(\rho \,c_{pt})_m \frac{\partial T}{\partial t} 
 
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \  \nabla T 
 
 - \nabla (\lambda_t \nabla T) 

- \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial p_\alpha}{\partial t}  
 
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg)  \nabla p

=  \frac{\delta E_H}{ \delta V \delta t}

\rho \, c_p \, \frac{\partial T}{\partial t} 
- {\bf \nabla}\, \left( \lambda \, {\bf \nabla} T \right)  + \rho \, c_p \, {\bf u} \, {\bf \nabla} T 
-  \phi \, \rho \ c_p \ \eta_s \ \frac{\partial p}{\partial t} 
+ \rho \ c_p \ \epsilon \ {\bf u} {\bf \nabla} p
=  \frac{\delta E_H}{ \delta V \delta t}