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+^*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 y₂ only via the parameters of state, i. e. U (S^* (y₂), V, N (y₂) ) 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.