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Hartmann Number

A dimensionless magnetohydrodynamic number combining field strength, fluid conductivity and viscosity, and a declared length to compare magnetic and viscous flow scales.

Version
v1 · 2026-10-03 · History
Domain-specific #
13295
Domain group
Natural Sciences
Origin domain
Physics
Subdomain
Magnetohydrodynamics → Physics

Core Idea

The Hartmann number, \(Ha\), compares characteristic magnetic and viscous scales in a flowing electrically conducting fluid. For magnetic flux density \(B\), a declared characteristic length \(L\), electrical conductivity \(\sigma\), and dynamic viscosity \(\mu=\rho\nu\), it is \(Ha=BL\sqrt{\sigma/\mu}=BL\sqrt{\sigma/(\rho\nu)}\). Here \(\rho\) is density and \(\nu\) kinematic viscosity. The combination is dimensionless; \(\mu\) is not magnetic permeability.[1]

In the usual low-magnetic-Reynolds-number, quasi-static scaling, \(Ha^2\) characterizes a Lorentz-force scale relative to a viscous-force scale. That is a modeling comparison, not an exact pointwise force ratio in every conducting flow. Current closure, wall electrical behavior, geometry, inertia, and induction regime affect what a given \(Ha\) means physically.[1][2]

Structural Signature

Sig role-phrases:

  • Conducting fluid properties — \(\sigma\) supports induced current; \(\mu\), or the equivalent \(\rho\nu\), supplies the viscous scale.[1]
  • Specified magnetic field \(B\) — an imposed flux-density scale for the flow being modeled, not merely the word “magnetic.”
  • Characteristic length \(L\) — a stated radius, half-width or other relevant transverse length; changing its convention changes the reported number.[1]
  • Dimensionless construction — the four quantities combine as \(BL\sqrt{\sigma/\mu}\). A field magnitude by itself does not make the comparison.[1]
  • Interpretive regime — low magnetic Reynolds number and declared wall/field geometry are needed before inferring classical Hartmann-flow profiles. These are conditions on an inference, not extra factors in the arithmetic definition.[1][3]

What It Is Not

It is not Reynolds number, which compares inertial and viscous scales. Nor is it magnetic Reynolds number, which concerns magnetic-field transport relative to diffusion. The interaction or Stuart parameter instead compares magnetic forcing with inertia and, under consistent standard definitions, scales as \(Ha^2/Re\). A report of \(Ha\) alone does not specify all these balances.[1]

It is also not a theorem that a strong field always laminarizes a flow. Studies of electromagnetic braking in continuous casting find qualitatively different outcomes when wall electrical boundary conditions change. Suppression, instability and intermediate behavior can all arise; \(Ha\) does not encode that boundary condition.[2]

Scope of Application

Hartmann number is used in liquid-metal duct models, fusion-blanket cooling, electromagnetic brakes in metal casting, and other magnetohydrodynamic flows. In an insulated fusion-blanket duct, Hua and Walker considered high Hartmann and interaction parameters under low magnetic Reynolds number assumptions; those further specifications are part of the physical problem, not part of the number's definition.[3]

In a continuous caster, Vakhrushev and colleagues modeled an applied electromagnetic brake over cases reaching approximately \(Ha=600\). Their reported reverse meniscus-flow behavior is a result of the modeled coupled flow and field, not a generic consequence of the numeral 600.[4]

Clarity

Units are a useful error check: \(B\) has tesla, \(L\) metres, \(\sigma\) siemens per metre, and \(\mu\) pascal-seconds, so \(BL\sqrt{\sigma/\mu}\) has no units. Substituting a magnetic permeability for \(\mu\) would describe a different quantity. If a paper instead reports \(\rho\) and \(\nu\), multiply them to recover dynamic viscosity.[1]

Because \(Ha\) is proportional to \(L\), two authors using duct radius versus full width can report different numerical values for the same field and fluid. Comparison requires identifying the chosen length and field scale before comparing “high-\(Ha\)” regimes.

Manages Complexity

One number condenses four dimensional inputs into a comparable magnetic–viscous scale. In the classical low-induction channel problem, it helps organize solutions from nearly viscous behavior to a relatively flat core with thin wall layers. The University of Wisconsin derivation obtains the large-\(Ha\) Hartmann-layer thickness proportional to \(L/Ha\) only for the specified transverse-field plate geometry.[1]

Compression has a cost. \(Ha\) does not carry Reynolds number, electrical wall conductivity, field orientation, or the path by which induced current closes. This is why seemingly similar magnetic-braking cases can have different velocity and turbulence responses.[2]

Abstract Reasoning

Choose the flow geometry and magnetic field scale first. Declare \(L\), obtain fluid \(\sigma\) and \(\mu\) at the relevant conditions, compute \(Ha\), and check the units. Only then ask whether the assumed model is quasi-static, whether the wall/field orientation matches a textbook Hartmann-flow solution, and what other nondimensional controls matter.[1]

For a force-scale interpretation, induced current is estimated from the moving conducting fluid in the imposed field; Lorentz and viscous terms are then nondimensionalized on the same velocity and length scales. This produces a coefficient \(Ha^2\) in the standard low-magnetic-Reynolds formulation. It is not a license to read \(Ha^2\) as a measured local force ratio in a complicated casting mold.[1]

Knowledge Transfer

The fusion-duct and casting-brake cases retain the same construction: field, fluid conductivity, viscosity and length yield one dimensionless number. The engineering questions differ. The fusion analysis asks about a liquid-metal duct under insulated-wall and low-induction assumptions; the casting analysis asks how an electromagnetic brake reorganizes molten-metal flow. One case's pressure-loss or wall-layer result does not transfer simply because the other has a similar \(Ha\).[3][4]

Transfer therefore means using \(Ha\) to organize a declared magnetic–viscous comparison, while bringing geometry, wall conductivity, Reynolds number and current closure along for the interpretation.[2]

Examples

Liquid-metal fusion-blanket duct

Hua and Walker's primary analysis treats magnetohydrodynamic flow in an insulating circular duct relevant to fusion blankets, with high Hartmann and interaction parameters and low magnetic Reynolds number.[3] Mapped back: the liquid metal supplies conductivity and viscosity; the imposed field supplies \(B\); duct radius or another explicitly defined transverse dimension supplies \(L\); their high-\(Ha\) regime summarizes magnetic relative to viscous scaling; the insulated wall and low-induction model delimit conclusions about velocity and pressure. A different wall closure would require more than carrying over the numerical \(Ha\).

Continuous-caster electromagnetic brake

Vakhrushev and colleagues modeled a continuous-casting brake with cases up to roughly \(Ha=600\), including a reverse meniscus-flow response.[4] Mapped back: conducting molten metal supplies \(\sigma\) and \(\mu\); the brake field sets \(B\); mold/flow geometry sets the declared \(L\); \(Ha\) indexes the magnetic–viscous scale; casting turbulence, wall/shell conductivity and the chosen simulation geometry govern the observed flow. Separate casting research reports that different electrical wall conditions change whether braking damps or destabilizes the flow.[2]

Structural Tensions

Compact scale versus omitted physics. \(Ha\) is portable because it removes dimensions, but it omits inertia and electrical boundary conditions. Diagnostic: What Reynolds/interaction parameters and current-closure assumptions accompany the quoted \(Ha\)?[2]

Core braking versus wall-layer shear. In classical transverse-field flow between plates, large \(Ha\) flattens the core yet confines gradients to layers of order \(L/Ha\). That redistribution is not a geometry-free law. Diagnostic: Are the relevant walls normal to the field, with the required low-induction, steady-flow assumptions?[1]

Numerical comparison versus length convention. Reusing a familiar label can conceal different radius, half-width or full-width choices. Diagnostic: Which exact \(L\), field component and material-property state were used?[1]

Structural–Framed Character

Evaluative weight. A larger Hartmann number is not inherently better, nor does it alone determine flow stability or turbulence suppression. Human-practice bound. Analysts choose characteristic field, length and material properties; dimensional consistency and the stated MHD regime constrain interpretation.[1][2]

Institutional origin. Magnetohydrodynamic flow studies use the number in ducts and casting systems; no one apparatus defines it. Vocabulary travel. Dimensionless scale comparison is general, while Lorentz forcing, viscosity and electrical conductivity are conducting-fluid quantities.[1][3]

Import versus recognition. A new case qualifies by its declared MHD scales and the exact Hartmann formula, not by merely reporting a magnetic-field strength or any force ratio. Its character: mixed-structural—a dimensionless physical comparison whose scale choices and consequences are regime-bound.[1]

Structural Core vs. Domain Accent

Portable skeleton. Live Ratio supplies the ordered comparison of characteristic quantities; the staged edge is composition/presupposes because \(Ha\) uses a ratio beneath a square root but is not itself simply the Ratio prime's quotient operation.[1]

Domain-bound mechanism. The Hartmann number combines magnetic field, conductivity, fluid density/viscosity and characteristic length under declared scales. Its squared value has a Lorentz-to-viscous interpretation only within appropriate low-induction/quasi-static scaling; thin layers, flatter cores or casting-flow changes require further geometry and Reynolds-regime assumptions.[1][3][2]

Why not prime. Normalized comparison travels across science, but no social or thermal cost ratio inherits the MHD formula or its conditional force balance. The portable comparison is already live as Ratio; this number's identity remains specific to conducting-fluid dynamics.

Live Reynolds Number is a related named metric rather than a parent because it compares inertia with viscosity. This is a proposal only; no canonical edge has been changed.

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

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Magnetic-field magnitude alone, which lacks \(L\), \(\sigma\) and \(\mu\).
  • Reynolds or magnetic Reynolds number, which compare different processes.
  • The Stuart/interaction parameter, which compares magnetic effects with inertia.
  • Universal laminarization, turbulence suppression or a universal \(L/Ha\) wall layer.
  • Magnetic permeability substituted for the dynamic-viscosity \(\mu\).

References

[1] 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; low-magnetic-Reynolds assumptions, \(Ha\) definition and the restricted plate-flow solution. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r

[2] 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; abstract contrasts instability, damping and transitional behavior under different electrical wall conditions. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[3] 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; low-induction, insulated-duct high-Hartmann case. A distinct journal version appeared in 1989. registry ↩a ↩b ↩c ↩d ↩e ↩f

[4] A. Vakhrushev et al., “Generation of Reverse Meniscus Flow by Applying An Electromagnetic Brake”, Metallurgical and Materials Transactions B 52, 3193–3207 (2021), abstract and model setup; continuous-casting electromagnetic brake cases to approximately \(Ha=600\). registry ↩a ↩b ↩c