On the characterization of constitutive equations for third grade viscous Korteweg fluids

We consider a model of a third-grade viscous Korteweg-type fluid in three space dimensions and apply the extended Liu procedure in order to explicitly solve the constraints imposed by the entropy principle on the nonlocal constitutive relations. We detail the algorithm we use and are able to characterize the material functions involved in the constitutive equations. In a natural way, the application of the extended Liu procedure allows us to recover an extra-term in the entropy flux, preserving all the features of third-grade viscous Korteweg-type fluids. Moreover, a further constraint, in order to avoid that at equilibrium only very special phase boundaries are admissible, is investigated.