# 🧲 MHD#

MHD (Magnetohydrodynamics) (also called magneto-fluid dynamics or hydromagnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single continuous medium. It is primarily concerned with the low-frequency, large-scale, magnetic behavior in plasmas and liquid metals and has applications in numerous fields including geophysics, astrophysics, and engineering.

—wikipedia

## Incompressible MHD#

In a connected, bounded domain $$\Omega \subset \mathbb{R}^{d}$$, $$d\in\left\lbrace2,3\right\rbrace$$ with Lipschitz boundary $$\partial \Omega$$, the incompressible constant density magnetohydrodynamic (or simply incompressible MHD) equations are given as

(2)#\begin{split}\begin{equation} \begin{aligned} \rho \left[ \partial_t\boldsymbol{u}^* + \left(\boldsymbol{u}^* \cdot \nabla\right)\boldsymbol{u}^* \right] - \tilde{\mu} \Delta \boldsymbol{u}^* - \boldsymbol{j}^* \times \boldsymbol{B}^* + \nabla p^* &= \rho \boldsymbol{f}^*, \\ \nabla\cdot \boldsymbol{u}^* &= 0 ,\\ \partial_t \boldsymbol{B}^* + \nabla\times \boldsymbol{E}^* &= \boldsymbol{0} ,\\ \boldsymbol{j}^* - \sigma \left(\boldsymbol{E}^* + \boldsymbol{u}^*\times\boldsymbol{B}^*\right) &= \boldsymbol{0} , \\ \boldsymbol{j}^* - \nabla\times \boldsymbol{H}^* &= \boldsymbol{0} ,\\ \boldsymbol{B}^* &= \mu \boldsymbol{H}^*, \end{aligned} \end{equation}\end{split}

where

• $$\boldsymbol{u}^*$$ fluid velocity

• $$\boldsymbol{j}^*$$ electric current density

• $$\boldsymbol{B}^*$$ magnetic flux density

• $$p^*$$ hydrodynamic pressure

• $$\boldsymbol{f}^*$$ body force

• $$\boldsymbol{E}^*$$ electric field strength

• $$\boldsymbol{H}^*$$ magnetic field strength

subject to material parameters the fluid density $$\rho$$, the dynamic viscosity $$\tilde{\mu}$$, the electric conductivity $$\sigma$$, and the magnetic permeability $$\mu$$.

By selecting the characteristic quantities of length $$L$$, velocity $$U$$, and magnetic flux density $$B$$, a non-dimensional formulation of (2) is

(3)#\begin{split}\begin{equation} \begin{aligned} \partial_t\boldsymbol{u} + \left(\boldsymbol{u} \cdot \nabla\right)\boldsymbol{u} - \mathrm{R}_f^{-1} \Delta \boldsymbol{u} - \mathrm{A}_l^{-2}\boldsymbol{j} \times \boldsymbol{B} + \nabla p &= \boldsymbol{f}, \\ \nabla\cdot \boldsymbol{u} &= 0 ,\\ \partial_t \boldsymbol{B} + \nabla\times \boldsymbol{E} &= \boldsymbol{0} ,\\ \mathrm{R}_m^{-1}\boldsymbol{j} - \left(\boldsymbol{E} + \boldsymbol{u}\times\boldsymbol{B}\right) &= \boldsymbol{0} , \\ \boldsymbol{j} - \nabla\times \boldsymbol{B} &= \boldsymbol{0} ,\\ \end{aligned} \end{equation}\end{split}

where $$\boldsymbol{u}$$, $$\boldsymbol{j}$$, $$\boldsymbol{B}$$, $$p$$, $$\boldsymbol{f}$$, and $$\boldsymbol{E}$$ are the non-dimensional variables, and $$\mathrm{R}_f = \dfrac{\rho U L}{\tilde{\mu}} = \dfrac{U L}{\nu}$$ (with $$\nu=\dfrac{\tilde{\mu}}{\rho}$$ being the kinematic viscosity) is the fluid Reynolds number, $$\mathrm{A}_l = \dfrac{U\sqrt{\rho \mu}}{B} = \dfrac{U}{U_A}$$ (with $$U_A = \dfrac{B}{\sqrt{\rho\mu}}$$ being the Alfvén speed), and $$\mathrm{R}_m = \mu\sigma U L$$ is the magnetic Reynolds number.

If we further introduce $$\boldsymbol{\omega}:=\nabla\times\boldsymbol{u}$$ and $$P:=p+\frac{1}{2}\boldsymbol{u}\cdot \boldsymbol{u}$$, (3) can be written into the rotational form:

(4)#\begin{split}\begin{equation} \begin{aligned} \partial_t\boldsymbol{u} + \boldsymbol{\omega}\times\boldsymbol{u} - \mathrm{R}_f^{-1} \Delta \boldsymbol{u} - \mathrm{A}_l^{-2}\boldsymbol{j} \times \boldsymbol{B} + \nabla P &= \boldsymbol{f}, \\ \boldsymbol{\omega} - \nabla\times\boldsymbol{u} &= \boldsymbol{0} ,\\ \nabla\cdot \boldsymbol{u} &= 0 ,\\ \partial_t \boldsymbol{B} + \nabla\times \boldsymbol{E} &= \boldsymbol{0} ,\\ \mathrm{R}_m^{-1}\boldsymbol{j} - \left(\boldsymbol{E} + \boldsymbol{u}\times\boldsymbol{B}\right) &= \boldsymbol{0} , \\ \boldsymbol{j} - \nabla\times \boldsymbol{B} &= \boldsymbol{0} .\\ \end{aligned} \end{equation}\end{split}

## Numerical Examples#

For numerical examples of MHD, see

↩️ Back to Gallery🖼.