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.
$${\frac {\partial B}{\partial t}}=\nabla \times (u\times B-\eta \nabla B)$$	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.
$$E+u\times B=\eta J$$	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.
$$\rho \left({\frac {D\varepsilon }{Dt}}\right)=-P\nabla \cdot u+\nabla \cdot (k\nabla T)+\eta J^{2}$$	MHD Energy Equation	Describes the conservation of energy in magnetized fluids. Applied in studies of magnetic confinement devices, astrophysical plasmas, and space weather modeling.
$${\frac {D\rho }{Dt}}+\rho \nabla \cdot u=0$$	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
$\rho$	Fluid Density	Density of the fluid in the MHD equations.
${\frac {Du}{Dt}}$	Material Derivative	Rate of change of a quantity moving with the fluid.
$P$	Pressure	Fluid pressure in the MHD equations.
$J$	Current Density	Current density vector in the MHD equations.
$B$	Magnetic Field	Magnetic field vector in the MHD equations.
$\eta$	Magnetic Diffusivity	Magnetic diffusivity in the MHD equations.
$\nabla$	Nabla Operator	Vector differential operator (gradient, divergence, or curl) in the MHD equations.
$\times$	Cross Product	Cross product of two vectors in the MHD equations.
$E$	Electric Field	Electric field vector in the MHD equations.
$u$	Fluid Velocity	Velocity vector of the fluid in the MHD equations.
${\frac {\partial B}{\partial t}}$	Time Derivative of Magnetic Field	Rate of change of the magnetic field with respect to time.
$\varepsilon$	Specific Internal Energy	Specific internal energy of the fluid in the MHD equations.
$k$	Thermal Conductivity	Thermal conductivity of the fluid in the MHD equations.
$\nabla \cdot$	Divergence Operator	Divergence of a vector field in the MHD equations.
${\frac {D\varepsilon }{Dt}}$	Material Derivative of Specific Internal Energy	Rate of change of specific internal energy moving with the fluid.
$\nabla \times$	Curl Operator	Curl of a vector field in the MHD equations.
${\frac {D\rho }{Dt}}$	Material Derivative of Density	Rate of change of fluid density moving with the fluid.