Magnetohydrodynamics

Magnetohydrodynamics
Magnetohydrodynamics
Overview
Abbreviation	MHD
Domain	Plasma physics · fluid mechanics · electromagnetism
Key Equation	ρ(Dv/Dt) = −∇P + J×B + η∇²v
Applications	Propulsion · power generation · astrophysics
Tho'ra Use	Magneto Speeder + Star Speeder primary thrust
Key Parameters
Reynolds_m	Rm = μ₀σvL (magnetic Reynolds number)
Hartmann	Ha = BL√(σ/η) (flow-field coupling)
Alfvén Speed	v_A = B/√(μ₀ρ)
	Foundation of MHD Tech

Magnetohydrodynamics (MHD) is the study of electrically conducting fluids — plasmas, liquid metals, ionized gases — under the influence of magnetic fields. It unifies fluid mechanics with electromagnetism via the Lorentz force 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 \mathbf{J} \times \mathbf{B}} , creating a rich physics that governs phenomena from stellar dynamos to submarine propulsion.

In Tho'ra technology, MHD provides the primary atmospheric thrust mechanism for the Magneto Speeder and a key subsystem of the Star Speeder's integrated propulsion.

Governing Equations

The MHD system is described by coupled partial differential equations combining the Navier-Stokes equations with Maxwell's equations:

Momentum Equation

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{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v}\right) = -\nabla P + \mathbf{J} \times \mathbf{B} + \eta\nabla^2\mathbf{v}}

The 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 \mathbf{J} \times \mathbf{B}} term is the Lorentz body force — this is what generates thrust. When current flows perpendicular to a magnetic field, the resulting force accelerates the conducting fluid, and by Newton's third law, the vehicle.

Induction Equation

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 \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) + \frac{\eta_m}{\mu_0}\nabla^2\mathbf{B}}

where 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_m = 1/(\mu_0\sigma)} is the magnetic diffusivity. The first term represents advection (flux frozen into the fluid), and the second represents diffusion (field decay due to resistivity).

Ohm's Law (MHD form)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} ={\frac {\mathbf {J} }{\sigma }}}

In ideal MHD (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma \rightarrow \infty } ): $\mathbf {E} =-\mathbf {v} \times \mathbf {B}$ , and magnetic field lines are "frozen" into the fluid.

Energy Equation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho \left({\frac {\partial \varepsilon }{\partial t}}+\mathbf {v} \cdot \nabla \varepsilon \right)=-P\nabla \cdot \mathbf {v} +\nabla \cdot (k\nabla T)+{\frac {J^{2}}{\sigma }}}

The Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J^{2}/\sigma } term represents Ohmic (Joule) heating — significant for MHD thruster thermal management.

Continuity Equation

${\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {v} )=0$

Dimensionless Numbers

Key MHD Dimensionless Parameters
Number	Definition	Physical Meaning	Magneto Speeder Value
Magnetic Reynolds (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R_{m}} )	$\mu _{0}\sigma vL$	Advection vs. diffusion of B-field	~10–100 (ionized air)
Hartmann (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Ha} )	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle BL{\sqrt {\sigma /\eta }}}	EM force vs. viscous force	~50–500
Stuart (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N} )	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 \sigma B^2 L / (\rho v)}	EM force vs. inertial force	~0.1–10
Alfvén Mach (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 M_A} )	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 v/v_A = v\sqrt{\mu_0\rho}/B}	Flow vs. Alfvén wave speed	~0.01–1

Alfvén Waves

MHD supports transverse waves propagating along magnetic field lines:

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 v_A = \frac{B}{\sqrt{\mu_0 \rho}}}

For the Magneto Speeder's ionized air channel (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 = 2\,\text{T}} , 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 = 0.1\,\text{kg/m}^3} ):

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 v_A = \frac{2}{\sqrt{4\pi \times 10^{-7} \times 0.1}} \approx 5{,}640\,\text{m/s}}

Alfvén waves carry energy and information through the MHD propulsion channel, enabling distributed thrust control.

MHD Propulsion Engineering

Principle of Operation

An MHD thruster accelerates a conducting fluid using the Lorentz force:

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 \mathbf{F}_{\text{thrust}} = \int_V (\mathbf{J} \times \mathbf{B})\,dV}

For a channel of length 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 L} , width 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 w} , height 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 h} , with uniform 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} and 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} :

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 F = J \cdot B \cdot L \cdot w \cdot h = \sigma(E + vB) \cdot B \cdot V_{\text{channel}}}

Atmospheric MHD (Magneto Speeder)

The Magneto Speeder ionizes air ahead of the vehicle using:

UV photoionization array (excimer laser, 172 nm)
Microwave breakdown (2.45 GHz magnetron, ~1 kW)
Seed injection (cesium or potassium vapor for lowered ionization potential)

Achievable ionization fraction: 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 \alpha \approx 10^{-4}} to 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 10^{-2}}

Effective air conductivity with cesium seeding: ^[1]

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 \sigma_{\text{air}} = \frac{n_e e^2}{m_e \nu_{en}} \approx 10 - 100\,\text{S/m}\quad\text{(with seeding at 2000–3000 K)}}

Compare: seawater 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 \sigma \approx 5\,\text{S/m}} , copper 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 \sigma \approx 5.8 \times 10^7\,\text{S/m}} .

Seawater MHD (Historical Precedent)

The Yamato 1 (1992) demonstrated seawater MHD propulsion: ^[2]

4 Tesla superconducting magnets
Seawater as working 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 \sigma \approx 5\,\text{S/m}} )
Achieved ~8 knots (limited by low conductivity and electrode drag)
Demonstrated the principle; atmospheric MHD with ionized air achieves higher 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 \sigma}

MHD Thruster Performance

Estimated MHD Thruster Parameters (Magneto Speeder)
Parameter	Value	Notes
Magnetic field (B)	2–5 T	HTS magnets (REBCO tape)
Channel volume	0.1 m³ (total, 4 nozzles)	Distributed around vehicle
Air conductivity (σ)	50 S/m (seeded, ionized)	Cs-seeded at ~2500 K
Current density (J)	10⁴ A/m²	External drive + self-excitation
Thrust per nozzle	~2.5 kN	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 F = JBV = 10^4 \times 3 \times 0.025 \times 3}
Total thrust	~10 kN	4 nozzles, full power
Specific impulse	~3,000–8,000 s	Working fluid is ambient air
Power input	~50 kW	From Micro Fusion Fuel Cells
Thrust-to-power	~200 N/kW	Competitive with ion thrusters

Efficiency

MHD thruster electrical efficiency:

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_{\text{MHD}} = \frac{F \cdot v}{P_{\text{input}}} = \frac{\text{thrust power}}{\text{electrical input}}}

At the Magneto Speeder's cruise speed of Mach 1 (~340 m/s):

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 = \frac{10{,}000 \times 340}{50{,}000} = 68\%}

This increases with vehicle speed (unlike propeller propulsion which decreases), making MHD especially efficient at high Mach numbers.

Symbol Definitions

MHD Equation Symbols
Symbol	Name	Units
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	kg/m³
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 \mathbf{v}}	Fluid velocity	m/s
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	Pa
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 \mathbf{J}}	Current density	A/m²
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 \mathbf{B}}	Magnetic field	T
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 \mathbf{E}}	Electric field	V/m
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}	Dynamic viscosity	Pa·s
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 \sigma}	Electrical conductivity	S/m
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 \mu_0}	Vacuum permeability	H/m
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	W/(m·K)
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	J/kg

References

↑ Rosa, R.J. (1987). Magnetohydrodynamic Energy Conversion. Hemisphere Publishing. ISBN 0-89116-690-9.
↑ Motora, S. et al. (1992). "An Experimental Study on Superconducting MHD Ship Propulsion." J. Ship Research 36(4), 361–367.

[1] Rosa, R.J. (1987). Magnetohydrodynamic Energy Conversion. Hemisphere Publishing. ISBN 0-89116-690-9.

[2] Motora, S. et al. (1992). "An Experimental Study on Superconducting MHD Ship Propulsion." J. Ship Research 36(4), 361–367.

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Magnetohydrodynamics

Contents

Governing Equations

Momentum Equation

Induction Equation

Ohm's Law (MHD form)

Energy Equation

Continuity Equation

Dimensionless Numbers

Alfvén Waves

MHD Propulsion Engineering

Principle of Operation

Atmospheric MHD (Magneto Speeder)

Seawater MHD (Historical Precedent)

MHD Thruster Performance

Efficiency

Symbol Definitions

Related Topics

See Also

References

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