MHD Fluid: Difference between revisions
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\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"> | ||
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\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"> | ||
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\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 | ||
</math> | </math> | ||
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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
| Symbol | Name(s) | Definition |
|---|---|---|
| Fluid Density | Density of the fluid in the MHD equations. | |
| 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. |
