🧲 MHD

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🧲 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 $Ω \subset R^{d}$ , $d \in {2, 3}$ , the incompressible constant density magnetohydrodynamic (or simply incompressible MHD) equations are given as

(2)#

\begin{array}{r} \begin{aligned} ρ [\partial_{t} u^{*} + (u^{*} \cdot \nabla) u^{*}] - \tilde{μ} Δ u^{*} - j^{*} \times B^{*} + \nabla p^{*} & = ρ f^{*}, \\ \nabla \cdot u^{*} & = 0, \\ \partial_{t} B^{*} + \nabla \times E^{*} & = 0, \\ j^{*} - σ (E^{*} + u^{*} \times B^{*}) & = 0, \\ j^{*} - \nabla \times H^{*} & = 0, \\ B^{*} & = μ H^{*}, \end{aligned} \end{array}

where

$u^{*}$ fluid velocity
$j^{*}$ electric current density
$B^{*}$ magnetic flux density
$p^{*}$ hydrodynamic pressure
$f^{*}$ body force
$E^{*}$ electric field strength
$H^{*}$ magnetic field strength

subject to material parameters the fluid density $ρ$ , the dynamic viscosity $\tilde{μ}$ , the electric conductivity $σ$ , and the magnetic permeability $μ$ .

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