🧲 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 \(\Omega \subset \mathbb{R}^{d}\), \(d\in\left\lbrace2,3\right\rbrace\), the incompressible constant density magnetohydrodynamic (or simply incompressible MHD) equations are given as

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

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 \(\mu\).

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