  The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface
model which is based on van der Waals theory of capillarity. Diffuse interface
methods have gained much interest to model two-phase flow in porous media.
However, for the numerical solution of the NSK equations two major challenges
have to be faced. First, an extended numerical stencil is required due to a
third-order term in the linear momentum and the total energy equations. In
addition, the dispersive contribution in the linear momentum equations prevents
the straightforward use of contact angle boundary conditions. Secondly, any
real gas equation of state is based on a non-convex Helmholtz free energy
potential which may cause the eigenvalues of the Jacobian of the first-order
fluxes to become imaginary numbers inside the spinodal region. In this work, a
thermodynamically consistent relaxation model is presented which is used to
approximate the NSK equations. The model is complimented by thermodynamically
consistent non-equilibrium boundary conditions which take contact angle effects
into account. Due to the relaxation approach, the contribution of the Korteweg
tensor in the linear momentum and total energy equations can be reduced to
second-order terms which enables a straightforward implementation of contact
angle boundary conditions in a numerical scheme. Moreover, the definition of a
modified pressure function enables to formulate first-order fluxes which remain
strictly hyperbolic in the entire spinodal region. The present work is a
generalization of a previously presented parabolic relaxation model for the
isothermal NSK equations.
