Particle-Laden Flow (Lagrangian-Eulerian Approach)

Mathematical Model

Consider the incompressible Navier-Stokes equations coupled with the Lagrangian motion of particles:

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

where \(\mathbf{F}_{\text{drag}}\) represents the drag force on the particles.

Boundary Conditions

• Inlet and outlet: Specify velocity or pressure conditions.

• Solid particle boundaries: Implement no-slip conditions.

Initial Conditions

Specify initial conditions for velocity, pressure, and particle positions.

Weak Formulation

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

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

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