MHD: The Lorentz Force in Fluid Flow

1. The coupling

A conductor moving with velocity u in a magnetic field B experiences a motional EMF u×B. By Ohm's law for a moving medium the induced current density is

and that current, sitting in the field, feels the Lorentz force per unit volume

which is added to the Navier–Stokes momentum equation as a body force. The fluid drags the field, the field brakes the fluid.

2. The boundary-layer simplification

Two standard assumptions tame the coupling for the stretching-sheet class of problems. First, the magnetic Reynolds number Re_m = μ₀σUL is small, so the flow's distortion of the applied field is negligible and B stays the imposed transverse field (0, B₀, 0). Second, no external electric field is applied, E = 0. Then with u = (u, v, 0):

The Lorentz force reduces to a linear drag on the streamwise velocity — mathematically the friendliest body force imaginable, which is why the MHD stretching-sheet problem still admits an exact solution.

3. The dimensionless groups

M compares magnetic braking with the stretching inertia a; Ha² compares the Lorentz force with viscous force. In the exact solution the wall shear scales as √(1+M): the field thins the momentum boundary layer and raises skin friction, while (in the heat problem) thickening the thermal layer and lowering the Nusselt number.

Key takeaway

For boundary-layer MHD, all of electromagnetism collapses to one extra term, −σB₀²u — one new parameter M, one new physical effect: controllable braking. The full derivation continues in the similarity-transformation notes.

5. Extended worked examples

Example 1 — Hartmann number of a liquid-metal duct. Liquid sodium (σ ≈ 1.0×10⁷ S/m, μ ≈ 6.8×10⁻⁴ Pa·s) in a 5 cm half-width duct under B₀ = 0.5 T: Ha = B₀L√(σ/μ) = 0.5×0.05×√(1.47×10¹⁰) ≈ 3 000. Lorentz forces dwarf viscosity by Ha² ≈ 10⁷: the velocity profile flattens into a plug with razor-thin Hartmann layers (thickness L/Ha ≈ 17 μm!) hugging the walls. Fusion-blanket engineers live at these numbers.

Example 2 — why seawater MHD propulsion disappoints. Seawater: σ ≈ 4 S/m — six orders below sodium. Even with a 1 T field over a 1 m duct, Ha = 1×1×√(4/10⁻³) = 63, and more damningly the thrust per watt is proportional to σ: the Yamato-1 demonstrator ship (1992) achieved ~8 knots with superconducting magnets and single-digit efficiency. Same equations, ruthless material constants.

Example 3 — the magnetic parameter in a stretching-sheet problem. Galinstan-like metal (ρ = 6 440, σ = 3.5×10⁶) stretched at a = 2 s⁻¹ under B₀ = 0.2 T: M = σB₀²/ρa = 3.5×10⁶×0.04/(6 440×2) ≈ 10.9. The exact solution then gives wall shear f″(0) = −√(1+M) = −3.45 — the field more than triples the drag on the sheet relative to M = 0. In water-based nanofluid papers σ is small and realistic M values come instead from strong fields and slow stretching; parameter sweeps M = 0…5 in the literature are spanning exactly this physics.

Example 4 — checking the low-Re_m assumption. A lab flow: U = 0.1 m/s, L = 0.1 m, σ = 10⁶ S/m. Re_m = μ₀σUL = 4π×10⁻⁷×10⁶×0.01 ≈ 0.013 ≪ 1 ✓ — the applied field is undisturbed by the flow, and the one-term Lorentz simplification of §2 is legitimate. Planetary dynamos sit at the opposite extreme (Re_m ~ 10²–10³), where the flow bends the field and the full induction equation must be kept.

6. Common misconceptions

"The magnetic field does no work, so it can't heat the fluid." The field does no work on free charges moving along it, but here the force acts on the fluid: with E = 0, the power density is (J×B)·u = −σB₀²u² < 0 — kinetic energy is drained, reappearing exactly as Joule heating J²/σ in the energy budget. Magnetic braking is an energy conversion, not a violation.

"The field always opposes the flow." Only in this generator-like configuration. Impose an external current (crossed E and B) and J×B becomes a pump — electromagnetic pumps move liquid sodium through fast-reactor loops with no moving parts for precisely this reason.

"M and Ha are the same parameter." Related but distinct nondimensionalisations: Ha² compares Lorentz to viscous force; M compares Lorentz to the stretching inertia a. In the sheet problem they connect through the flow's own scales; quoting one as the other is a common referee complaint.

"Stronger field, better heat transfer." Generally the opposite in this class of flows: braking thickens the thermal layer and lowers −θ′(0). Papers reporting Nusselt increases with M usually have another agent (suction, radiation, a second field orientation) doing the lifting — read the coupling, not the headline.

7. Where this shows up

Continuous casting of steel uses electromagnetic brakes — DC fields across the mould — to calm turbulent jets and improve slab quality: industrial MHD at the thousand-tonne scale. Czochralski growth of silicon crystals applies fields to suppress convective striations in the melt; fusion-reactor breeding blankets pump PbLi through magnetic fields where MHD pressure drop is the design constraint; electromagnetic flow meters (blood, liquid metals, slurries) read velocity directly from the u×B voltage. And astrophysics is MHD's home turf: sunspots are magnetically strangled convection, and accretion-disc turbulence is powered by the magnetorotational instability.

8. Practice problems

P1. Compute Ha for mercury (σ = 1.0×10⁶, μ = 1.5×10⁻³) in a 2 cm channel at B₀ = 1 T. Solution: Ha = 1×0.02×√(6.7×10⁸) = 0.02×25 900 ≈ 517.

P2. In Example 3, what B₀ would give M = 2? Solution: B₀ = √(Mρa/σ) = √(2×6 440×2/3.5×10⁶) = √(7.36×10⁻³) ≈ 0.086 T.

P3. Show that with E = 0 the Joule heating per volume equals σB₀²u² and verify it matches the kinetic-energy drain rate. Solution: J = σ(u×B) has magnitude σuB₀; J²/σ = σu²B₀² = −(J×B)·u ✓ — the books balance.

P4. A generator duct extracts power instead: with load field E opposing u×B, write J and identify the condition separating generator, brake and pump regimes. Solution: J = σ(E + uB₀); the sign of J·E (power to circuit) versus (J×B)·u (power from flow) partitions the K = E/uB₀ axis: 0 < K < 1 generator, K < 0 brake-plus-heater, K > 1 pump.

9. Going deeper

Keep the full coupling and the magnetic field acquires its own transport equation — the induction equation ∂B/∂t = ∇×(u×B) + (1/μ₀σ)∇²B: advection and stretching of field lines versus ohmic diffusion, with Re_m as the referee. At high Re_m, field lines are "frozen into" the fluid (Alfvén's theorem), tension along them supports Alfvén waves with speed B/√(μ₀ρ), and flows can amplify seed fields into dynamos — the origin of Earth's and the Sun's magnetism. The single −σB₀²u term of the boundary-layer literature is the low-Re_m shadow of this much larger theory; knowing what was discarded is what lets you trust what remains.

10. Historical context

MHD's founding experiment failed. Faraday, fresh from discovering induction, tried in 1832 to measure the voltage generated by the Thames flowing through Earth's magnetic field, stringing electrodes from Waterloo Bridge — the signal drowned in electrode noise, but the reasoning was exactly right (undersea cables later detected precisely this tidal voltage). Hartmann's Copenhagen experiments on mercury in strong magnets (1937) founded quantitative liquid-metal MHD — the flattened duct profiles and thin wall layers he measured carry his name. Alfvén's 1942 proposal that magnetised fluids support transverse waves was received sceptically (legend says Fermi's "of course" at a seminar changed the field's mind overnight) and earned the 1970 Nobel Prize — still the only one awarded for plasma astrophysics. The stretching-sheet MHD literature your thesis belongs to began when these classical ingredients met Crane's 1970 boundary-layer geometry in the late 1970s–80s, and the parameter M of these notes has been the field's dial ever since.

11. Another way to see it: where −σB₀²u really comes from

The term looks decreed; it is derived, with every step auditable. Ohm's law in a moving conductor: J = σ(E + u×B). Take B = B₀ŷ applied, flow u = u x̂. Then u×B = −uB₀ ẑ: charge carriers swept through the field feel a sideways push, driving current J_z = −σuB₀ (with E = 0 — no external circuit, and in a wide layer no charge build-up along z). That current, sitting in the field, feels the second Lorentz push: J×B = (−σuB₀ẑ)×(B₀ŷ) = −σuB₀²(ẑ×ŷ)·(−1)… computing the cross product ẑ×ŷ = −x̂ gives J×B = σuB₀²(−x̂)·… careful signs land on −σB₀²u x̂: directly opposing the motion, quadratic in B₀ because the field acts twice — once creating the current, once pushing on it. The same two-step explains every regime of P4 in §8: add an external E and the first step changes sign territory, turning brake into pump or generator. Nothing in the boundary-layer papers' "−(σB₀²/ρ)u" is a modelling whim; it is Faraday's and Ampère's laws composed, then linearised by the low-Re_m assumption checked in Example 4.

12. Frequently asked questions

Does Earth's magnetic field brake water pipelines? Immeasurably: σ_water ~ 0.05 S/m, B ~ 5×10⁻⁵ T give braking forces some fifteen orders below viscous ones. But the induced voltages are detectable — electromagnetic ocean-current monitoring and the Faraday-effect artefacts on ECGs inside MRI machines (blood flowing at 1.5–3 T) are real, routine physics.

Why does the ideal fluid conduct but carry no net charge? Quasi-neutrality: any charge imbalance in a good conductor relaxes in ~ε₀/σ seconds (10⁻¹⁸ s for metals). Currents flow; charge density stays essentially zero — which is why the electric force ρ_eE is dropped while J×B is kept.

Is M > 1 physically special? It marks Lorentz braking exceeding the stretching inertia — layer thickness and wall shear then scale as M^{−1/2} and M^{1/2} respectively via β = √(1+M). Nothing discontinuous happens at M = 1; it is a bookkeeping midpoint, not a phase boundary.

Can the field destabilise rather than calm a flow? Yes — orientation matters. Fields along the flow damp turbulence (the Hartmann tendency exploited in casting); in rotating shear flows a weak field triggers the magnetorotational instability that turns accretion discs turbulent. "Magnetic fields stabilise" is a configuration statement, not a law.

13. Further practice

P5. Blood (σ ≈ 0.7 S/m) flows at 0.5 m/s across a 3 T MRI field in a 2 cm vessel. Estimate the induced voltage and compare with ECG signals (~1 mV). Solution: V ≈ uBd = 0.5×3×0.02 = 30 mV — thirty times the ECG's R-wave; this is why MRI-compatible ECG leads and algorithms exist.

P6. For the exact solution with M = 3 versus M = 0, compute the ratio of wall heat-transfer-relevant layer thicknesses (∝ 1/β). Solution: β ratio 2:1 — the field halves the momentum layer; the thermal layer thickens on convective braking, the tension resolved in the heat-transfer chapter.

P7. An EM pump drives sodium with J = 5×10⁵ A/m² across B = 0.8 T. Pressure gradient produced? Solution: dp/dx = JB = 4×10⁵ Pa/m — 4 bar per centimetre of active length; why fast-reactor pumps are compact and have no shaft seals to leak.

14. Worked exam problem

Problem. A liquid-metal microcooler channel (galinstan: σ = 3.46×10⁶ S/m, μ = 2.4×10⁻³ Pa·s) has half-gap 1 mm and sits in a 0.3 T magnet. Find (a) the Hartmann number, (b) the Hartmann layer thickness, (c) state the MHD pressure-drop consequence.

Solution. (a) Ha = B₀L√(σ/μ) = 0.3×10⁻³×√(1.44×10⁹) = 3×10⁻⁴×3.8×10⁴ = 11.4. (b) δ_Ha = L/Ha = 1 mm/11.4 = 88 μm — the velocity climbs from wall to plug within a tenth of the gap. (c) The flattened plug loses the parabola's easy core: at these Ha the pressure gradient for a given flow rises by roughly a factor of Ha/3 relative to ordinary Poiseuille flow (with conducting walls the penalty is far worse, scaling with Ha²·wall-conductance) — the central design tax of every fusion-blanket cooling circuit, computed here at desktop scale.

15. Key takeaways

A conducting fluid crossing a magnetic field induces currents (Ohm's law with u×B) that feel the field in turn (J×B): with no external circuit the composite force is −σB₀²u — a brake, quadratic in B₀, converting kinetic energy to Joule heat. Ha measures Lorentz against viscosity; M measures it against stretching inertia; low Re_m justifies treating the applied field as fixed. Reverse the current externally and the same physics pumps, generates, or meters.

16. Where to go next

The −Mf′ term this chapter justified enters the stretching-sheet reduction and pushes β to √(1+M); its thermal consequences unfold in heat transfer. Compute Ha and M with the Reynolds number calculator and Reynolds number calculator calculators, both wired to this chapter's definitions.