MHD Fluid

$$\rho \left({\frac {Du}{Dt}}\right)=-\nabla P+J\times B+\eta \nabla ^{2}u$$

$${\frac {\partial B}{\partial t}}=\nabla \times (u\times B-\eta \nabla B)$$

$$E+u\times B=\eta J$$

$$\rho \left({\frac {D\varepsilon }{Dt}}\right)=-P\nabla \cdot u+\nabla \cdot (k\nabla T)+\eta J^{2}$$

$${\frac {D\rho }{Dt}}+\rho \nabla \cdot u=0$$

MHD Equations and Their Applications

Equation/Formula	Name	Usefulness and Applications
$$\rho \left({\frac {Du}{Dt}}\right)=-\nabla P+J\times B+\eta \nabla ^{2}u$$	MHD Momentum Equation	Describes the conservation of momentum in magnetized fluids. Applications include understanding fluid motion in plasmas, astrophysical phenomena, and magnetic confinement in fusion experiments.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial B}{\partial t} = \nabla \times (u \times B - \eta \nabla B)}	MHD Induction Equation	Governs the evolution of the magnetic field. Used in studies of magnetic reconnection, dynamo processes, and the behavior of magnetic fields in astrophysical systems.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E + u \times B = \eta J}	Ideal MHD Ohm's Law	Relates electric fields, fluid velocity, and magnetic fields. Essential for understanding the electrical behavior of magnetized plasmas in fusion research and space plasma physics.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \left(\frac{D\varepsilon}{Dt}\right) = -P \nabla \cdot u + \nabla \cdot (k \nabla T) + \eta J^2}	MHD Energy Equation	Describes the conservation of energy in magnetized fluids. Applied in studies of magnetic confinement devices, astrophysical plasmas, and space weather modeling.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{D\rho}{Dt} + \rho \nabla \cdot u = 0}	Ideal MHD Frozen-in Flux Equation	Expresses the conservation of mass and the 'frozen-in' property of magnetic flux in ideal MHD. Important for understanding plasma dynamics in fusion research, solar wind interactions, and astrophysical accretion processes.

Symbol Definitions in MHD Equations

Symbol	Name(s)	Definition
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho}	Fluid Density	Density of the fluid in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{Du}{Dt}}	Material Derivative	Rate of change of a quantity moving with the fluid.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}	Pressure	Fluid pressure in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J}	Current Density	Current density vector in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}	Magnetic Field	Magnetic field vector in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta}	Magnetic Diffusivity	Magnetic diffusivity in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla}	Nabla Operator	Vector differential operator (gradient, divergence, or curl) in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times}	Cross Product	Cross product of two vectors in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E}	Electric Field	Electric field vector in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u}	Fluid Velocity	Velocity vector of the fluid in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial B}{\partial t}}	Time Derivative of Magnetic Field	Rate of change of the magnetic field with respect to time.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon}	Specific Internal Energy	Specific internal energy of the fluid in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}	Thermal Conductivity	Thermal conductivity of the fluid in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot}	Divergence Operator	Divergence of a vector field in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{D\varepsilon}{Dt}}	Material Derivative of Specific Internal Energy	Rate of change of specific internal energy moving with the fluid.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times}	Curl Operator	Curl of a vector field in the MHD equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{D\rho}{Dt}}	Material Derivative of Density	Rate of change of fluid density moving with the fluid.