{\bf v}_n = \frac{R}{T},\end{aligned}$$ where $R$ is the dissipative function, $T$ is the temperature; finally, the conservation of the energy, $E$, reads $$\begin{aligned}
\frac{\partial E}{\partial t}  +  \nabla \cdot  Q = 0 ,\end{aligned}$$ where $ Q$ is the energy current. Equations above should be supplemented by the Euler equations for the superfluid and the normal fluid $$\begin{aligned}
\label{eq:AS:EULER_S}
\rho_S\left[ \frac{\partial {\bf v}_S}{\partial t} 
 + ({\bf v}_S\cdot
\nabla)\cdot {\bf v}_S\right]& = & -
\frac{\rho_S}{\rho}\nabla p -\rho_S\nabla\phi + F,\\
\label{eq:AS:EULER_N}
\rho_N\left[\frac{\partial {\bf v}_N }{\partial t} 
+ ({\bf v}_N\cdot\nabla)\cdot {\bf v}_N \right]&=&-\frac{\rho_N}{\rho}\nabla p
- \rho_N\nabla \phi +\eta_N\Delta {\bf v}_N -  F,\end{aligned}$$ where $ F$ is the mutual friction force, and $\eta_N$ is the viscosity of the normal fluid, $\phi$ is the Newtonian gravitational potential. To determine the unknowns in the hydrodynamic equations, let us write the total energy of the fluid in the frame in which the normal fluid is at rest as $$\begin{aligned}
\label{eq:ASAS_ENERGY}
E =\frac{1}{2}\rho v_S^2 
+ ( J -  \rho {\bf v}_S) \cdot {\bf v}_S + {\cal E},\end{aligned}$$ where the internal energy ${\cal E}$ is given by the second law of thermodynamics as $$\begin{aligned}
d{\cal E} = TdS + \mu d\rho 
+ ({\bf v}_N-{\bf v}_S) \cdot d( J -  \rho 