Hartmann Number¶
A dimensionless magnetohydrodynamic number combining field strength, fluid conductivity and viscosity, and a declared length to compare magnetic and viscous flow scales.
Core Idea¶
The Hartmann number, \(Ha=BL\sqrt{\sigma/\mu}=BL\sqrt{\sigma/(\rho\nu)}\), is a dimensionless magnetic–viscous scale for an electrically conducting fluid. \(B\) is magnetic flux density, \(L\) a declared characteristic length, \(\sigma\) electrical conductivity, and \(\mu=\rho\nu\) dynamic viscosity. In standard low-magnetic-Reynolds, quasi-static scaling, \(Ha^2\) is a characteristic Lorentz-to-viscous force-scale ratio, not a universal pointwise force ratio.[^ref-03ce0225fcd0]
Scope of Application¶
Hua and Walker studied liquid-metal flow in insulating circular ducts for fusion blankets with high Hartmann and interaction parameters but low magnetic Reynolds number.[^ref-fa1761e72c54] Vakhrushev and colleagues modeled an electromagnetic brake in a continuous caster with cases reaching approximately \(Ha=600\). Their reverse meniscus-flow result depends on field, geometry and induced-current closure; it is not a consequence of that number alone.[^ref-2389d8c61367]
Clarity¶
The formula requires the chosen field, length and fluid properties. Its \(\mu\) means viscosity, not magnetic permeability. Since \(Ha\) grows linearly with \(L\), a duct radius and a full width produce different numerical values; compare studies only after checking conventions. It is distinct from Reynolds number, which compares inertia with viscosity, and magnetic Reynolds number, which compares magnetic advection with diffusion.[^ref-03ce0225fcd0]
Manages Complexity¶
One dimensionless number condenses four dimensional inputs. In the classical steady low-induction flow between transverse-field plates, large \(Ha\) produces a flatter core and wall layers of thickness proportional to \(L/Ha\). Those predictions require that geometry and regime. Electromagnetic-brake research shows that electrically insulated, conducting and solid-shell walls can yield different turbulence responses, so high \(Ha\) alone does not guarantee suppression.[ref-03ce0225fcd0][ref-2eb1805de535]
Abstract Reasoning¶
Declare the magnetic field and characteristic length, obtain fluid conductivity and viscosity at the relevant state, compute \(Ha\), then check whether a proposed physical inference uses compatible wall, induction, inertia and flow assumptions. \(Ha\) organizes the magnetic–viscous comparison; it does not replace the rest of the model.
Knowledge Transfer¶
Fusion ducts and casting brakes share the same formula, but they need not share velocity profiles or turbulence effects. Carry the number with its field, length and material definitions, then re-evaluate wall conductivity, current closure and Reynolds-related conditions in the new setting.[ref-fa1761e72c54][ref-2eb1805de535]
[^ref-03ce0225fcd0]: University of Wisconsin–Madison Magnetohydrodynamics group, “Hartmann Flow,” Lecture 5, Classic Problems in MHD, §§5.1–5.4, especially equations 5.12–5.14 and 5.24–5.27.
[^ref-fa1761e72c54]: T. Q. Hua and J. S. Walker, “MHD Flow in Insulating Circular Ducts for Fusion Blankets”, Argonne National Laboratory conference manuscript CONF-881031-70 (1988), abstract and §II.
[^ref-2389d8c61367]: A. Vakhrushev et al., “Generation of Reverse Meniscus Flow by Applying An Electromagnetic Brake”, Metallurgical and Materials Transactions B 52, 3193–3207 (2021).
[^ref-2eb1805de535]: Alexander Vakhrushev, Abdellah Kharicha, Zhongqiu Liu, Menghuai Wu, Andreas Ludwig, Gerald Nitzl, Yong Tang, Gernot Hackl and Josef Watzinger, “Electric Current Distribution During Electromagnetic Braking in Continuous Casting”, Metallurgical and Materials Transactions B 51, 2811–2828 (2020), DOI: 10.1007/s11663-020-01952-3.
Neighborhood in Abstraction Space¶
Hartmann Number sits in a sparse region of the domain-specific corpus (75th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Electromagnetic Fields & Responses (11 abstractions)
Nearest neighbors
- Monin–Obukhov Length — 0.85
- Stuart Number — 0.85
- Stokes's law — 0.84
- Nusselt number — 0.83
- Magnetic resonance velocimetry — 0.82
Computed from structural-signature embeddings · 2026-10-08