On compressible Korteweg fluid-like materials

We provide a thermodynamic basis for compressible fluids of a Korteweg type that are characterized by the presence of the dyadic product of the density gradients ∇ϱ ⊗ ∇ϱ in the constitutive equation for the Cauchy stress. Our approach does not need to introduce any new or non-standard concepts such as multipolarity or interstitial working and is based on prescribing the constitutive equations for two scalars: the entropy and the entropy production. In comparison with the Navier–Stokes–Fourier fluids we suppose that the entropy is not only a function of the internal energy and the density but also of the density gradient. The entropy production takes the same form as for a Navier–Stokes–Fourier fluid. For a Navier–Stokes–Fourier fluid one can express the entropy production equivalently in terms of either thermodynamic affinities or thermodynamic fluxes. Following the ideas of K.R. Rajagopal concerning the systematic development of implicit constitutive theory and primary role of thermodynamic fluxes (such as force) that are cause of effects in thermodynamic affinities (such as deformation) in considered processes, we further proceed with a constitutive equation for entropy production expressed in terms of thermodynamic fluxes. The constitutive equation for the Cauchy stress is then obtained by maximizing the form of the rate of entropy production with respect to thermodynamic fluxes keeping as the constraint the equation expressing the fact that the entropy production is the scalar product of thermodynamic fluxes and thermodynamic affinities. We also look at how the form of the constitutive equation changes if the material in question is incompressible or if the processes take place at constant temperature. In addition, we provide several specific examples for the form of the internal energy and make the link to models proposed earlier. Starting with fully implicit constitutive equation for the entropy production, we also outline how the methodology presented here can be extended to non-Newtonian fluid models containing the Korteweg tensor in a straightforward manner.