Governing Equations

Abstract

The set of equations governing the viscous incompressible fluid motion under the influence of an external MF are the momentum and mass conservation equations, given by

$$\begin{aligned} \rho \left( \frac{\partial {\varvec{u}}}{\partial t}+{\varvec{u}}\cdot \nabla {\varvec{u}}\right) = -\nabla p +\nabla \cdot \mathbb {S}+{\varvec{F}}_{s}+{\varvec{F}}_{l}+{\varvec{S}} \end{aligned}$$

$$\begin{aligned} \nabla \cdot {\varvec{u}}=0 \end{aligned}$$

where $\mathbb {S}$ is the viscous stress tensor written as

$$\begin{aligned} \mathbb {S}=2\mu \mathbb {D}=2\mu \left( \frac{1}{2}(\nabla {\varvec{u}}+\nabla {\varvec{u}}^{T})\right) \end{aligned}$$

with density $\rho$, pressure p and dynamics viscosity $\mu$. ${\varvec{F}}_{s}$ stands for the surface tension which just acts on the interface and is formulated as a volume force based on the CSF technique.