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

()#

\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 $μ$ .

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

()#

\begin{array}{r} \begin{aligned} \partial_{t} u + (u \cdot \nabla) u - R_{f}^{- 1} Δ u - A_{l}^{- 2} j \times B + \nabla p & = f, \\ \nabla \cdot u & = 0, \\ \partial_{t} B + \nabla \times E & = 0, \\ R_{m}^{- 1} j - (E + u \times B) & = 0, \\ j - \nabla \times B & = 0, \end{aligned} \end{array}

where $u$ , $j$ , $B$ , $p$ , $f$ , and $E$ are the non-dimensional variables, and $R_{f} = \frac{ρ U L}{\tilde{μ}} = \frac{U L}{ν}$ (with $ν = \frac{\tilde{μ}}{ρ}$ being the kinematic viscosity) is the fluid Reynolds number, $A_{l} = \frac{U \sqrt{ρ μ}}{B} = \frac{U}{U_{A}}$ (with $U_{A} = \frac{B}{\sqrt{ρ μ}}$ being the Alfvén speed), and $R_{m} = μ σ U L$ is the magnetic Reynolds number.

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

()#

\begin{array}{r} \begin{aligned} \partial_{t} u + ω \times u - R_{f}^{- 1} Δ u - A_{l}^{- 2} j \times B + \nabla P & = f, \\ ω - \nabla \times u & = 0, \\ \nabla \cdot u & = 0, \\ \partial_{t} B + \nabla \times E & = 0, \\ R_{m}^{- 1} j - (E + u \times B) & = 0, \\ j - \nabla \times B & = 0 . \end{aligned} \end{array}

Numerical Examples#

For numerical examples of MHD, see

Orszag-Tang Vortex

↩️ Back to Gallery🖼.

🧲 MHD

On this page

🧲 MHD#

Incompressible MHD#

Numerical Examples#