2 parents · 2 chains
Knowledge card
Unsteadiness Term
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The transient momentum term — the unsteadiness parameter times its velocity group.
About levels
Why the levels load one at a time
Only level 0 travels with this article; the expansion runs to level 8. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them.
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:times.
Open full derivation chain: 3 stepsDerivation
- 1
Open Step 1: Given / definition · The source relation this block transforms: the substituted unsteady acceleration — the growth s…
The source relation this block transforms: the substituted unsteady acceleration — the growth scale times the transient velocity group.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.
- 2
Open Step 2: Non-dimensionalisation · Divide the whole momentum balance by the convective scale c²x/(1−γt)²: γ(cx/(1−γt)²) becomes th…
Divide the whole momentum balance by the convective scale c²x/(1−γt)²: γ(cx/(1−γt)²) becomes the raw ratio γ/c, so the term reads (γ/c)(f′ + ηf″/2) — every explicit x and t has cancelled.Non-dimensionalisation Defining rulex* = x/x_ref ⟹ dimensionless groups (Π-theorem)Scaling every variable by a reference quantity so the equation is left with named dimensionless groups as its only coefficients.Divisor · c²x/(1−γt)² Operand nodeConvective Momentum ScaleThe dimensional scale of u∂u/∂x — the common factor every momentum term carries after substitution, whose cancellation IS the self-similarity.Divisor: The source is divided by the operand, leaving a named remainder.Open term-change ledger: 1 records
Open record 1: cx/(1−γt)² → γ/c
Consumes:L0Produces:L0Dividing γ(cx/(1−γt)²) by the convective scale c²x/(1−γt)² leaves the raw ratio γ/c; the transient velocity group f′ + ηf″/2 is carried unchanged.γ/cResulting expressionTransient-to-stretch RatioThe raw ratio of the transient rate to the stretching rate that emerges when the unsteady term is divided by the convective scale — the quantity the unsteadiness parameter A names.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 3: Parameter definition · Identify the raw ratio γ/c BY DEFINITION as the unsteadiness parameter A.
Identify the raw ratio γ/c BY DEFINITION as the unsteadiness parameter A.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: γ/c → A
Consumes:L0Produces:L0AThe raw ratio γ/c is DEFINED as the unsteadiness parameter A = γ/c; substituting the name gives the dimensionless unsteadiness term A(f′ + ηf″/2).AResulting expressionUnsteadiness Parameter AA — the dimensionless rate of transient change; larger A means a faster-evolving boundary layer.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
Set known values, calculate unknowns, inspect dependencies, and run numerical solves.
Calculator
Equation Workspace
Set known values, calculate unknowns, inspect dependencies, and run numerical solves.
Equation workspace
Set the values you know, then calculate the remaining quantities in Unsteadiness Term.
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 (3 steps)
Step 1 given
Expand checked this step.Step 2 non_dimensionalise
Expand checked this step.Step 3 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: A(f′+(η/2)f″)
“A(f′+(η/2)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 (3 steps)
Step 1 — given
Step 2 — non dimensionalise
Step 3 — parameter definition
Essence
S1The scale-divided unsteady term with its raw coefficient exposed: (γ/c)(f′ + ηf″/2).
- S1.1essenceThe product of γ/c and f′+f″(η/2).
S2The unsteadiness term of the reduced ODE: A(f′ + ηf″/2).
- S2.1essenceThe product of A and f′+f″(η/2).
Dimension I
Definition
What it is — and what it is not
- What it is
- S3The substituted unsteady acceleration: γ times the growth scale cx/(1−γt)² times (f′ + ηf″/2).
- S3.1essenceThe product of γ and cx/(1−γt)².
- S3.2essenceThe product of (γ · cx/(1−γt)²) and f′+f″(η/2).
- S3The substituted unsteady acceleration: γ times the growth scale cx/(1−γt)² times (f′ + ηf″/2).
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.