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Knowledge card
Magnetic Damping Term
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The Lorentz braking term — the conductivity-to-density ratio times the magnetic parameter times the dimensionless velocity.
About levels
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To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
In words:divided bytimestimes.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · The source relation this block transforms: the substituted Lorentz braking — the magnetic coeff…
The source relation this block transforms: the substituted Lorentz braking — the magnetic coefficient times U_w f′.Given / definition Defining ruleThe starting relation of a derivation — the definition or established equation every later step transforms; it justifies no change itself.Level 0 · more availableBrief — named quantitiesAbout levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
- ★
Open Step 2: Parameter definition · Divide by the convective scale c²x/(1−γt)² and insert B(t)² = B₀²/(1−γt): the residual transien…
Divide by the convective scale c²x/(1−γt)² and insert B(t)² = B₀²/(1−γt): the residual transient factor cancels exactly (the very reason the B(t) artifact is chosen), and σ_hnf B₀²/(ρ_hnf c) is identified BY DEFINITION as (φ_σ/φ_ρ)M.Parameter definition Defining ruleΠ ≔ (grouped physical quantities)Naming a recurring group of quantities as a single parameter (A ≔ γ/c, M ≔ σB₀²/(ρc), …), replacing the group by its name.Open term-change ledger: 1 records
Open record 1: σB²/ρ + Uw → M
Consumes:L0L0UwProduces:L0MDividing by the convective scale c²x/(1−γt)² and inserting B(t)² = B₀²/(1−γt): the (1−γt) factors cancel exactly, and the residue σ_hnf B₀²/(ρ_hnf c) is DEFINED as (φ_σ/φ_ρ)M — a Hartmann-type magnetic parameter.MResulting expressionMagnetic Parameter MThe strength of the Lorentz force relative to inertia (a Hartmann-type group); its definition is composed on demand.Level 0 · more availableBrief — named quantitiesAbout levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
Equation Workspace
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Calculator
Equation Workspace
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Equation workspace
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Domain Analysis
Unit status, dimensional trails, per-step checks, and custom unit expressions.
Domain Analysis
Unit status, dimensional trails, per-step checks, and custom unit expressions.
Step-by-step unit check (2 steps)
Step 1 given
Expand checked this step.Step 2 parameter_definition
Expand checked this step.
Try your own — the unit calculator
Ever wondered whether a formula “adds up”? This tool answers one plain question: do the units match? Pick a couple of quantities (distance, time, a speed…), join them with ×, ÷, +, or =, and press Run. It works out the resulting unit — for example distance ÷ time gives a speed, LT⁻¹ (metres per second) — and flags anything that can’t be right, like adding a length to a time.
Reads as: (φ_σ/φ_ρ) M f′
“(φ_σ/φ_ρ) M f′” is analysed by expanding its full defining equation (its children and their relations), not as a bare symbol. Use Check this concept’s units above for the same result.
Mathematical Analysis
Object type, lawful operations, conditions, comparisons, and derivation-step checks.
Mathematical Analysis
Object type, lawful operations, conditions, comparisons, and derivation-step checks.
Check one step at a time (2 steps)
Step 1 — given
Step 2 — parameter definition
Essence
S1The magnetic damping term of the reduced ODE: (φ_σ/φ_ρ)M f′.
- S1.1essenceφ_σ divided by φ_ρ.
- S1.2essenceThe product of (φ_σ / φ_ρ) and M.
- S1.3essenceThe product of ((φ_σ / φ_ρ) · M) and f′.
Dimension I
Definition
What it is — and what it is not
- What it is
- S2The substituted Lorentz braking: (σ_hnf B²/ρ_hnf)U_w f′.
- S2.1essenceThe product of σB²/ρ and Uw.
- S2.2essenceThe product of (σB²/ρ · Uw) and f′.
- S2The substituted Lorentz braking: (σ_hnf B²/ρ_hnf)U_w f′.
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.