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+μ^*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.