Internal energy in compressible Poiseuille flow

We analyse a compressible Poiseuille flow of ideal gas in a plane channel. We provide the form of internal energy U for a non-equilibrium stationary state (NESS) that includes viscous dissipation and pressure work. We demonstrate that U depends strongly on the ratio {\Delta}p/p_0, where {\Delta}p is the pressure difference between inlet and outlet and p_0 is the outlet's pressure. In addition, U depends on two other variables: the channel aspect ratio and the parameter equivalent to Reynolds number. The stored internal energy, {\Delta}U=U-U0, is small compared to the internal energy U0 of the equilibrium state (ES) for a moderate range of values of {\Delta}p/p_0. However, {\Delta}U can become large for big {\Delta}p or close to vacuum conditions at the outlet (p_0~0 Pa).