The internal energy as a function of state parameters in steady and unsteady Poiseuille flows

We studied planar compressible Poiseuille flows of an ideal gas, both in steady and unsteady states, to identify the minimal number of state parameters required to describe changes in internal energy. In previous work (Phys. Rev. E 104, 055107 (2021)), five parameters were needed for steady flows. Here, using global non-equilibrium thermodynamics, we reduce this number to three: non-equilibrium entropy S *, volume V, and number of particles N. The internal energy U(S *, V, N) of such systems in stationary and non-stationary states is the function of non-equilibrium entropy S *, volume V and number of particles N in the system irrespective of any processes, number of boundary conditions or imposed constraints. We tested this by placing a cylinder inside the channel, finding that U depends on the cylinder’s location y c only via the state parameters S *(y c ) and N(y c ) for V = const. Moreover, in cases where the flow becomes unstable and parameters such as velocity and pressure oscillate, U depends on time t only through S *(t) and N(t) for V = const. These results demonstrate that this formulation of internal energy remains robust and consistent, even in unsteady flows with varying boundary conditions.