Two-Phase Flow (Immersed Boundary Method)

Mathematical Model

Consider the incompressible Navier-Stokes equations with the Immersed Boundary Method for simulating two-phase flows:

\[\begin{split} \begin{align} \rho \frac{\partial \mathbf{v}}{\partial t} + \rho (\mathbf{v} \cdot \nabla) \mathbf{v} &= -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}_{\text{IB}} \\ \nabla \cdot \mathbf{v} &= 0 \end{align} \end{split}\]

where \(\mathbf{f}_{\text{IB}}\) is the immersed boundary force.

Boundary Conditions

• Inlet and outlet: Specify velocity or pressure conditions.

• Immersed boundary conditions to model the interface.

Initial Conditions

Specify initial conditions for velocity, pressure, and the immersed boundary.

Weak Formulation

Find \(\mathbf{v} \in H_0^1(\Omega)\) and \(p \in L^2(\Omega)\) such that:

\[\begin{split} \begin{align} &\int_{\Omega} \rho \frac{\partial \mathbf{v}}{\partial t} \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \mathbf{f}_{\text{IB}} \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega \\ &= 0 \end{align} \end{split}\]

for all test functions \(\mathbf{w} \in H_0^1(\Omega)\) and \(q \in L^2(\Omega)\).