Energy Density Conservation Equation

Thermal energy evolution equation
$\frac{\partial U}{\partial t}=-\bm{\nabla \cdot q}-\bm{\nabla \cdot} (U\bm{v})-\bm{\nabla \cdot }(\bm{{\rm P}\cdot v}) + \bm{v \cdot}(\bm{\nabla \cdot {\rm P}})$

$U$ is the thermal energy density. $t$ is the time. $\bm{q}$ is the heat flux vector. $\bm{v}$ is the bulk velocity. $\bm{\rm P}$ is the pressure tensor.

Continuity equation
$\frac{\partial n}{\partial t} + \bm{\nabla \cdot} (n \bm{v}) = 0$

Where $n$ is the number density of a species.

Because
$U=\frac{1}{2}(P_{xx}+P_{yy}+P_{zz})$

We consider the kinetic temperature $T$
$T=\frac{1}{3}\frac{P_{xx}+P_{yy}+P_{zz}}{n}$

Then, the relationship between thermal energy $U$ and temperature $T$
$U=\frac{3}{2}nT$

Then
$\frac{\partial U}{\partial t} = \frac{\partial}{\partial t}(\frac{3}{2}nT)=\frac{3}{2}T\frac{\partial n}{\partial t} + \frac{3}{2}n\frac{\partial T}{\partial t}=\frac{3}{2}n\frac{\partial T}{\partial t}-\frac{3}{2}T\bm{\nabla \cdot}(n \bm{v})$

And
$\bm{\nabla \cdot }(\bm{{\rm P} \cdot v})=\bm{v \cdot } (\bm{\nabla \cdot {\rm P}})+(\bm{{\rm P}\cdot \nabla})\bm{\cdot v}$

Then
$\frac{\partial U}{\partial t}=-\bm{\nabla \cdot q}-\bm{\nabla \cdot} (U\bm{v})-(\bm{{\rm P}\cdot \nabla})\bm{\cdot v}$

And
$\bm{\nabla \cdot} (U\bm{v})=\bm{\nabla \cdot} (\frac{3}{2}nT\bm{v})=\frac{3}{2}T\bm{\nabla \cdot}(n\bm{v})+\frac{3}{2}n\bm{v \cdot \nabla}T$

Then
$\frac{3}{2}n\frac{\partial T}{\partial t}-\frac{3}{2}T\bm{\nabla \cdot}(n \bm{v}) = -\bm{\nabla \cdot q} - \frac{3}{2}T\bm{\nabla \cdot}(n\bm{v}) - \frac{3}{2}n\bm{v \cdot \nabla}T - (\bm{{\rm P}\cdot \nabla})\bm{\cdot v}$

$\frac{3}{2}n\frac{\partial T}{\partial t} = -\bm{\nabla \cdot q} - \frac{3}{2}n\bm{v \cdot \nabla}T - (\bm{{\rm P}\cdot \nabla})\bm{\cdot v}$

Then
$\frac{3}{2}n(\frac{\partial T}{\partial t} + \bm{v \cdot \nabla}T) = -\bm{\nabla \cdot q} - (\bm{{\rm P}\cdot \nabla})\bm{\cdot v}$

Let
$\bm{{\rm P=P}}-p\bm{{\rm I}}+p\bm{{\rm I}} = \bm{{\rm P'}}+p\bm{{\rm I}}$

Where $\bm{\rm I}$ is the unit tensor, and $p$ is the scalar pressure.

Then
$\frac{3}{2}n(\frac{\partial T}{\partial t} + \bm{v \cdot \nabla}T) = -\bm{\nabla \cdot q} - (\bm{{\rm P'}\cdot \nabla})\bm{\cdot v} - p(\bm{{\rm I}\cdot \nabla})\bm{\cdot v}$

Finally, the energy density conservation equation is
$\frac{3}{2}n (\frac{\partial T}{\partial t}+\bm{v} \bm{\cdot \nabla}T)+p\bm{\nabla \cdot v}=-\bm{\nabla \cdot q}-(\bm{{\rm P'} \cdot \nabla})\bm{\cdot v}$