MHD Flow Analysis
Theory: the brake a field puts on flow
Overview
I've heard liquid-metal cooling in fusion reactors needs huge pumping power. Why?
Liquid metal conducts well, so flowing across a strong field induces current perpendicular to both flow and field. That current, sitting in the field, feels a Lorentz force opposing the flow — a brake. The brake is strongest where flow is fastest, so the profile flattens in the centre and velocity change is squeezed into thin layers at the walls. Friction in those layers is large, so the pressure needed for the same flow rises hundreds or thousands of times. Flow between two parallel plates (Hartmann flow) has an exact solution that shows the size of the effect.
Hartmann number and velocity profile
y is distance from the centre; walls are electrically insulating. δ_Ha is the thickness of the Hartmann layer at the wall.
Flow rate and pressure drop
Mean-velocity ratio at equal pressure gradient; its inverse is the pressure-drop increase factor at equal flow rate.
Hartmann's mercury experiments
In 1937 the Dane Hartmann showed, through experiments and theory on mercury flowing in a channel in a magnetic field, that the field flattens the flow and increases resistance — the origin of the terms Hartmann flow and Hartmann number. Around the same time Alfvén predicted waves in which conducting fluid and field move together (Alfvén waves), opening the study of astrophysical plasmas. MHD analysis is now indispensable in designing liquid-metal blankets for fusion reactors.
Worked examples
Example 1: fluid and field combinations
| Case | Hartmann number | Hartmann layer | Pressure-drop increase |
|---|---|---|---|
| Mercury, 0.1 T, 10 mm | 26 | 0.38 mm | 9.0× |
| Mercury, 1 T, 10 mm | 261 | 0.038 mm | 87× |
| PbLi, 5 T, 50 mm | 5,000 | 0.010 mm | about 1,670× |
Example 2: Hartmann number and profile flatness
| Ha | Mean/centre velocity | Region above 95% of centre speed | Pressure-drop increase | Ha/3 |
|---|---|---|---|---|
| 1 | 0.677 | 23% | 1.1 | 0.3 |
| 10 | 0.900 | 70% | 3.7 | 3.3 |
| 100 | 0.990 | 97% | 33.7 | 33.3 |
| 1,000 | 0.999 | 99.7% | 334 | 333 |
At Ha 5,000 the boundary layer is just 10 μm. Capturing that numerically sounds hard.
It is hard. In a 50 mm half-width channel you need several grid points within 10 μm of the wall, cells thousands of times finer than at the centre. Coarsening this layer changes wall friction and current paths, badly mispredicting pressure drop. Real channels are rectangular ducts, and side walls parallel to the field develop thicker layers (roughly a/√Ha) that can carry fast jets. Conducting walls let current close through them, raising pressure drop further; fusion designs consider electrically insulating flow-channel inserts to suppress this. Numerical work should first be checked against this exact Hartmann solution to confirm boundary layers are resolved.
Analysis workflow
- Estimate field effects with the Hartmann number and interaction parameter.
- Build grids resolving Hartmann layers (a/Ha) and side layers (a/√Ha).
- Compute induced current with a charge-conserving formulation (e.g. electric potential method).
- Include wall conductivity and thickness so current through walls is captured.
- Verify against exact solutions such as Hartmann flow before real geometry.
“Refining the grid doubled the pressure drop”
In one liquid-metal channel analysis, the initial grid was coarse near the walls with no points inside the Hartmann layer, giving about half the theoretical pressure drop. Refining toward the walls to put five or more points in the Hartmann layer brought flat-plate sections within a few percent of the exact solution, and total pressure drop roughly doubled. Calculations that don't resolve boundary layers can look plausible yet badly underestimate pressure drop.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Unresolved Hartmann layers | Pressure drop underestimated | Several points in layer |
| Assuming insulating walls | Pressure drop underestimated | Include wall conductivity |
| Non-conservative current | Unphysical forces | Conservative discretisation |
| Missing side-wall jets | Wrong heat transfer | Analyse in 3D |
| Standard turbulence models | Field suppression missed | MHD-aware models |
I'd like to learn related topics.
Related pages include Lorentz force, electromagnetic forming, magnetic bearing analysis, EMAT simulation and magnetostriction.