Interpretation of Second Law of Thermodynamics in the presence of interfaces

In 1996, Muschik and Ehrentraut (J. Non-Equilib. Thermodyn. 21:175–192, 1996) proposed an amendment to the classical Second Law of Thermodynamics, which asserts that, except in equilibria, reversible process directions in state space do not exist. As a consequence of this statement, they proved that the Second Law of Thermodynamics necessarily restricts the constitutive equations and not the thermodynamic processes. In this way, the classical Coleman–Noll approach to the exploitation of Second Law (Coleman and Noll in Arch. Rational Mech. Anal. 13:167–178, 1963) follows by a rigorous proof. In the present paper, we generalize the amendment, in order to encompass the case, not considered in Muschik and Ehrentraut (J. Non-Equilib. Thermodyn. 21:175–192, 1996), in which there are surfaces across which the unknown fields suffer jump discontinuities. Due to the generalization above, we prove that the same conclusions of Muschik and Ehrentraut (J. Non-Equilib. Thermodyn. 21:175–192, 1996) can be achieved also in the presence of non-regular processes. As an application, we study the thermodynamics of a Kortweg-type fluid interface.