thermodynamics - Conservation of heat equation: what represent heat, enthalpy or internal energy?

I'm trying to write the heat transfer equation in an arbitrary fluid (compressible and viscous). Consider an adiabatic system where the only heat generated is due to the internal friction/viscosity. In a fluid, the dissipation function is:

$$\Phi_v= \check{\tau}:\check{\nabla}^T\check{\nu} \tag{1}$$

$\boldsymbol{\tau}$ is defined as

$$\check{\tau}=\check{\sigma}-P\check{I} \tag{2}$$

Where $\boldsymbol{\sigma}$ is the Cauchy stress tensor and $P$ is the hydrostatic pressure.

Assuming that conduction and radiation are negligible, and convection is the only form of heat transfer, I think I can write the conservation of heat as:

$$\frac{\partial}{\partial t}\left( q \right) +\check{\nabla}\left( q \, \check{\nu}^T \right)= \check{\tau}: \check{\nabla}^T \check{\nu} \tag{3}$$

Now what I do not understand is which one represent heat: enthalpy $h$ or internal energy $e$? For example I'm trying to write the equation for an ideal gas. Can I use enthalpy and write the conservation of heat as:

$$\frac{\partial}{\partial t}\left( \rho T \right) +\check{\nabla}\left( \rho T \, \check{\nu}^T \right)= \frac{1}{c_P} \check{\tau}:\check{\nabla}^T\check{\nu} \tag{4}$$

Or should I use the internal energy:

$$\frac{\partial}{\partial t}\left( \rho T \right) +\check{\nabla}\left( \rho T \, \check{\nu}^T \right)= \frac{1}{c_{\nu}} \check{\tau}:\check{\nabla}^T\check{\nu} \tag{5}$$

Which one of the equations 4 or 5 is the correct form? Or maybe they are both wrong?

Notation:

• $\check{A}$ is the matrix representation of tensor $\boldsymbol{A}$


• $:$ is the the double dot product of two square matrices (i.e. $\check{A}:\check{B}=a_{ij}b_{ji}$ in Einstein notation form)


• $\check{a}^T\check{b}$ (Where $\check{a}$ and $\check{b}$ are row matrices) is the dyadic product of first rank tensors $\boldsymbol{a}$ and $\boldsymbol{b}$, usually noted as $\boldsymbol{a}\boldsymbol{b}$. Also known as outer product $\vec{a} \otimes \vec{b}$ in vector form.


• $\check{A}\check{B}$ is the matrix multiplication and for row matrices $\check{a}\check{b}^T \equiv\vec{a}.\vec{b}$ dot product in vector form