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:

$continuity of mass$,

conservation of momentum:

$momentum conservation$,

and equation for the entropy density $\rho s$:

$entropy equation$,

where $\rho$ is the medium mass density, $u_\alpha$ is the velocity, $s$ is the specific entropy (entropy per unit mass), $T$ is the temperature, $p$ is the thermodynamic pressure, and $K$ 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 $a_{\alpha}b_{\alpha} = a_xb_x + a_yb_y + a_zb_z$ is understood; $\D{\alpha}$ stands for $\partial/\partial x_{\alpha}$ and $\D{t}$ denotes partial derivative with respect to the time $t$.

The viscous stress tensor $\sigma^{(v)}_{\alpha\beta}$ is supposed to be of the Newtonian form with bulk viscosity set to zero

$viscous stress$.

The heat flux $q_{\alpha}$ is given by the Fourier Law

$heat flux$.

In the last formula $c_v$ is the specific heat for constant volume, $n$ is the particle density (i.e., $\rho=m n$, where $m$ is the particle mass). Kinematic viscosity $\nu$ as well as thermal diffusivity $\chi$ are assumed constant.

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

$capillary stress$,

which models the forces associated with the interface.

The above system of equations must be supplemented by two equations of state: $p=p(n,T)$ and $s=s(n,T)$. We use van der Waals approximate formulas with parameters characteristic for argon in that place.