# 1.10. OpenFOAM and driftFluxFoam¶

These are notes on driftFluxFoam (Brennan 2001).

## 1.10.1. Two-fluids Model¶

### 1.10.1.1. Conservation of Mass¶

$\begin{split}{{\partial } \over {\partial t}}(\alpha_{n} \rho_n) + \nabla \cdot (\alpha_{n} \rho_n \vec{u}_{n}) = 0 \\\end{split}$

### 1.10.1.2. Conservation of Momentum¶

${{\partial } \over {\partial t}} (\alpha_{n} \rho_n \vec{u}_{n}) + \nabla \cdot ( \alpha_{n} \rho_n \vec{u}_{n} \vec{u}_{n}) = - {{\alpha_{n}}} \nabla {p_n} + \nabla \cdot ( \alpha_{n} \vec{\vec{\tau}}_{n} ) + \alpha_{n} \rho_n \vec{g} + \vec{\vec{M}}$

## 1.10.2. Mixture Model¶

• Consider a mixture, not two phases: one continuity equation, one momentum equation and one convective diffuction equation for the dispersed phase

• Increased efficiency: reduces number of equations from 4 to 3

• Increased robustness: interphase momentum transfer term disappears (as momentum equations are added and the terms are equal and opposite)

• Physically realistic: main slip between phases is gravitational settling, the settling velocity

• New concepts: mixture density, centre of mass of mixture velocity, and diffusion velocities with respect to centre of mass of mixture velocity

### 1.10.2.1. Mixture Density¶

$\rho_m = \alpha_1 \rho_1 + \alpha_2 \rho_2$

### 1.10.2.2. Centre of Mass of Mixture Velocity¶

$u_m = {{\alpha_1 \rho_1 u_1 + \alpha_2 \rho_2 u_2} \over {\rho_m}}$

### 1.10.2.3. Diffusion Velocities wrt Centre of Mass of Mixture Velocity¶

$u_{1m} = u_1 - u_m$
$u_{2m} = u_2 - u_m$

### 1.10.2.4. Mixture Conservation of Mass¶

The above concepts can be used to derive a new continuity equation for the mixture model:

${{\partial \rho_m} \over {\partial t}} + \nabla \cdot (\rho_m \vec{u}_m) = 0$