On a thermodynamic framework for developing boundary conditions for Korteweg-type fluids

Abstract

We provide a derivation of several classes of boundary conditions for fluids of Korteweg- type using a simple and transparent thermodynamic approach that automatically guaran- tees that the derived boundary conditions are compatible with the second law of thermo- dynamics. The starting assumption of our approach is to describe the boundary of the do- main as the membrane separating two different continua, one inside the domain, and the other outside the domain. With this viewpoint one may employ the framework of con- tinuum thermodynamics involving singular surfaces. This approach allows us to identify, for various classes of surface Helmholtz free energies, the corresponding surface entropy production mechanisms. By establishing the constitutive relations that guarantee that the surface entropy production is non-negative, we identify a new class of boundary condi- tions, which on one hand generalizes in a nontrivial manner the Navier’s slip boundary conditions, and on the other hand describes dynamic and static contact angle conditions. We explore the general model in detail for a particular case of a Korteweg fluid where the Helmholtz free energy in the bulk is that of a van der Waals fluid. We perform a series of numerical experiments to document the basic qualitative features of the novel boundary conditions and their practical applicability to model phenomena such as the contact angle hysteresis.