Knowledge card
Wall Slip Condition
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The wall condition relating the velocity to the wall shear through the slip parameter.
In words:equalsplus.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · The fluid partially slips on the stretching wall: u at y = 0 is the wall velocity U_w plus the …
The fluid partially slips on the stretching wall: u at y = 0 is the wall velocity U_w plus the slip length L₀√(1−γt) times the wall shear rate (Navier slip) — the dimensional boundary 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
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- ★
Open Step 2: Apply boundary condition · Insert the map u = U_w f′ and divide by U_w: the √(1−γt) decay of the slip length cancels the g…
Insert the map u = U_w f′ and divide by U_w: the √(1−γt) decay of the slip length cancels the growth of the shear scale, leaving the constant slip parameter L = L₀√(c/ν_f).Apply boundary condition Defining ruleu|_{y=0} = u_w, u|_{y→∞} = u_∞Imposing the constraint the solution must satisfy on a boundary; evaluating the balance there can eliminate interior terms.Open term-change ledger: 3 records
Open record 1: u(0) → f′(0)
Consumes:L0Produces:L0The map u = U_w f′ turns the wall value u(0) into U_w f′(0); dividing by U_w leaves f′(0).f′(0)Resulting expressionWall VelocityThe dimensionless velocity evaluated at the wall.Open record 2: Uw → 1
Consumes:L0UwProduces:L01Dividing the wall-velocity term by U_w itself leaves exactly 1 — the unit stretching velocity of the similarity frame.1Resulting expressionOneThe multiplicative identity: the unit reference for the normalised wall stretching and the baseline from which the transient and thermal deficits are measured.Open record 3: L₀√(1−t)·∂u/∂y(0) → L f″(0)
Consumes:L0Produces:L0Dividing the slip excess L₀√(1−γt)·∂u/∂y(0) by U_w and inserting the shear-rate map: the √(1−γt) decay of the slip length cancels the (1−γt)^{−1/2} growth of the shear scale exactly, leaving the constant group L f″(0) — the new slip parameter L = L₀√(c/ν_f) is named here.L f″(0)Resulting expressionSlip Velocity ContributionThe slip parameter times the wall shear — the partial-slip addition to the wall velocity.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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Equation workspace
Set the values you know, then calculate the remaining quantities in Wall Slip 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 boundary_condition
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)=1+Lf″(0)
“f′(0)=1+Lf″(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 — boundary condition
Dimension I
Definition
What it is — and what it is not
- What it is
- S1The dimensional partial-slip condition: u(0) = U_w + L₀√(1−γt) · ∂u/∂y(0).
- S1.1essenceThe sum of Uw and L₀√(1−t)·∂u/∂y(0).
- S1.2essenceu(0) set equal to (Uw + L₀√(1−t)·∂u/∂y(0)).
- S1The dimensional partial-slip condition: u(0) = U_w + L₀√(1−γt) · ∂u/∂y(0).
- Wisdoms
- S2The slip condition in similarity variables: f′(0) = 1 + L f″(0).
- S2.1essenceThe sum of 1 and L f″(0).
- S2.2essencef′(0) set equal to (1 + L f″(0)).
- S2The slip condition in similarity variables: f′(0) = 1 + L f″(0).
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.