Stuart Number¶

The Stuart number compares characteristic magnetic Lorentz forcing with fluid inertia in an electrically conducting flow.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13649
Domain group: Natural Sciences
Origin domain: Physics
Subdomain: Magnetohydrodynamics → Physics
Aliases: Magnetic interaction parameter, MHD interaction parameter

Core Idea¶

The Stuart number, \(N\), also called the magnetic interaction parameter, compares a characteristic Lorentz-force scale with the inertial scale of an electrically conducting fluid moving through an imposed magnetic field. With conductivity \(\sigma\), field magnitude \(B\), characteristic length \(L\), density \(\rho\), and speed \(U\), its common form is \(N=\sigma B^{2}L/(\rho U)\). In the usual low-magnetic-Reynolds-number scaling, motion across the field induces current and a magnetic reaction on the flow. \(N\) estimates whether that reaction is small or important relative to inertia; it is not a local force meter or a guaranteed flow-state classifier.^[1]

The ratio is algebraically \(N=Ha^{2}/Re\) if \(Ha=BL\sqrt{\sigma/\mu}\) and \(Re=\rho UL/\mu\) are formed with consistent length, speed, field and dynamic-viscosity \(\mu\) conventions. This identity connects three different comparisons: Hartmann concerns magnetic versus viscous scales, Reynolds concerns inertial versus viscous scales, and Stuart concerns magnetic versus inertial scales. One cannot substitute a high \(Ha\) or a low \(Re\) for a physical account of how current closes through the actual geometry and walls.^[1]

The seed's tempting gloss—large \(N\) means the field brakes flow, suppresses turbulence and aligns structures—is too categorical. A large characteristic ratio can motivate a magnetic-dominated model term, but the observed flow depends on field orientation, induction, boundary conditions and forcing. An original continuous-casting study of an electromagnetic brake found instability in one insulated-mold model, strong damping in a conductive mold, and transitional behavior with a modeled solid shell.^[2]

Structural Signature¶

Conducting medium: \(\sigma\) and \(\rho\) describe how current can arise and how much inertia the fluid carries.
Imposed field: \(B\) is a declared characteristic magnetic flux-density scale; the standard ratio depends on its square.
Flow scales: \(U\) and \(L\) identify which inertia and spatial gradient are being compared. They must be reported, not guessed.
Dimensionless construction: \(N=\sigma B^{2}L/(\rho U)\) is the magnetic-to-inertial characteristic ratio.
Consistent companion numbers: under matched conventions, \(Ha^{2}/Re=N\), while each number asks a different question.
Model envelope: low-induction assumptions, field direction, wall electrical properties and current closure condition downstream interpretations.^[3]^[4]^[2]

Sig role-phrases: conducting medium; imposed magnetic field; declared flow scales; magnetic-to-inertial ratio; compatible companion numbers; boundary-conditioned interpretation.

Condensed: conducting flow + field and flow scales → magnetic/inertial ratio; boundary physics decides the actual response.

What It Is Not¶

Not Hartmann number. \(Ha\) compares magnetic and viscous scales; \(N\) compares magnetic and inertial scales.
Not ordinary Reynolds number. \(Re\) compares inertia and viscosity.
Not magnetic Reynolds number. That number compares magnetic-field advection and diffusion and helps determine whether the low-induction approximation is appropriate.
Not a theorem that \(N\gg1\) always laminarizes or aligns flow. Electrical boundary conditions can materially change the outcome.^[2]
Not the magnetic field strength alone. A fixed \(B\) yields different \(N\) for fluids or velocity/length scales with different \(\sigma,\rho,U,L\).
Not an exact pointwise ratio of local Lorentz and inertial vectors. The formula is a chosen nondimensional scale balance; local cancellation, direction and current distribution require a governing model.^[3]

Scope of Application¶

In liquid-metal fusion-blanket ducts, applied fields and conducting coolants make electromagnetic pressure-drop and flow-structure questions central. An original insulating-duct analysis explicitly works with high interaction and Hartmann parameters while assuming low magnetic Reynolds number. The value of \(N\) organizes a regime, but the duct wall and current path remain part of the explanation.^[4]

In continuous steel casting, an electromagnetic brake imposes a field to influence melt flow. The Lorentz-to-inertia ratio helps express the strength of that influence, but a casting model must also specify mold electrical conditions and current-loop closure. The cited original comparison of insulated and conductive molds demonstrates why “strong magnetic interaction” is not synonymous with one universal turbulence outcome.^[2]

In MHD model comparison, \(N\), \(Ha\) and \(Re\) can be checked against each other as a consistency test. A published \(N\) cannot be compared meaningfully to another study's value without knowing whether both used the same length and velocity definitions and compatible physical regimes.^[3]

Clarity¶

The name “interaction parameter” can sound broader than its calculation. Here the interaction is specifically the imposed-field electromagnetic reaction relative to inertia in a conducting flow. Write down the characteristic geometry and field before attaching the words “high” or “low” to \(N\). A number can be high at one scale and not at another. When \(U\) is very small, the formula also grows; that does not itself prove a high-speed magnetic braking process—its chosen inertial denominator is vanishing.

The relation \(N=Ha^{2}/Re\) follows directly by canceling viscosity from the two definitions. It is not an independent empirical law. If another paper defines \(Ha\) using duct half-width but \(Re\) using full width, or uses a different \(U\), the simple numerical equality will not apply until conventions are reconciled.^[3]

Manages Complexity¶

Five dimensional quantities collapse into one magnetic-to-inertial comparison. This helps a modeler decide whether to retain the Lorentz term when organizing an MHD momentum balance and helps readers compare scale regimes across experiments. The compression does not carry current distribution, wall conduction, field direction, induced-field strength or turbulence closure. Treating \(N\) as a screening number rather than a full prediction preserves its usefulness without smuggling in the missing physics.^[3]^[2]

Abstract Reasoning¶

First identify the flow, imposed field and chosen characteristic \(L\) and \(U\). Obtain \(\sigma\) and \(\rho\) for the relevant fluid conditions; calculate \(N\) and verify that units cancel. If \(Ha\) and \(Re\) are available, calculate \(Ha^{2}/Re\) with the same scales as a cross-check. Then ask which physical inference is actually wanted: pressure drop, velocity profile, turbulence response or wall-layer behavior. Specify induction regime, walls and current closure before drawing that inference. If the desired conclusion depends on them, \(N\) alone does not decide it.^[3]^[4]^[2]

Knowledge Transfer¶

The same nondimensional force-scale comparison can be used in a fusion-blanket duct and in a casting mold: conductivity, density, field and chosen flow scales supply \(N\). Their actual responses cannot be copied across settings because wall conduction, flow geometry, driving and field orientation differ. The transfer is the question “how large is electromagnetic forcing relative to inertia under this model?”, not the answer “turbulence must be suppressed.”^[4]^[2]

Examples¶

Transparent illustrative nondimensionalization¶

This is an author-constructed parameter set, not measurements from a named experiment or a named fluid. Let \(\sigma=10^{6}\ {\rm S\,m^{-1}}\), \(B=0.5\ {\rm T}\), \(L=0.1\ {\rm m}\), \(\rho=10^{3}\ {\rm kg\,m^{-3}}\) and \(U=1\ {\rm m\,s^{-1}}\). Applying the source's scale definition gives \(N=(10^{6})(0.5)^2(0.1)/[(10^{3})(1)]=25\). The numerator and denominator each have units \({\rm kg\,m^{-2}\,s^{-1}}\), so their quotient is dimensionless. If only \(U\) rises to \(5\ {\rm m\,s^{-1}}\), the same construction gives \(N=5\), not 25. The change says the selected magnetic force scale is smaller relative to inertia; it does not calculate an actual pressure loss or turbulence state.^[1]

Mapped back: assigned conductivity and density → specified field, length and speed → executed magnetic/inertial quotient → limited scale interpretation.

Original continuous-casting boundary comparison¶

Vakhrushev and coauthors modeled an electromagnetic brake using a GaInSn experiment and a steady \(0.31\ {\rm T}\) field. They reported induced loop-current density up to \(350\ {\rm kA\,m^{-2}}\), then varied the electrical boundary: a perfectly insulating mold, a perfectly conducting mold, and a conductive solid shell experimentally represented by \(0.5\ {\rm mm}\) brass plates. The reported response was field-aligned unstable vortices for the insulating mold, strong damping for the conducting mold, and transitional behavior for the shell condition. Thus the same imposed-field experiment did not license the single sentence “magnetic interaction damps turbulence”: loop closure changes the response. The paper does not give one matched \(\sigma,\rho,L,U\) set here from which this entry can calculate a study-specific \(N\); its separate illustrative steel-alloy magnetic-Reynolds calculation must not be mixed with the GaInSn brake field.^[2]

Mapped back: conducting flow and imposed field → induced-current loops → electrical-boundary choices → three observed model responses; no fabricated study-specific ratio.

Structural Tensions¶

No intrinsic two-sided tradeoff belongs to the Stuart number itself: once scales and model assumptions are fixed, \(N\) is a calculated ratio. The choice between compact regime screening and a more detailed boundary-resolving model is an analyst's modeling decision, not an opposing cost built into this parameter. The casting example shows why the screening number cannot by itself supply a flow prediction.^[2]

Structural–Framed Character¶

The Stuart number lies chiefly on the structural side of the spectrum: under the declared MHD scaling, its algebraic numerator and denominator identify physical force scales. It is not an evaluative score of whether a casting machine is “good,” though a practitioner may evaluate the modeled outcome against process goals. The number depends on human practice where an analyst selects \(L\), \(U\), a representative \(B\), material properties and the low-induction envelope; those selections make a reported value reproducible or incomparable. No institution constitutes the force ratio, although laboratory conventions, engineering reports and the historical naming of the parameter stabilize its vocabulary. “Interaction parameter” can travel from fusion-blanket ducts to casting molds because both have conducting flow in an imposed field; the actual wall-current closure and geometry cannot travel merely with the name. A use in another setting counts as recognition of the same abstraction only if its force-scale numerator, inertial denominator and declared scales match. Merely importing the label onto a generic magnetic effect would not suffice. Its character: a physically grounded, structurally defined force ratio whose numerical interpretation is model-framed, not an institutional grade or universal flow outcome.

Structural Core vs. Domain Accent¶

The portable skeleton is an ordered ratio of two characteristic physical effects with dimensions that cancel. Here the domain-bound mechanism is Lorentz forcing arising from current induced in a moving conducting fluid, compared against inertia by \(\sigma B^{2}L/(\rho U)\) under matched scales and nonzero \(U\). Remove conductivity, magnetic field and the fluid momentum balance, and the named Stuart object disappears rather than becoming a free-standing prime about all ratios. Its portability across ducts and molds is real but stays within magnetohydrodynamics, so this entry does not clear the prime bar for a domain-general abstraction. Live Dimensionless Quantity supplies the dimension-one physical-quantity genus and live prime Ratio supplies ordered division; neither alone captures the MHD differentia. The live Hartmann Number and Reynolds Number are algebraically related neighbors, not strict genera.

This entry is a kind of Dimensionless Quantity and is a kind of Ratio.

Two independently tested strict parent edges are staged: Dimensionless Quantity for the unit-invariant physical quantity and prime Ratio for the ordered Lorentz-to-inertia quotient. An angle can be dimensionless without being a force ratio, and a speed ratio can be a ratio without being this magnetic parameter. These nonredundant parents do not turn the domain-bound child into a prime. Hartmann and Reynolds numbers compare different force pairs; algebraic identity with \(Ha^{2}/Re\) is not a genus claim.

Relationships to Other Abstractions¶

Current abstraction Stuart Number Domain-specific

Parents (2) — more general patterns this builds on

Stuart Number is a kind of Dimensionless Quantity Domain-specific

Stuart Number is a named dimension-one physical quantity formed from matched MHD force scales.
Stuart Number is a kind of Ratio Prime

Stuart Number divides a magnetic-force scale by an inertial-force scale.

Hierarchy paths (2) — routes to 2 parentless roots

Stuart Number → Dimensionless Quantity → Physical quantity → Measurement

Show alternative path (1)

Neighborhood in Abstraction Space¶

Stuart Number sits in a sparse region of the domain-specific corpus (94^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

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

Not to Be Confused With¶

Hartmann number scales magnetic influence against viscosity. Reynolds number scales inertia against viscosity. Magnetic Reynolds number concerns transport of magnetic field against diffusion. Stuart number selects the magnetic-versus-inertia comparison; none of these alone supplies boundary conditions or a full turbulence prediction.^[3]^[2]

References¶

[1] University of Wisconsin, “Hartmann Flow,” Classic Problems in MHD, Eq. 5.28, explicit \(N=\sigma B_0^2a/(\rho U)=Ha^2/Re\) under common conventions. registry ↩a ↩b ↩c

[2] Original research, “Electric Current Distribution During Electromagnetic Braking in Continuous Casting”, modeled electrical-boundary comparison. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j

[3] “Magnetic Action at a Distance: Fields, Gradients and Currents in Fluids”, author-written open-access MHD chapter, equations 2.20–2.24 and neighboring definitions; publisher full HTML directly checked. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g

[4] T. Q. Hua and J. S. Walker, “MHD Flow in Insulating Circular Ducts for Fusion Blankets,” conference manuscript prepared for the Eighth Topical Meeting on the Technology of Fusion Energy (1988), CONF-881031-70. https://www.osti.gov/servlets/purl/6034162 . The original manuscript treats high interaction/Hartmann and low magnetic Reynolds assumptions; do not conflate this 1988 manuscript with its later journal publication. registry ↩a ↩b ↩c ↩d