  We provide a derivation of several classes of boundary conditions for fluids
of Korteweg-type using a simple and transparent thermodynamic approach that
automatically guarentees that the derived boundary conditions are compatible
with the second law of thermodynamics. The starting assumption of our approach
is to describe the boundary of the domain 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 continuum 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 conditions, 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 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.
