The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion: Nonisothermal case (Q1112679)

We discuss the nature of a van der Waals fluid and the main assumptions of the system. We formulate the Riemann problem and then describe briefly the elementary waves which arise in the problem, namely, rarefaction waves, shocks, contact discontinuities and phase boundaries. We show, that there exists a one-parameter family of solutions to the Riemann problem in consideration. Then, we show that the consequence of the entropy rate admissibility criterion agrees with the result of classical thermodynamics. Namely, we compute the first and second derivatives of the entropy rate with respect to the parameter to see that the stationary phase boundary is admissible if the physical entropies on the left and on the right of the phase boundary at \(t=0\) are equal and is not admissible if they are not equal. In order to enforce the applicability of the entropy rate admissibility criterion, we shown an example of nontrivial solution which minimizes the entropy rate locally among the solutions assumed in this paper.