1.1. Governing equations

We will deal with an incompressible isothermal two-phase system of Newtonian fluids without phase change only. The momentum and continuity equations of each phase are given by

(1.1)\[ \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} = - \frac{\nabla p}{\rho} + \frac{\nabla \cdot \boldsymbol{\tau}}{\rho} + \mathbf{g} \]

(1.2)\[ \frac{1}{\rho} \frac{D \rho}{Dt} + \nabla \cdot \mathbf{v} = 0\]

where \(\mathbf{v}\) is the velocity, \(t\) is the time, \(p\) is the pressure, \(\rho\) is the density, \(\mathbf{g}\) is the acceleration of gravity, and \(\boldsymbol{\tau}\) is the viscous stress tensor given by

(1.3)\[ \boldsymbol{\tau} = \mu \left\{ \nabla \mathbf{v} + ( \nabla \mathbf{v} )^{T} \right\}\]

Here, \(\mu\) is the viscosity. For fluids with constant density and viscosity, these equations reduce to

(1.4)\[ \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} = - \frac{\nabla p}{\rho} + \nu \nabla^{2} \mathbf{v} + \mathbf{g} \]

(1.5)\[ \nabla \cdot \mathbf{v} = 0\]

where \(\nu~(=\mu / \rho)\) is the kinematic viscosity. These equations hold in each phase, while we need to account for boundary conditions at the interfaces between the two phases.