Knowledge card
Dimensionless Powell–Eyring Terms
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The Powell–Eyring diffusion and nonlinear retardation of the reduced ODE grouped as ONE block — the dimensionless form of the substituted Powell–Eyring stress it derives from. The verified grouping carries 1/φ_ρ onto BOTH stress groups.
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In words:minus.
Open full derivation chain: 3 stepsDerivation
- 1
Open Step 1: Given / definition · The source relation this block transforms: the substituted Powell–Eyring stress — linear diffus…
The source relation this block transforms: the substituted Powell–Eyring stress — linear diffusion and nonlinear retardation with explicit dimensional coefficients.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
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- 2
Open Step 2: Non-dimensionalisation · Divide both stress parts by the convective scale c²x/(1−γt)²: the diffusion coefficient becomes…
Divide both stress parts by the convective scale c²x/(1−γt)²: the diffusion coefficient becomes (ν_hnf + 1/(ρ_hnf βς))/ν_f and the retardation exposes its x², (1−γt)⁻³ scale — the raw coefficients the parameter step will name.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: 2 records
Open record 1: (ν_hnf+1/(ρ_hnf βς))U_w(c/(ν_f(1−γt)))f‴ → ((ν_hnf+1/(ρ_hnf βς))/ν_f)f‴
Consumes:L0Produces:L0Dividing the linear diffusion by the convective scale leaves the raw coefficient (ν_hnf + 1/(ρ_hnf βς))/ν_f times f‴.((ν_hnf+1/(ρ_hnf βς))/ν_f)f‴Resulting expressionScale-divided Powell–Eyring DiffusionThe linear stress after division — the dimensional coefficient per base viscosity, which the parameter step identifies as (φ_μ+ε)/φ_ρ.Open record 2: (1/(2ρ_hnf βς³))U_w³(c/(ν_f(1−γt)))²f″²f‴ → (1/(2ρ_hnf βς³))(c³x²/(ν_f²(1−γt)³))f″²f‴
Consumes:L0Produces:L0Dividing the nonlinear retardation by the convective scale exposes its residual x² and (1−γt)⁻³ dependence times f″²f‴ — the local-similarity artifact the material scaling must absorb.(1/(2ρ_hnf βς³))(c³x²/(ν_f²(1−γt)³))f″²f‴Resulting expressionScale-divided Powell–Eyring RetardationThe nonlinear stress after division — the retardation scale exposes the x², (1−γt)⁻³ dependence that the material scaling absorbs into the constant δ.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 each raw coefficient BY DEFINITION: the diffusion coefficient becomes (φ_μ + ε)/φ_ρ an…
Identify each raw coefficient BY DEFINITION: the diffusion coefficient becomes (φ_μ + ε)/φ_ρ and the retardation coefficient εδ/φ_ρ — the verified grouping carrying 1/φ_ρ on BOTH stress groups.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: 2 records
Open record 1: ((ν_hnf+1/(ρ_hnf βς))/ν_f)f‴ → c₁f‴
Consumes:L0Produces:L0(ν_hnf + 1/(ρ_hnf βς))/ν_f is identified as (φ_μ + ε)/φ_ρ by the definitions of φ_μ, φ_ρ and ε — dividing the balance by ρ_hnf puts 1/φ_ρ on the WHOLE group because ε is defined with base-fluid properties (the verified correction).c₁f‴Resulting expressionPowell–Eyring Diffusion TermThe highest-order diffusion term — the effective Powell–Eyring coefficient times the velocity curvature f‴.Open record 2: (1/(2ρ_hnf βς³))(c³x²/(ν_f²(1−γt)³))f″²f‴ → (εδ/φ_ρ)f″²f‴
Consumes:L0Produces:L0The retardation coefficient (1/(2ρ_hnf βς³)) times the retardation scale reduces to exactly εδ/φ_ρ by the definitions of ε and δ; the x² and (1−γt)⁻³ dependence is absorbed into δ (the local-similarity artifact made explicit).(εδ/φ_ρ)f″²f‴Resulting expressionNonlinear Retardation TermThe second-order Powell–Eyring term — the retardation coefficient εδ/φ_ρ times the squared velocity gradient times the velocity curvature. Because it multiplies f‴, the solvable form divides by (φ_μ+ε)/φ_ρ − (εδ/φ_ρ)f″², which must stay positive.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.
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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 (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.
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Reads as: ((φ_μ + ε)/φ_ρ) f‴ − ((εδ/φ_ρ))f″²f‴
“((φ_μ + ε)/φ_ρ) f‴ − ((εδ/φ_ρ))f″²f‴” 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 Powell–Eyring stress with raw coefficients exposed: ((ν_hnf+1/(ρ_hnf βς))/ν_f)f‴ − (1/(2ρ_hnf βς³))(c³x²/(ν_f²(1−γt)³))f″²f‴.
- S1.1essence((ν_hnf+1/(ρ_hnf βς))/ν_f)f‴ reduced by (1/(2ρ_hnf βς³))(c³x²/(ν_f²(1−γt)³))f″²f‴.
S2The dimensionless Powell–Eyring terms of the reduced ODE: (((φ_μ + ε)/φ_ρ))f‴ − ((εδ/φ_ρ))f″²f‴.
- S2.1essencec₁f‴ reduced by (εδ/φ_ρ)f″²f‴.
Dimension I
Definition
What it is — and what it is not
- What it is
- S3The substituted Powell–Eyring stress: the dimensional diffusion coefficient times f‴ minus the retardation coefficient times f″²f‴.
- S3.1essence(ν_hnf+1/(ρ_hnf βς))U_w(c/(ν_f(1−γt)))f‴ reduced by (1/(2ρ_hnf βς³))U_w³(c/(ν_f(1−γt)))²f″²f‴.
- S3The substituted Powell–Eyring stress: the dimensional diffusion coefficient times f‴ minus the retardation coefficient times f″²f‴.
Dimension II
In practice
How to deal with it
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