Knowledge card

Substituted Powell–Eyring Stress Terms

Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory

LT⁻²

Symbol

τ_{P} E (η)

The Powell–Eyring stress after substitution — its linear diffusion and nonlinear retardation parts grouped as ONE block, preserving the Powell–Eyring stress concept across the substitution. Derives from the Powell–Eyring stress terms.

Main equation

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((\frac{ν _{hn f} + 1}{ρ _{hn f} β ς}) U_{w} (\frac{c}{ν _{f} ( 1 - γ t )}) f^{'''} S2.1 - (\frac{1}{2 ρ _{hn f} β ς ^{3}}) U_{w}^{3} (\frac{c}{ν _{f} ( 1 - γ t )})^{2} f^{'' 2} f^{'''})

Dimension:LT⁻² (m s⁻²)Maths:ℝ scalar(·, ·, ·) — A single number (real numbers), depending on another quantity, another quantity, another quantityDimensionally verifiedMathematically verifiedFully reduced to base quantities

In words:minus.

Open full derivation chain: 3 stepsDerivation

1
Open Step 1: Given / definition · The source relation this block transforms: the Powell–Eyring stress contribution — linear visco…
The source relation this block transforms: the Powell–Eyring stress contribution — linear viscous diffusion (coefficient × ∂²u/∂y²) minus nonlinear retardation (coefficient × (∂u/∂y)² × ∂²u/∂y²).Given / definition
Level 0 · more available
Brief — named quantities
About 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.
$(P E^{1} S3.1 - P E^{2})$
2
Open Step 2: Substitution · Insert the velocity-gradient similarity maps ∂u/∂y = U_w√(c/(ν_f(1−γt)))f″ and ∂²u/∂y² = U_w(c/…
Insert the velocity-gradient similarity maps ∂u/∂y = U_w√(c/(ν_f(1−γt)))f″ and ∂²u/∂y² = U_w(c/(ν_f(1−γt)))f‴ into both stress parts. The linear diffusion, carrying one curvature, reaches (ν_hnf+1/(ρ_hnf βς))U_w(c/(ν_f(1−γt)))f‴ directly; the nonlinear retardation becomes the raw product (1/(2ρ_hnf βς³))(U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/(ν_f(1−γt)))f‴, its three wall-velocity scales not yet combined.Substitution Substitution · η
Open operand record: Substitution η
Substitution:L0 $η$ L2 $(\frac{c}{ν _{f} ( 1 - γ t )}) \sqrt ? y$
Open term-change ledger: 2 records
Open record 1: PE¹ → (ν_hnf+1/(ρ_hnf βς))U_w(c/(ν_f(1−γt)))f‴
Consumes:L0PE¹Produces:L0 $(ν_{h} n f + 1/ (ρ_{h} n f β ς)) U_{w} (c / (ν_{f} (1 - γ t))) f ‴$ The curvature map ∂²u/∂y² = U_w(c/(ν_f(1−γt)))f‴ substitutes the linear diffusion; its coefficient (ν_hnf + 1/(ρ_hnf βς)) stays explicit and the term is already in final form.(ν_hnf+1/(ρ_hnf βς))U_w(c/(ν_f(1−γt)))f‴
Open record 2: PE² → (1/(2ρ_hnf βς³))(U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/…
Consumes:L0PE²Produces:L0 $(1/ (2 ρ_{h} n f β ς^{3})) (U_{w} \sqrt (c / (ν_{f} (1 - γ t))) f ″)^{2} \cdot U_{w} (c / (ν_{f} (1 - γ t))) f ‴$ The shear-rate map (squared) and the curvature map substitute the nonlinear retardation, giving (1/(2ρ_hnf βς³))(U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/(ν_f(1−γt)))f‴ before the U_w scales are combined.(1/(2ρ_hnf βς³))(U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/(ν_f(1−γt)))f‴
Level 0 · more available
Brief — named quantities
About 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.
$((\frac{ν _{hn f} + 1}{ρ _{hn f} β ς}) U_{w} (\frac{c}{ν _{f} ( 1 - γ t )}) f^{'''} S1.1 - (\frac{1}{2 ρ _{hn f} β ς ^{3}}) (U_{w} \sqrt (\frac{c}{ν _{f} ( 1 - γ t )}) f^{''})^{2} \cdot U_{w} (\frac{c}{ν _{f} ( 1 - γ t )}) f^{'''})$
★
Open Step 3: Simplification · Combine the nonlinear retardation's wall-velocity scales: (U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/(ν_f(1…
Combine the nonlinear retardation's wall-velocity scales: (U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/(ν_f(1−γt)))f‴ = U_w³(c/(ν_f(1−γt)))²f″²f‴, giving (1/(2ρ_hnf βς³))U_w³(c/(ν_f(1−γt)))²f″²f‴. The linear diffusion is already in final form and carries over unchanged.Simplification
Open term-change ledger: 1 records
Open record 1: (1/(2ρ_hnf βς³))(U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/… → (1/(2ρ_hnf βς³))U_w³(c/(ν_f(1−γt)))²f″²f‴
Consumes:L0 $(1/ (2 ρ_{h} n f β ς^{3})) (U_{w} \sqrt (c / (ν_{f} (1 - γ t))) f ″)^{2} \cdot U_{w} (c / (ν_{f} (1 - γ t))) f ‴$ Produces:L0 $(1/ (2 ρ_{h} n f β ς^{3})) U_{w}^{3} (c / (ν_{f} (1 - γ t)))^{2} f ″^{2} f ‴$ Combining (U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/(ν_f(1−γt)))f‴ collects three wall-velocity scales into U_w³ and two similarity ratios into (c/(ν_f(1−γt)))², leaving U_w³(c/(ν_f(1−γt)))²f″²f‴.(1/(2ρ_hnf βς³))U_w³(c/(ν_f(1−γt)))²f″²f‴
Level 0 · more available
Brief — named quantities
About 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.
$((\frac{ν _{hn f} + 1}{ρ _{hn f} β ς}) U_{w} (\frac{c}{ν _{f} ( 1 - γ t )}) f^{'''} S2.1 - (\frac{1}{2 ρ _{hn f} β ς ^{3}}) U_{w}^{3} (\frac{c}{ν _{f} ( 1 - γ t )})^{2} f^{'' 2} f^{'''})$

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Reads as: τ_PE(η)

“τ_PE(η)” 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

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Step 2 — substitution

Step 3 — simplify

Essence
S1The Powell–Eyring stress after the gradient maps are inserted: the linear diffusion in final form, the nonlinear retardation still a raw product of substituted gradients.
S1.1essence(ν_hnf+1/(ρ_hnf βς))U_w(c/(ν_f(1−γt)))f‴ reduced by (1/(2ρ_hnf βς³))(U_w√(c/(ν_f(1−γt)))f″)²·U_w(c/(ν_f(1−γt)))f‴.
S2The substituted Powell–Eyring stress: the linear diffusion minus the retardation with its three wall-velocity scales combined into U_w³ and its similarity ratios into (c/(ν_f(1−γt)))².
S2.1essence(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‴.

Dimension I

Definition

What it is — and what it is not

What it is: S3The Powell–Eyring stress terms: linear viscous diffusion minus the nonlinear retardation, both carrying the velocity gradients ∂²u/∂y² and (∂u/∂y)².
S3.1essencePE¹ reduced by PE².

Dimension II

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Substituted Powell–Eyring Stress Terms

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