The Model

The idea of the diffuse interface goes back to van der Waals who proposed square-gradient theory of the interface. The dynamic generalizations of the models of that type have been proposed in works of B. U. Felderhof, J. S. Langer and L. A. Turski, D. M. Anderson, and, perhaps, others. The set of dynamical equations comprises conservation of mass:

,

conservation of momentum:

,

and equation for the entropy density :

,

where is the medium mass density, is the velocity, is the specific entropy (entropy per unit mass), is the temperature, is the thermodynamic pressure, and is the constant related to the liquid-vapor surface tension. Greek indices label vector and tensor components in Cartesian coordinates and the familiar summation convention is understood; stands for and denotes partial derivative with respect to the time .

The viscous stress tensor is supposed to be of the Newtonian form with bulk viscosity set to zero

.

The heat flux is given by the Fourier Law

.

In the last formula is the specific heat for constant volume, is the particle density (i.e., , where is the particle mass). Kinematic viscosity as well as thermal diffusivity are assumed constant.

Momentum equation coincides with the classical Navier-Stokes equation except of the capillary stress tensor

,

which models the forces associated with the interface.

The above system of equations must be supplemented by two equations of state: and . We use van der Waals approximate formulas with parameters characteristic for argon in that place.