MHD

Contents

MHD#

In magnetohydrodynamics (MHD), finite element analysis can be applied to solve a variety of problems related to the behavior of electrically conducting fluids (plasmas or liquid metals) in the presence of magnetic fields. Here are some common problems in magnetohydrodynamics that can be addressed using finite element methods.

Finite element analysis provides a powerful and flexible framework for addressing these complex problems in magnetohydrodynamics by discretizing the governing equations and solving them numerically. The specific application will determine the details of the problem setup and the required numerical techniques.

MHD Equations#

Magnetohydrodynamics (MHD) is a branch of physics and fluid dynamics that studies the magnetic properties and behavior of electrically conducting fluids, such as plasmas, liquid metals, and saltwater. The MHD equations describe the coupled interactions between the magnetic field, fluid flow, and electric current. Here are the mathematical models for the MHD equations:

  1. MHD Continuity Equation: \( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \) where:

    • \(\rho\) is the fluid density,

    • \(\mathbf{v}\) is the fluid velocity.

  2. MHD Momentum Equation: \( \rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \rho \mathbf{g} + \nabla \cdot \boldsymbol{\tau} + \mathbf{J} \times \mathbf{B} \) where:

    • \(\frac{D\mathbf{v}}{Dt}\) is the material derivative of velocity,

    • \(p\) is the pressure,

    • \(\mathbf{g}\) is the gravitational acceleration,

    • \(\boldsymbol{\tau}\) is the stress tensor,

    • \(\mathbf{J}\) is the current density,

    • \(\mathbf{B}\) is the magnetic field.

  3. MHD Induction Equation: \( \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \) where:

    • \(\eta\) is the magnetic diffusivity.

  4. MHD Ohm’s Law: \( \mathbf{J} = \sigma (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \) where:

    • \(\sigma\) is the electrical conductivity,

    • \(\mathbf{E}\) is the electric field.

  5. MHD Energy Equation: \( \rho C_p \frac{DT}{Dt} = -p \nabla \cdot \mathbf{v} + \nabla \cdot (k \nabla T) + \frac{1}{\sigma}(\mathbf{J} \cdot \mathbf{E}) \) where:

    • \(C_p\) is the specific heat at constant pressure,

    • \(T\) is the temperature,

    • \(k\) is the thermal conductivity.

These equations, when solved together with appropriate boundary conditions, describe the behavior of magnetized fluids in the presence of electric and magnetic fields. Keep in mind that the specific form of these equations may vary based on the assumptions made and the type of fluid being considered (e.g., ideal MHD vs. resistive MHD).

Contents