Steady-state thermodynamics of a system with heat and mass flow coupling

Equilibrium thermodynamics describes the energy exchange of a body with its environment. Here, we describe the global energy exchange of an ideal gas in the Coutte flow in a thermodynamic-like manner. We derive a fundamental relation between internal energy as a function of parameters of state. We analyze a non-equilibrium transition in the system and postulate the extremum principle, which determines stable steady states in the system. The steady-state thermodynamic framework resembles equilibrium thermodynamics.