The Riemann problem for fluid flow of real materials is examined. An arbitrary equation of state is allowed, subject only to the physical requirements of thermodynamics. The properties of the isentropes and the shock Hugoniot loci that follow from conditions imposed on the equation of state are reviewed systematically. Important properties of these wave curves are determined by three dimensionless variables characterizing the equation of state: the adiabatic exponent $\gamma$, the Grüneisen coefficient $\Gamma$, and the fundamental derivative $\mathcal{G}$. Standard assumptions on these variables break down near phase transitions. The result is an anomalous wave structure: either shock waves split into multiple waves, or composite waves form. Additional questions related to shock stability and nonuniqueness of the solution of the Riemann problem are discussed.

DOI:https://doi.org/10.1103/RevModPhys.61.75