The first law of thermodynamics in hydrodynamic steady and unsteady flows

We studied planar compressible flows of ideal gas as models of a non-equilibrium thermodynamic system. We demonstrate that internal energy $U(S^{*},V,N)$ of such systems in stationary and non-stationary states is the function of only three parameters of state, i.e. non-equilibrium entropy $S^{*}$, volume $V$ and number of particles $N$ in the system. Upon transition between different states, the system obeys the first thermodynamic law, i.e. $dU=T^{*}dS^{*}-p^{*}dV+{\mu}^{*}dN$, where $U=3/2 NRT^{*}$ and $p^{*}V=NRT^{*}$. Placing a cylinder inside the channel, we find that U depends on the location of the cylinder $y_{c}$ only via the parameters of state, i.e. $U(S^{*}(y_{c}),V,N(y_{c}))$ at V=const. Moreover, when the flow around the cylinder becomes unstable, and velocity, pressure, and density start to oscillate as a function of time, t, U depends on t only via the parameters of state, i.e. $U(S^{*}(t),V,N(t))$ for V=const. These examples show that such a form of internal energy is robust and does not depend on the particular boundary conditions even in the unsteady flow.