Magnetohydrodynamics (MHD) Flow

Mathematical Model

Consider the MHD equations coupling the Navier-Stokes equations with Maxwell’s equations:

\[\begin{split} \begin{align} \rho \frac{\partial \mathbf{v}}{\partial t} + \rho (\mathbf{v} \cdot \nabla) \mathbf{v} &= -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{J} \times \mathbf{B} \\ \frac{\partial \mathbf{B}}{\partial t} &= \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \\ \nabla \cdot \mathbf{v} &= 0 \\ \nabla \cdot \mathbf{B} &= 0 \end{align} \end{split}\]

where \(\mathbf{J} = \nabla \times \mathbf{B}\) is the current density.

Boundary Conditions

• Inlet and outlet: Specify velocity or pressure conditions.

• Magnetic field conditions depending on the problem.

Initial Conditions

Specify initial conditions for velocity, pressure, and the magnetic field.

Weak Formulation

Find \(\mathbf{v} \in H_0^1(\Omega)\), \(p \in L^2(\Omega)\), and \(\mathbf{B} \in H(\text{curl}; \Omega)\) such that:

\[\begin{split} \begin{align} &\int_{\Omega} \rho \frac{\partial \mathbf{v}}{\partial t} \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \frac{\partial \mathbf{B}}{\partial t} \cdot \boldsymbol{\psi} \, d\Omega - \int_{\Omega} \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \cdot \boldsymbol{\psi} \, d\Omega \\ &+ \int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega + \int_{\Omega} \nabla \cdot \mathbf{B} r \, d\Omega \\ &= 0 \end{align} \end{split}\]

for all test functions \(\mathbf{w} \in H_0^1(\Omega)\), \(q \in L^2(\Omega)\), \(\boldsymbol{\psi} \in H(\text{curl}; \Omega)\), and \(r \in L^2(\Omega)\).