Knowledge card
Powell–Eyring Rheology
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
A non-Newtonian constitutive law capturing shear-thinning between Newtonian limits through the material constant β and characteristic shear rate ς; the stretching rate c is unrelated.
About levels
Why the levels load one at a time
Only level 0 travels with this article; the expansion runs to level 4. 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:minus.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · The exact Powell–Eyring shear stress: a Newtonian part plus the inverse-hyperbolic-sine kernel …
The exact Powell–Eyring shear stress: a Newtonian part plus the inverse-hyperbolic-sine kernel — shear-thinning between two Newtonian limits.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: Taylor expansion / truncation · Truncate the kernel: sinh⁻¹(z) ≈ z − z³/6 for |z| ≪ 1, i.e. (1/ς)|∂u/∂y| ≪ 1 — the truncated la…
Truncate the kernel: sinh⁻¹(z) ≈ z − z³/6 for |z| ≪ 1, i.e. (1/ς)|∂u/∂y| ≪ 1 — the truncated law feeding the momentum equation. Outside that shear-rate range the model understates the true stress.Taylor expansion / truncation Defining rulef(x) = Σₙ f⁽ⁿ⁾(a)(x−a)ⁿ/n!; e.g. sinh⁻¹(z) ≈ z − z³/6 for |z| ≪ 1Expanding a function as a power series about a point; truncating the series introduces the leading terms and discards the higher-order remainder within a stated validity range.Open term-change ledger: 1 records
Open record 1: μ∂u/∂y + (1/β)sinh⁻¹((1/ς)∂u/∂y) → (μ+1/(βς))∂u/∂y + (1/(6βς³))(∂u/∂y)³
Consumes:L0L0Produces:L0L0sinh⁻¹(z) ≈ z − z³/6 for |z| ≪ 1: the kernel’s linear part 1/(βς)·∂u/∂y joins the Newtonian stress to form the effective linear stress (μ + 1/(βς))∂u/∂y, and its cubic remainder becomes the shear-thinning correction (1/(6βς³))(∂u/∂y)³; higher-order terms are discarded within the stated validity range.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 Powell–Eyring Rheology.
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 taylor_expansion
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: Powell–Eyring Rheology
“Powell–Eyring Rheology” 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 — taylor expansion
Essence
S1It interpolates between Newtonian behaviour at low and high shear through β and the characteristic shear rate ς; c belongs only to the stretching kinematics. [1.1]
Dimension I
Definition
What it is — and what it is not
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.