## 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.