MHD Fluid: Difference between revisions

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<math display="block">
\rho \left(\frac{Du}{Dt}\right) = -\nabla P + J \times B + \eta \nabla^2 u
\rho \left(\frac{Du}{Dt}\right) = -\nabla P + J \times B + \eta \nabla^2 u
</math>
</math><math display="block">
 
<math display="block">
\frac{\partial B}{\partial t} = \nabla \times (u \times B - \eta \nabla B)
\frac{\partial B}{\partial t} = \nabla \times (u \times B - \eta \nabla B)
</math>
</math><math display="block">
 
<math display="block">
E + u \times B = \eta J
E + u \times B = \eta J
</math>
</math><math display="block">
 
<math display="block">
\rho \left(\frac{D\varepsilon}{Dt}\right) = -P \nabla \cdot u + \nabla \cdot (k \nabla T) + \eta J^2
\rho \left(\frac{D\varepsilon}{Dt}\right) = -P \nabla \cdot u + \nabla \cdot (k \nabla T) + \eta J^2
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\frac{D\rho}{Dt} + \rho \nabla \cdot u = 0
\frac{D\rho}{Dt} + \rho \nabla \cdot u = 0
</math>
</math>
{| class="wikitable"
|+ Symbol Definitions in MHD Equations
! Symbol !! Name(s) !! Definition
|-
| <math>\rho</math> || Fluid Density || Density of the fluid in the MHD equations.
|-
| <math>\frac{Du}{Dt}</math> || Material Derivative || Rate of change of a quantity moving with the fluid.
|-
| <math>P</math> || Pressure || Fluid pressure in the MHD equations.
|-
| <math>J</math> || Current Density || Current density vector in the MHD equations.
|-
| <math>B</math> || Magnetic Field || Magnetic field vector in the MHD equations.
|-
| <math>\eta</math> || Magnetic Diffusivity || Magnetic diffusivity in the MHD equations.
|-
| <math>\nabla</math> || Nabla Operator || Vector differential operator (gradient, divergence, or curl) in the MHD equations.
|-
| <math>\times</math> || Cross Product || Cross product of two vectors in the MHD equations.
|-
| <math>E</math> || Electric Field || Electric field vector in the MHD equations.
|-
| <math>u</math> || Fluid Velocity || Velocity vector of the fluid in the MHD equations.
|-
| <math>\frac{\partial B}{\partial t}</math> || Time Derivative of Magnetic Field || Rate of change of the magnetic field with respect to time.
|-
| <math>\varepsilon</math> || Specific Internal Energy || Specific internal energy of the fluid in the MHD equations.
|-
| <math>k</math> || Thermal Conductivity || Thermal conductivity of the fluid in the MHD equations.
|-
| <math>\nabla \cdot</math> || Divergence Operator || Divergence of a vector field in the MHD equations.
|-
| <math>\frac{D\varepsilon}{Dt}</math> || Material Derivative of Specific Internal Energy || Rate of change of specific internal energy moving with the fluid.
|-
| <math>\nabla \times</math> || Curl Operator || Curl of a vector field in the MHD equations.
|-
| <math>\frac{D\rho}{Dt}</math> || Material Derivative of Density || Rate of change of fluid density moving with the fluid.
|}

Revision as of 15:23, 11 February 2024




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{Du}{Dt}\right) = -\nabla P + J \times B + \eta \nabla^2 u } 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) } 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 } 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 }


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 }

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.
Current Density Current density vector in the MHD equations.
Magnetic Field Magnetic field vector in the MHD equations.
Magnetic Diffusivity Magnetic diffusivity in the MHD equations.
Nabla Operator Vector differential operator (gradient, divergence, or curl) in the MHD equations.
Cross Product Cross product of two vectors in the MHD equations.
Electric Field Electric field vector in the MHD equations.
Fluid Velocity Velocity vector of the fluid in the MHD equations.
Time Derivative of Magnetic Field Rate of change of the magnetic field with respect to time.
Specific Internal Energy Specific internal energy of the fluid in the MHD equations.
Thermal Conductivity Thermal conductivity of the fluid in the MHD equations.
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.
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