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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, or magnetic interaction parameter, is \(N=\sigma B^{2}L/(\rho U)\): an estimate of imposed-field Lorentz forcing relative to inertia in a conducting-fluid flow under declared characteristic scales. Under compatible definitions \(N=Ha^{2}/Re\); the three numbers still ask different force-balance questions. The University of Wisconsin MHD treatment gives both forms explicitly.[^ref-fd87f7777b80]

Scope of Application

It organizes MHD analyses of liquid-metal fusion ducts and electromagnetic braking of casting flows. For a transparent author-constructed parameter set—not an observed fluid—with \(\sigma=10^6\ {\rm S\,m^{-1}}\), \(B=0.5\ {\rm T}\), \(L=0.1\ {\rm m}\), \(\rho=10^3\ {\rm kg\,m^{-3}}\), and \(U=1\ {\rm m\,s^{-1}}\), the source formula gives \(N=25\); changing only \(U\) to \(5\ {\rm m\,s^{-1}}\) gives \(N=5\). Neither value predicts a flow outcome by itself. In Vakhrushev and coauthors' original 0.31 T GaInSn brake study, induced currents reached 350 kA/m²; insulating, conducting and shell boundary models produced unstable, damped and transitional responses respectively. The study does not provide one matched parameter set here from which this entry can compute its own study-specific \(N\).[ref-fd87f7777b80][ref-2eb1805de535]

Clarity

It is not Hartmann number (magnetic versus viscous), Reynolds number (inertial versus viscous), or magnetic Reynolds number (field advection versus diffusion). Large \(N\) does not guarantee laminarization; an original casting study found different outcomes under distinct electrical boundaries.[^ref-2eb1805de535]

Manages Complexity

The dimensionless ratio compresses conductivity, density, field strength and flow scales into one comparison, but intentionally omits geometry and boundary physics. It screens model terms, not full flow outcomes. There is no intrinsic two-sided tradeoff in the number itself; compact screening versus a detailed boundary-resolving model is the analyst's separate modeling choice.

Abstract Reasoning

Choose \(L,U,B,\sigma,\rho\), calculate \(N\), and check that the result is dimensionless. Compare \(Ha^{2}/Re\) only if those numbers use matching scale conventions. Then identify induction and wall assumptions before predicting a pressure drop or turbulence change. An explicit study-specific \(N\) requires a matched set of material and flow values; do not mix illustrative values for one metal with a field reported for another.[ref-fd87f7777b80][ref-2eb1805de535]

Knowledge Transfer

The force-ratio question transfers between fusion and casting settings; their numerical scales and boundary-conditioned outcomes do not. This is a strict instance of both Dimensionless Quantity, because coherent units cancel, and Ratio, because the declared magnetic and inertial scales occupy ordered numerator and denominator roles. Hartmann and Reynolds numbers remain algebraic neighbors rather than parents.

[^ref-1124603edb6b]: Original author-written MHD chapter, Eq. 2.24; publisher open-access full HTML directly checked. [^ref-fd87f7777b80]: University of Wisconsin, “Hartmann Flow”, Eq. 5.28. [^ref-fa1761e72c54]: 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 [^ref-2eb1805de535]: Original continuous-casting electromagnetic-brake study.

Relationships to Other Abstractions

Local relationship map for Stuart NumberParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Stuart NumberDOMAINDomain-specific abstraction: Dimensionless Quantity — is a kind ofDimensionlessQuantityDOMAINPrime abstraction: Ratio — is a kind ofRatioPRIME

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

Neighborhood in Abstraction Space

Stuart Number sits in a sparse region of the domain-specific corpus (94th 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