0\\ 0& \left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& 0\end{array}\right]$$ Next, $\frac{\partial {\widehat{F}}_v}{\partial a^2}\frac{\partial \left(a^2\right)}{\partial Q_v}$

But first $\frac{\partial \left(a^2\right)}{\partial \xi }$ must be placed in terms of the viscous solution vector.

$$\frac{\partial \left(a^2\right)}{\partial \xi }=\frac{\partial \left(\frac{\gamma p}{\rho }\right)}{\partial \xi }=\frac{\gamma }{\rho }\frac{\partial p}{\partial \xi }-\frac{\gamma p}{{\rho }^2}\frac{\partial \rho }{\partial \xi }$$ $$p=\left(\gamma -1\right)\left(\rho e_0-\frac{1}{2}\rho \left(u^2+v^2+w^2\right)\right)$$ $$\frac{\partial p}{\partial \xi }=\left(\gamma -1\right)\left(\frac{\partial \left(\rho e_0\right)}{\partial \xi }-\frac{1}{2}\left(u^2+v^2+w^2\right)\frac{\partial \rho }{\partial \xi }-\rho u\frac{\partial u}{\partial \xi }-\rho v\frac{\partial v}{\partial \xi }-\rho w\frac{\partial w}{\partial \xi }\right)$$ $$\frac{\partial \left(a^2\right)}{\partial \xi }=\frac{\gamma \left(\gamma -1\right)}{\rho }\left(\frac{\partial \left(\rho e_0\right)}{\partial \xi }-\frac{1}{2}\left(u^2+v^2+w^2\right)\frac{\partial \rho }{\partial \xi }-\rho u\frac{\partial u}{\partial \xi }-\rho v\frac{\partial v}{\partial \xi }-\rho w\frac{\partial w}{\partial \xi }\right)-\frac{\gamma \left(\gamma -1\right)}{\rho }\frac{\left(\rho e_0-\frac{1}{2}\rho \left(u^2+v^2+w^2\right)\right)}{\rho }\frac{\partial \rho }{\partial \xi }$$ $$\frac{\partial \left(a^2\right)}{\partial \xi }=\frac{\gamma \left(\gamma -1\right)}{\rho }\left[\frac{\partial \left(\rho e_0\right)}{\partial \xi }-\rho u\frac{\partial u}{\partial \xi }-\rho v\frac{\partial v}{\partial \xi }-\rho w\frac{\partial w}{\partial \xi }-\frac{\left(\rho e_0\right)}{\rho }\frac{\partial \rho }{\partial \xi }\right]$$ By inspection it can be seen that:

$$\frac{\partial {\widehat{F}}_v}{\partial a^2}\frac{\partial \left(a^2\right)}{\partial Q_v}=\kappa \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{\gamma \left(\gamma -1\right)}{\rho }\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -\frac{\left(\rho e_0\right)}{\rho }& -\rho u& -\rho v& -\rho w& 1\end{array}\right]$$ Therefore the derivative of the viscous flux with respect to the viscous solution vector when taking into account that $\kappa =\frac{\mu }{\mathrm{Pr}\left(\gamma -1\right)}$, becomes:

$$\frac{\partial {\widehat{F}}_v}{\partial Q_v}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z& 0\\ 0& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z& 0\\ 0& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1& 0\\ 0& \left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& 0\end{array}\right]+\frac{\mu }{\mathrm{Pr}}\left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{\gamma }{\rho }\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -\frac{\left(\rho e_0\right)}{\rho }& -\rho u& -\rho v& -\rho w& 1\end{array}\right]$$ Next, dealing with $\frac{\partial Q_v}{\partial Q}$.

$$\frac{\partial Q_v}{\partial Q}=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& \frac{1}{\rho }& 0& 0& 0\\ 0& 0& \frac{1}{\rho }& 0& 0\\ 0& 0& 0& \frac{1}{\rho }& 0\\ 0& 0& 0& 0& 1\end{array}\right]-\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ \frac{\left(\rho u\right)}{{\rho }^2}& 0& 0& 0& 0\\ \frac{\left(\rho v\right)}{{\rho }^2}& 0& 0& 0& 0\\ \frac{\left(\rho w\right)}{{\rho }_2}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$$ $$\frac{\partial Q_v}{\partial Q}=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ -\frac{\left(\rho u\right)}{{\rho }^2}& \frac{1}{\rho }& 0& 0& 0\\ -\frac{\left(\rho v\right)}{{\rho }^2}& 0& \frac{1}{\rho }& 0& 0\\ -\frac{\left(\rho w\right)}{{\rho }^2}& 0& 0& \frac{1}{\rho }& 0\\ 0& 0& 0& 0& 1\end{array}\right]$$ Putting it all together.

$$\frac{\partial {\widehat{F}}_v}{\partial Q_v}\frac{\partial Q_v}{\partial Q}=\left[\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z& 0\\ 0& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z& 0\\ 0& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1& 0\\ 0& \left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& 0\end{array}\right]+\frac{\mu }{\mathrm{Pr}}\left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{\gamma }{\rho }\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -\frac{\left(\rho e_0\right)}{\rho }& -\rho u& -\rho v& -\rho w& 1\end{array}\right]\right]\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ -\frac{\left(\rho u\right)}{{\rho }^2}& \frac{1}{\rho }& 0& 0& 0\\ -\frac{\left(\rho v\right)}{{\rho }^2}& 0& \frac{1}{\rho }& 0& 0\\ -\frac{\left(\rho w\right)}{{\rho }^2}& 0& 0& \frac{1}{\rho }& 0\\ 0& 0& 0& 0& 1\end{array}\right]$$ Results in:

$$\frac{\partial {\widehat{F}}_v}{\partial Q}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{\rho }\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ -\left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z& 0\\ -\left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z& 0\\ -\left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1& 0\\ -\left(\frac{1}{3}{\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)}^2+\overrightarrow{U}·\overrightarrow{U}\right)& \left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& 0\end{array}\right]+\frac{\mu }{\mathrm{Pr}}\left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{\gamma }{\rho }\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -\left(\frac{\left(\rho e_0\right)}{\rho }-\overrightarrow{U}·\overrightarrow{U}\right)& -u& -v& -w& 1\end{array}\right]$$ The eigenvalues for this are:

$$\lambda =\frac{\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)}{\rho }\left(0\text{,}1\text{,}1\text{,}\frac{4}{3}\text{,}\frac{\gamma }{\mathrm{Pr}}\right)$$

---

Diagonal Viscous Jacobian

$$\frac{\partial {\widehat{F}}_v}{\partial Q}=T_k{T}_k^{-1}\left[{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_U+{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_a\right]T_k{T}_k^{-1}$$ Velocity part.

$$T_k^{-1}{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_U{T}_k=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{\rho }{T}_k^{-1}\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ -\left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z& 0\\ -\left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z& 0\\ -\left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1& 0\\ -\left(\frac{1}{3}{\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)}^2+\overrightarrow{U}·\overrightarrow{U}\right)& \left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& 0\end{array}\right]\left[\begin{array}{ccccc}{\tilde{k}}_x& {\tilde{k}}_y& {\tilde{k}}_z& \alpha & \alpha \\ {\tilde{k}}_{x}u& {\tilde{k}}_{y}u-{\tilde{k}}_z\rho & {\tilde{k}}_{z}u+{\tilde{k}}_y\rho & \alpha \left(u+{\tilde{k}}_{x}c\right)& \alpha \left(u-{\tilde{k}}_{x}c\right)\\ {\tilde{k}}_{x}v+{\tilde{k}}_z\rho & {\tilde{k}}_{y}v& {\tilde{k}}_{z}v-{\tilde{k}}_x\rho & \alpha \left(v+{\tilde{k}}_{y}c\right)& \alpha \left(v-{\tilde{k}}_{y}c\right)\\ {\tilde{k}}_{x}w-{\tilde{k}}_y\rho & {\tilde{k}}_{y}w+{\tilde{k}}_x\rho & {\tilde{k}}_{z}w& \alpha \left(w+{\tilde{k}}_{z}c\right)& \alpha \left(w-{\tilde{k}}_{z}c\right)\\ \left[{\tilde{k}}_x\frac{{\varphi }^2}{\left(\gamma -1\right)}+\rho \left({\tilde{k}}_{z}v-{\tilde{k}}_{y}w\right)\right]& \left[{\tilde{k}}_y\frac{{\varphi }^2}{\left(\gamma -1\right)}+\rho \left({\tilde{k}}_{x}w-{\tilde{k}}_{z}u\right)\right]& \left[{\tilde{k}}_z\frac{{\varphi }^2}{\left(\gamma -1\right)}+\rho \left({\tilde{k}}_{y}u-{\tilde{k}}_{x}v\right)\right]& \alpha \left[\frac{{\varphi }^2+c^2}{\left(\gamma -1\right)}+c\tilde{\theta }\right]& \alpha \left[\frac{{\varphi }^2+c^2}{\left(\gamma -1\right)}-c\tilde{\theta }\right]\end{array}\right]$$ $$T_k^{-1}{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_U{T}_k=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{\rho }{T}_k^{-1}\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& -{\tilde{k}}_z\rho & {\tilde{k}}_y\rho & \frac{4}{3}\alpha {\tilde{k}}_{x}c& -\frac{4}{3}\alpha {\tilde{k}}_{x}c\\ {\tilde{k}}_z\rho & 0& -{\tilde{k}}_x\rho & \frac{4}{3}\alpha {\tilde{k}}_{y}c& -\frac{4}{3}\alpha {\tilde{k}}_{y}c\\ -{\tilde{k}}_y\rho & {\tilde{k}}_x\rho & 0& \frac{4}{3}\alpha {\tilde{k}}_{z}c& -\frac{4}{3}\alpha {\tilde{k}}_{z}c\\ \rho \left({\tilde{k}}_{z}v-{\tilde{k}}_{y}w\right)& \rho \left({\tilde{k}}_{x}w-{\tilde{k}}_{z}u\right)& \rho \left({\tilde{k}}_{y}u-{\tilde{k}}_{x}v\right)& \left(\frac{4}{3}\alpha \left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)c\right)& \left(-\frac{4}{3}\alpha \left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)c\right)\end{array}\right]$$ $$T_k^{-1}{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_U{T}_k=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{\rho }\left[\begin{array}{ccccc}\left[{\tilde{k}}_x\left(1-\frac{{\varphi }^2}{c^2}\right)-{\rho }^{-1}\left({\tilde{k}}_{z}v-{\tilde{k}}_{y}w\right)\right]& {\tilde{k}}_x\left(\gamma -1\right)u{c}^{-2}& \left[{\tilde{k}}_z{\rho }^{-1}+{\tilde{k}}_x\left(\gamma -1\right)v{c}^{-2}\right]& \left[-{\tilde{k}}_y{\rho }^{-1}+{\tilde{k}}_x\left(\gamma -1\right)w{c}^{-2}\right]& -{\tilde{k}}_x\left(\gamma -1\right)c^{-2}\\ \left[{\tilde{k}}_y\left(1-\frac{{\varphi }^2}{c^2}\right)-{\rho }^{-1}\left({\tilde{k}}_{x}w-{\tilde{k}}_{z}u\right)\right]& \left[-{\tilde{k}}_z{\rho }^{-1}+{\tilde{k}}_y\left(\gamma -1\right)u{c}^{-2}\right]& {\tilde{k}}_y\left(\gamma -1\right)v{c}^{-2}& \left[{\tilde{k}}_x{\rho }^{-1}+{\tilde{k}}_y\left(\gamma -1\right)w{c}^{-2}\right]& {-\tilde{k}}_y\left(\gamma -1\right)c^{-2}\\ \left[{\tilde{k}}_z\left(1-\frac{{\varphi }^2}{c^2}\right)-{\rho }^{-1}\left({\tilde{k}}_{y}u-{\tilde{k}}_{x}v\right)\right]& \left[{\tilde{k}}_y{\rho }^{-1}+{\tilde{k}}_z\left(\gamma -1\right)u{c}^{-2}\right]& \left[-{\tilde{k}}_x{\rho }^{-1}+{\tilde{k}}_z\left(\gamma -1\right)v{c}^{-2}\right]& {\tilde{k}}_z\left(\gamma -1\right)w{c}^{-2}& -{\tilde{k}}_z\left(\gamma -1\right)c^{-2}\\ \beta \left({\varphi }^2-c\tilde{\theta }\right)& \beta \left[{\tilde{k}}_{x}c-\left(\gamma -1\right)u\right]& \beta \left[{\tilde{k}}_{y}c-\left(\gamma -1\right)v\right]& \beta \left[{\tilde{k}}_{z}c-\left(\gamma -1\right)w\right]& \beta \left(\gamma -1\right)\\ \beta \left({\varphi }^2+c\tilde{\theta }\right)& -\beta \left[{\tilde{k}}_{x}c+\left(\gamma -1\right)u\right]& -\beta \left[{\tilde{k}}_{y}c+\left(\gamma -1\right)v\right]& -\beta \left[{\tilde{k}}_{z}c+\left(\gamma -1\right)w\right]& \beta \left(\gamma -1\right)\end{array}\right]\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& -{\tilde{k}}_z\rho & {\tilde{k}}_y\rho & \frac{4}{3}\alpha {\tilde{k}}_{x}c& -\frac{4}{3}\alpha {\tilde{k}}_{x}c\\ {\tilde{k}}_z\rho & 0& -{\tilde{k}}_x\rho & \frac{4}{3}\alpha {\tilde{k}}_{y}c& -\frac{4}{3}\alpha {\tilde{k}}_{y}c\\ -{\tilde{k}}_y\rho & {\tilde{k}}_x\rho & 0& \frac{4}{3}\alpha {\tilde{k}}_{z}c& -\frac{4}{3}\alpha {\tilde{k}}_{z}c\\ \rho \left({\tilde{k}}_{z}v-{\tilde{k}}_{y}w\right)& \rho \left({\tilde{k}}_{x}w-{\tilde{k}}_{z}u\right)& \rho \left({\tilde{k}}_{y}u-{\tilde{k}}_{x}v\right)& \left(\frac{4}{3}\alpha \left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)c\right)& \left(-\frac{4}{3}\alpha \left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)c\right)\end{array}\right]$$ $$T_k^{-1}{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_U{T}_k=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{\rho }\left[\begin{array}{ccccc}1-{\tilde{k}}_x{\tilde{k}}_x& -{\tilde{k}}_y{\tilde{k}}_x& -{\tilde{k}}_z{\tilde{k}}_x& 0& 0\\ -{\tilde{k}}_x{\tilde{k}}_y& 1-{\tilde{k}}_y{\tilde{k}}_y& -{\tilde{k}}_z{\tilde{k}}_y& 0& 0\\ -{\tilde{k}}_x{\tilde{k}}_z& -{\tilde{k}}_y{\tilde{k}}_z& 1-{\tilde{k}}_z{\tilde{k}}_z& 0& 0\\ 0& 0& 0& \frac{2}{3}& -\frac{2}{3}\\ 0& 0& 0& -\frac{2}{3}& \frac{2}{3}\end{array}\right]$$ $${\lambda }_{U\text{,}U}=1\text{,}1\text{,}0$$ $${\lambda }_{U\text{,}a}=\frac{4}{3}\text{,}0$$ Speed of sound part.

$$T_k^{-1}{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_a{T}_k=\frac{\mu }{\mathrm{Pr}}\left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{\gamma }{\rho }{T}_k^{-1}\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -\left(\frac{\left(\rho e_0\right)}{\rho }-\overrightarrow{U}·\overrightarrow{U}\right)& -u& -v& -w& 1\end{array}\right]\left[\begin{array}{ccccc}{\tilde{k}}_x& {\tilde{k}}_y& {\tilde{k}}_z& \alpha & \alpha \\ {\tilde{k}}_{x}u& {\tilde{k}}_{y}u-{\tilde{k}}_z\rho & {\tilde{k}}_{z}u+{\tilde{k}}_y\rho & \alpha \left(u+{\tilde{k}}_{x}c\right)& \alpha \left(u-{\tilde{k}}_{x}c\right)\\ {\tilde{k}}_{x}v+{\tilde{k}}_z\rho & {\tilde{k}}_{y}v& {\tilde{k}}_{z}v-{\tilde{k}}_x\rho & \alpha \left(v+{\tilde{k}}_{y}c\right)& \alpha \left(v-{\tilde{k}}_{y}c\right)\\ {\tilde{k}}_{x}w-{\tilde{k}}_y\rho & {\tilde{k}}_{y}w+{\tilde{k}}_x\rho & {\tilde{k}}_{z}w& \alpha \left(w+{\tilde{k}}_{z}c\right)& \alpha \left(w-{\tilde{k}}_{z}c\right)\\ \left[{\tilde{k}}_x\frac{{\varphi }^2}{\left(\gamma -1\right)}+\rho \left({\tilde{k}}_{z}v-{\tilde{k}}_{y}w\right)\right]& \left[{\tilde{k}}_y\frac{{\varphi }^2}{\left(\gamma -1\right)}+\rho \left({\tilde{k}}_{x}w-{\tilde{k}}_{z}u\right)\right]& \left[{\tilde{k}}_z\frac{{\varphi }^2}{\left(\gamma -1\right)}+\rho \left({\tilde{k}}_{y}u-{\tilde{k}}_{x}v\right)\right]& \alpha \left[\frac{{\varphi }^2+c^2}{\left(\gamma -1\right)}+c\tilde{\theta }\right]& \alpha \left[\frac{{\varphi }^2+c^2}{\left(\gamma -1\right)}-c\tilde{\theta }\right]\end{array}\right]$$ $$T_k^{-1}{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_a{T}_k=\frac{\mu }{\mathrm{Pr}}\left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{\gamma }{\rho }{T}_k^{-1}\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -{\tilde{k}}_x\frac{c^2}{\gamma \left(\gamma -1\right)}& -{\tilde{k}}_y\frac{c^2}{\gamma \left(\gamma -1\right)}& -{\tilde{k}}_z\frac{c^2}{\gamma \left(\gamma -1\right)}& -\frac{\alpha c^2}{\gamma }& -\frac{\alpha c^2}{\gamma }\end{array}\right]$$ $$T_k^{-1}{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_a{T}_k=\frac{\mu }{\mathrm{Pr}}\left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{\rho }\left[\begin{array}{ccccc}\left[{\tilde{k}}_x\left(1-\frac{{\varphi }^2}{c^2}\right)-{\rho }^{-1}\left({\tilde{k}}_{z}v-{\tilde{k}}_{y}w\right)\right]& {\tilde{k}}_x\left(\gamma -1\right)u{c}^{-2}& \left[{\tilde{k}}_z{\rho }^{-1}+{\tilde{k}}_x\left(\gamma -1\right)v{c}^{-2}\right]& \left[-{\tilde{k}}_y{\rho }^{-1}+{\tilde{k}}_x\left(\gamma -1\right)w{c}^{-2}\right]& -{\tilde{k}}_x\left(\gamma -1\right)c^{-2}\\ \left[{\tilde{k}}_y\left(1-\frac{{\varphi }^2}{c^2}\right)-{\rho }^{-1}\left({\tilde{k}}_{x}w-{\tilde{k}}_{z}u\right)\right]& \left[-{\tilde{k}}_z{\rho }^{-1}+{\tilde{k}}_y\left(\gamma -1\right)u{c}^{-2}\right]& {\tilde{k}}_y\left(\gamma -1\right)v{c}^{-2}& \left[{\tilde{k}}_x{\rho }^{-1}+{\tilde{k}}_y\left(\gamma -1\right)w{c}^{-2}\right]& {-\tilde{k}}_y\left(\gamma -1\right)c^{-2}\\ \left[{\tilde{k}}_z\left(1-\frac{{\varphi }^2}{c^2}\right)-{\rho }^{-1}\left({\tilde{k}}_{y}u-{\tilde{k}}_{x}v\right)\right]& \left[{\tilde{k}}_y{\rho }^{-1}+{\tilde{k}}_z\left(\gamma -1\right)u{c}^{-2}\right]& \left[-{\tilde{k}}_x{\rho }^{-1}+{\tilde{k}}_z\left(\gamma -1\right)v{c}^{-2}\right]& {\tilde{k}}_z\left(\gamma -1\right)w{c}^{-2}& -{\tilde{k}}_z\left(\gamma -1\right)c^{-2}\\ \beta \left({\varphi }^2-c\tilde{\theta }\right)& \beta \left[{\tilde{k}}_{x}c-\left(\gamma -1\right)u\right]& \beta \left[{\tilde{k}}_{y}c-\left(\gamma -1\right)v\right]& \beta \left[{\tilde{k}}_{z}c-\left(\gamma -1\right)w\right]& \beta \left(\gamma -1\right)\\ \beta \left({\varphi }^2+c\tilde{\theta }\right)& -\beta \left[{\tilde{k}}_{x}c+\left(\gamma -1\right)u\right]& -\beta \left[{\tilde{k}}_{y}c+\left(\gamma -1\right)v\right]& -\beta \left[{\tilde{k}}_{z}c+\left(\gamma -1\right)w\right]& \beta \left(\gamma -1\right)\end{array}\right]\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -{\tilde{k}}_x\frac{c^2}{\left(\gamma -1\right)}& -{\tilde{k}}_y\frac{c^2}{\left(\gamma -1\right)}& -{\tilde{k}}_z\frac{c^2}{\left(\gamma -1\right)}& -\alpha c^2& -\alpha c^2\end{array}\right]$$ $$T_k^{-1}{\left(\frac{\partial {\widehat{F}}_v}{\partial Q}\right)}_a{T}_k=\frac{\mu }{\mathrm{Pr}}\left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{\rho }\left[\begin{array}{ccccc}{k}_x{k}_x& k_x{k}_y& k_x{k}_z& k_x\alpha \left(\gamma -1\right)& k_x\alpha \left(\gamma -1\right)\\ k_y{k}_x& k_y{k}_y& k_y{k}_z& k_y\alpha \left(\gamma -1\right)& k_y\alpha \left(\gamma -1\right)\\ k_z{k}_x& k_z{k}_y& k_z{k}_z& k_z\alpha \left(\gamma -1\right)& k_z\alpha \left(\gamma -1\right)\\ -k_x\beta c^2& -k_y\beta c^2& -k_z\beta c^2& -\frac{\left(\gamma -1\right)}{2}& -\frac{\left(\gamma -1\right)}{2}\\ -k_x\beta c^2& -k_y\beta c^2& -k_z\beta c^2& -\frac{\left(\gamma -1\right)}{2}& -\frac{\left(\gamma -1\right)}{2}\end{array}\right]$$ $${\lambda }_a=2-\gamma \text{,}0\text{,}0\text{,}0\text{,}0$$

References

1) Krist, S. L., Biedron, R. T., and Rumsey, C. L., "CFL3D User's Manual (Version 5.0)," NASA TM 1998-208444, June 1998.