The version of the second law of thermodynamics stated as "Entropy always increases" (a version which, according to A. Eddington, holds "the supreme position among the laws of Nature") is in fact a theorem deduced by Clausius in 1865:
Jos Uffink, Bluff your Way in the Second Law of Thermodynamics, p. 37: "Hence we obtain: THE ENTROPY PRINCIPLE (Clausius' version) For every nicht umkehrbar [irreversible] process in an adiabatically isolated system which begins and ends in an equilibrium state, the entropy of the final state is greater than or equal to that of the initial state. For every umkehrbar [reversible] process in an adiabatical system, the entropy of the final state is equal to that of the initial state." http://philsci-archive.pitt.edu/archive/00000313/
Clausius' deduction was based on three postulates:
Postulate 1 (implicit): The entropy is a state function.
Postulate 2: Clausius' inequality (formula 10 on p. 33 in Uffink's paper) is correct.
Postulate 3: Any irreversible process can be closed by a reversible process to become a cycle.
All the three postulates remain totally unjustified even nowadays. Postulate 1 can easily be disproved by considering cycles (heat engines) converting heat into work in ISOTHERMAL conditions. Postulate 3 is almost obviously false:
Uffink, p.39: "A more important objection, it seems to me, is that Clausius bases his conclusion that the entropy increases in a nicht umkehrbar [irreversible] process on the assumption that such a process can be closed by an umkehrbar [reversible] process to become a cycle. This is essential for the definition of the entropy difference between the initial and final states. But the assumption is far from obvious for a system more complex than an ideal gas, or for states far from equilibrium, or for processes other than the simple exchange of heat and work. Thus, the generalisation to all transformations occurring in Nature is somewhat rash."
Note that, even if Clausius's theorem were correct (it is not), it only holds for "an adiabatically isolated system which begins and ends in an equilibrium state". This means that (even if Clausius's theorem were correct) all applications of "Entropy always increases" to processes which do not begin and end in equilibrium would be still unjustified!