Hartmann Number Calculator

The Hartmann number, Ha, is a dimensionless number that compares the strength of magnetic (Lorentz) forces to viscous forces in a conducting fluid or plasma flow through a magnetic field. It plays the same organizing role in magnetohydrodynamic (MHD) duct flow that the Reynolds number plays in ordinary fluid dynamics.

Ha = BL√(σ/μ), where B is the magnetic field strength, L is a characteristic length (often the channel half-width), σ is the fluid's electrical conductivity, and μ is its dynamic viscosity.

A high Hartmann number means magnetic forces strongly dominate over viscous forces, producing a distinctively flattened ('Hartmann') velocity profile across most of the channel, with the velocity change concentrated in thin boundary layers near the walls, quite different from the smooth parabolic profile of ordinary viscous pipe flow.

Liquid-metal blanket and coolant systems in fusion reactor designs operate inside extremely strong confining magnetic fields, giving very high Hartmann numbers. This dramatically increases flow resistance (MHD pressure drop) and suppresses turbulence, both of which are critical engineering considerations for pumping power requirements and heat transfer in these systems.

All three are MHD dimensionless numbers built from similar ingredients, but they compare different pairs of effects: Hartmann number compares magnetic to viscous forces, magnetic Reynolds number compares advection to resistive diffusion of the field, and Lundquist number is the magnetic Reynolds number evaluated specifically at the Alfvén speed. Each is useful in a different regime of MHD flow analysis.

Hartmann layers are the thin viscous boundary layers that form near channel walls in high-Hartmann-number MHD duct flow, with thickness scaling as roughly 1/Ha times the channel width. Nearly all the velocity variation in the flow is confined to these thin layers, while the core of the channel moves at a nearly uniform (flattened) velocity.

No, unlike the Reynolds number, the Hartmann number as defined here does not depend on flow velocity at all, it depends only on the magnetic field, geometry, conductivity, and viscosity. It characterizes the flow's underlying force balance rather than any particular flow rate.

Fusion reactor liquid-metal blanket designs can reach Hartmann numbers in the thousands to tens of thousands, reflecting the combination of strong confining fields, sizable channel dimensions, and highly conductive liquid metal coolants like lithium-lead alloys.

Yes, it applies broadly to any MHD duct or channel flow, including electromagnetic pumps, liquid-metal batteries, metallurgical processing (electromagnetic stirring and casting), and geophysical flows like the liquid outer core of planets, wherever a conducting fluid moves through a significant magnetic field.

This calculator uses dynamic viscosity μ (in pascal-seconds), consistent with the standard Ha = BL√(σ/μ) form. If only kinematic viscosity ν is available, it can be converted using μ = ρν, where ρ is the fluid's mass density.