2.2.2. Governing Equations for the Fluid Flow

In this study, two immiscible, incompressible, isothermal Newtonian fluids are considered. Hence, one can describe the two-phase flow by the Navier-Stokes equations in the following form:

$$\nabla \cdot \mathbf{u} = 0 \tag{7}$$

$$\frac{\partial(\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla p - \nabla \cdot \left[ \mu (\nabla \mathbf{u} + (\nabla \mathbf{u})^{\mathsf{T}}) \right] + \mathbf{f}\_{\sigma} + \rho \mathbf{g} \tag{8}$$

The density and viscosity fields depend on the order parameter as

$$
\rho = \frac{1+\mathcal{C}}{2}\rho\_L + \frac{1-\mathcal{C}}{2}\rho\_{G\prime} \tag{9}
$$

$$
\mu = \frac{1+\mathcal{C}}{2}\mu\_L + \frac{1-\mathcal{C}}{2}\mu\_{G\prime} \tag{10}
$$

where *ρL*,*<sup>G</sup>* and *μL*/*<sup>G</sup>* are the density and viscosity of the pure phases. In Equation (8), *p* denotes the pressure field and **f***σ* the surface tension force which reads

$$\mathbf{f}\_{\sigma} = -\mathbb{C}\nabla\Phi.\tag{11}$$
