Heat equation for a compressible fluid

Question

I'm trying to answer the question I have asked here. It is really confusing as different references contradict each other as I have explained here. So far I have seen two different versions of heat equation for compressible fluids:

$\rho \frac{D e}{D t}=-P\left( \check{\nabla}\check{\nu}^T \right)+ \check{\tau}:\check{\nabla}\check{\nu} \tag{1}$

and 

$\rho \frac{D h}{D t}=-\frac{D P}{D t}+ \check{\tau}:\check{\nabla}\check{\nu} \tag{2}$

I would appreciate if you could help me know what is the correct form and if the ones above are equal how I can drive one from the other.