Direction of spontaneous processes in non-equilibrium systems with movable/permeable internal walls

The second law of equilibrium thermodynamics explains the direction of spontaneous processes in a system after removing internal constraints. When the system only exchanges energy with the environment as heat, the second law states that spontaneous processes at constant temperature satisfy: d U - δ Q ≤ 0. Here, d U is the infinitesimal change of the internal energy, and δ Q is the infinitesimal heat exchanged in the process. We will consider ideal gas, van der Waals gas, and a binary mixture of ideal gases in a heat flow. We will divide each system into two subsystems by a movable wall. We will show that the direction of the motion of the wall, after release, at constant boundary conditions is determined by the same inequality as in equilibrium thermodynamics. The only difference between equilibrium and non-equilibrium is the dependence of the net heat change, δ Q, on the state parameters of the system. We will study the influence of the gravitational field. The inequality determining the direction of motion of the internal wall at constant boundary conditions is d E - δ Q - δ W_p ≤ 0, where d E is the change of the total energy (internal and gravitational), and δ W_p is the infinitesimal work performed by gravity. We will also consider a wall thick and permeable to gas particles and derive Archimedes' principle in the heat flow. Finally, we will study the ideal gas's Couette flow, where the direction of the motion of the internal wall follows from the inequality d E - δ Q - δ W_s ≤ 0, with d E being the infinitesimal change of the total energy (internal and kinetic) and δ W_s the infinitesimal work exchanged with the environment due to shear force. Ultimately, we will synthesize all these cases in a framework of the second law of non-equilibrium thermodynamics.