Knowledge card
Free-stream Velocity Condition
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The far-field condition driving the dimensionless velocity to zero.
In words:equalszero.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · The far field is quiescent: the axial velocity dies out away from the plate — the dimensional f…
The far field is quiescent: the axial velocity dies out away from the plate — the dimensional far-field condition to reduce.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: Similarity transformation · Insert the map u = U_w f′: since U_w ≠ 0, the dimensionless velocity itself must vanish.
Insert the map u = U_w f′: since U_w ≠ 0, the dimensionless velocity itself must vanish.Similarity transformation Defining ruleu(x, y, t) = U(x, t)·F(η), η = y/δ(x, t)A change of variables that collapses several independent variables into one similarity coordinate, reducing a PDE to an ODE; each substituted term splits into a dimensional scale times a dimensionless group.Substitution · u = U_w f′(η) Operand nodeAxial Velocity Similarity Definition u(x, y, t) = U_w(x, t) f′(η)The Problem 1 similarity definition mapping the dimensional axial velocity field u(x, y, t) to the wall-velocity scale U_w(x, t) times the dimensionless velocity f′(η).Substitution: The operand (a map or model) replaces the source directly.Open operand record: Substitution u = U_w f′(η)
Substitution:L0L2Open term-change ledger: 1 records
Open record 1: u(∞) → f′(∞)
Consumes:L0Produces:L0The map u = U_w f′ turns the far-field value u(∞) into U_w f′(∞); since U_w ≠ 0, the condition passes to the dimensionless velocity f′(∞).f′(∞)Resulting expressionFree-stream VelocityThe dimensionless velocity evaluated in the free stream.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 Free-stream Velocity Condition.
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 similarity_transform
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: f′(∞)=0
“f′(∞)=0” 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 — similarity transform
Dimension I
Definition
What it is — and what it is not
- What it is
- S1The dimensional far-field condition: u → 0 as y → ∞.
- S1.1essenceu(∞) set equal to 0.
- S1The dimensional far-field condition: u → 0 as y → ∞.
- Wisdoms
- S2The far-field condition in similarity variables: f′(∞) = 0.
- S2.1essencef′(∞) set equal to 0.
- S2The far-field condition in similarity variables: f′(∞) = 0.
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.