Starting from the compressible Navier-Stokes equations, I want to derive the standard form the instationary heat equation.
The energy equation in general form can be written as
$$ \begin{align} \frac{\partial E}{\partial t}+\nabla\cdot \left(H\vec{v}\right)&=\nabla\cdot\left(\mathbf{\sigma}\cdot\vec{v}-\vec{q}\right) \end{align} $$
or in a more specific form
$$ \begin{align} \frac{\partial \left(\rho \epsilon + \frac{1}{2} \rho |\vec{v}|^2\right)}{\partial t} + \nabla \cdot \left(\left[ \rho \epsilon + \frac{1}{2} \rho |\vec{v}|^2 + p \right] \vec{v} \right) = \dots \\ \dots \nabla \cdot \Bigg\langle \left( \eta \left[ \left( \nabla\otimes\vec{v} \right)^\top+\nabla\otimes\vec{v} \right]-\frac{2}{3}\eta \left(\nabla\cdot\vec{v}\right)\mathbf{1}\right)\cdot\vec{v}+\lambda \nabla T \Bigg\rangle. \end{align} $$
Using the caloric ideal EoS
$$ \begin{align} \epsilon=c_v T, \quad h=c_p T,\quad h=\epsilon + p/ \rho \end{align} $$
we get
$$ \begin{align} \frac{\partial \left(\rho c_{v} T + \frac{1}{2} \rho |\vec{v}|^2\right)}{\partial t} + \nabla \cdot \left(\left[ \rho c_{p} T + \frac{1}{2} \rho |\vec{v}|^2 \right] \vec{v} \right) = \dots \\ \dots \nabla \cdot \Bigg\langle \left( \eta \left[ \left( \nabla\otimes\vec{v} \right)^\top+\nabla\otimes\vec{v} \right]-\frac{2}{3}\eta \left(\nabla\cdot\vec{v}\right)\mathbf{1}\right)\cdot\vec{v}+\lambda \nabla T \Bigg\rangle. \end{align} $$
Now assuming
- Zero velocity, $\vec{v}=0$,
- Zero viscosity, $\eta=0$,
I end up with the following heat equation
$$ \begin{align} \frac{\partial (\rho c_v T)}{\partial t} = \left( \lambda T_{x} \right)_{x} + \left( \lambda T_{y} \right)_{y} + \left( \lambda T_{z} \right)_{z}, \end{align} $$
and not
$$ \begin{align} \frac{\partial (\rho c_p T)}{\partial t} = \left( \lambda T_{x} \right)_{x} + \left( \lambda T_{y} \right)_{y} + \left( \lambda T_{z} \right)_{z}. \end{align} $$
as given in most literature, see Wikipedia.
Any hints are appreciated, thanks!
