On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system
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Date
2012-06-15
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Abstract
A new method will be introduced for the derivation of thermodynamically consistent boundary
conditions for the full Cahn-Hilliard-Navier-Stokes-Fourier system for two immiscible fluids, where the phase
field variable (order parameter) is given in terms of concentrations or partial densities. Five different types
of models will be presented and discussed. The article can be considered as a continuation of a previous work
by Heida, Málek and Rajagopal [16], which focused on the derivation and generalization of Cahn-Hilliard-
Navier-Stokes models. The method is based on the assumption of maximum rate of entropy production by
Rajagopal and Srinivasa [30]. This assumption will be generalized to surfaces of bounded domains using
an integral formulation of the balance of entropy. Following [30], the calculations are based on constitutive
equations for the bulk energy, the surface energy and the rates of entropy production in the bulk and on
the surface. The resulting set of boundary conditions will consist of dynamic boundary conditions for the
Cahn-Hilliard equation and either generalized Navier-slip, perfect slip or no-slip boundary conditions for the
balance of linear momentum. Additionally, we will find that we also have to impose a boundary condition on
the normal derivative of the normal component of the velocity field. The new approach has the advantage
that the calculations are very transparent, the resulting equations come up very naturally and it is obvious
how the calculations can be generalized to more than two fluids or more general constitutive assumptions
for the energies. Additionally to former approaches, the approach also yields the full balance of energy for
thewhole system. Finally, a possible explanation will be given for the “rolling” movement of the contact line,
first observed in Dussan and Davis [8].