Modeling of multiphase multicomponent flow with phase transitions
by the means of density functional theory
The basic concepts and equations are presented. The proof of equations is omitted.
1. Multiphase flow of gas and liquid with constant temperature
The dynamics is governed by the conservation equations for the mixture chemical components and the momentum. The tensor notations are used (indices i, j, k denote the mixture component numbers, indices a, b denote the Cartesian coordinate numbers; the summation over repeated indices are implied).


(1) 


(2) 
Here
 molar density of the i–th mixture component,
 flow vector of th component,
 average mass velocity,
 mass density,
 molar mass of i–th component,
 stress tensor in the mixture,
 gravitational potential.
The flow of th component are represented as
,
where
 vector of ith component diffusion flow.
By definition the stress tensor are constructed of the static stress tensor
and the viscous stress tensor :
.
For the viscous stress tensor the NavierStokes linear viscous model is used


(3) 
where and  volume and shear viscosity correspondingly.
In the classical NavierStokes liquid model the static stress tensor is where pthe pressure in the liquid. In the density functional theory this tensor has another expression:


(4) 
where
 Helmholz energy per unit volume,
 Helmholz energy of the homogenous mixture per unit volume,
 coefficients of the positive symmetric matrix,
 generalized chemical potentials of the mixture components.
The diffusion flow is determined by the generalized Fick’s law


(5) 
where  the symmetric nonnegative matrix which satisfies the condition
.
2. Multiphase isothermal flow with presence of solid phase
The dynamics is still governed by the equations (1), (2). However, in this case the elastic stresses inside solid phase must be taken into account. This causes certain modifications of static stress tensor (4), though the expressions for viscous stress tensor (3) and diffusion flows (5) remain the same.
The expression of static stress tensor is
,
 Helmholz energy of solid phase per unit volume,
 Lame coefficients (which vanish inside liquid phase),
 tensor of finite deformations,
.
The evolution of deformations is determined by the velocity field in accordance with the equation
