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 $\mathbf {J} \times \mathbf {B}$, 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

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

The $\mathbf {J} \times \mathbf {B}$ 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

${\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} )+{\frac {\eta _{m}}{\mu _{0}}}\nabla ^{2}\mathbf {B}$

where $\eta _{m}=1/(\mu _{0}\sigma )$ 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)

$\mathbf {E} +\mathbf {v} \times \mathbf {B} ={\frac {\mathbf {J} }{\sigma }}$

In ideal MHD ($\sigma \rightarrow \infty$): $\mathbf {E} =-\mathbf {v} \times \mathbf {B}$, and magnetic field lines are "frozen" into the fluid.

Energy Equation

$\rho \left({\frac {\partial \varepsilon }{\partial t}}+\mathbf {v} \cdot \nabla \varepsilon \right)=-P\nabla \cdot \mathbf {v} +\nabla \cdot (k\nabla T)+{\frac {J^{2}}{\sigma }}$

The $J^{2}/\sigma$ 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 ($R_{m}$)	$\mu _{0}\sigma vL$	Advection vs. diffusion of B-field	~10–100 (ionized air)
Hartmann ($Ha$)	$BL{\sqrt {\sigma /\eta }}$	EM force vs. viscous force	~50–500
Stuart ($N$)	$\sigma B^{2}L/(\rho v)$	EM force vs. inertial force	~0.1–10
Alfvén Mach ($M_{A}$)	$v/v_{A}=v{\sqrt {\mu _{0}\rho }}/B$	Flow vs. Alfvén wave speed	~0.01–1

Alfvén Waves

MHD supports transverse waves propagating along magnetic field lines:

$v_{A}={\frac {B}{\sqrt {\mu _{0}\rho }}}$

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

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

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:

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

For a channel of length $L$, width $w$, height $h$, with uniform $J$ and $B$:

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

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: $\alpha \approx 10^{-4}$ to $10^{-2}$

Effective air conductivity with cesium seeding:

$\sigma _{\text{air}}={\frac {n_{e}e^{2}}{m_{e}\nu _{en}}}\approx 10-100\,{\text{S/m}}\quad {\text{(with seeding at 2000–3000 K)}}$

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

Seawater MHD (Historical Precedent)

The Yamato 1 (1992) demonstrated seawater MHD propulsion:

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

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	$F=JBV=10^{4}\times 3\times 0.025\times 3$
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:

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

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

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

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
$\rho$	Fluid density	kg/m³
$\mathbf {v}$	Fluid velocity	m/s
$P$	Pressure	Pa
$\mathbf {J}$	Current density	A/m²
$\mathbf {B}$	Magnetic field	T
$\mathbf {E}$	Electric field	V/m
$\eta$	Dynamic viscosity	Pa·s
$\sigma$	Electrical conductivity	S/m
$\mu _{0}$	Vacuum permeability	H/m
$k$	Thermal conductivity	W/(m·K)
$\varepsilon$	Specific internal energy	J/kg

Related Topics

• MHD Fluids, Quantum Mechanics and Spin Waves

See Also

• MHD Core
• Magnetogravitics
• Electrogravitics
• Magneto Speeder
• Star Speeder
• MHD Tech
• Clan Tho'ra