Knowledge card
Wall Mass-transfer Condition
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The wall condition setting the similarity function equal to the suction/injection parameter.
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In words:equals.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · The wall is porous: the normal velocity at y = 0 is the prescribed mass-transfer velocity v_w =…
The wall is porous: the normal velocity at y = 0 is the prescribed mass-transfer velocity v_w = −S√(cν_f/(1−γt)) (suction for S > 0, injection for S < 0) — 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
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: Apply boundary condition · Insert the map v = −√(cν_f/(1−γt)) f and cancel the common factor −√(cν_f/(1−γt)) from both sid…
Insert the map v = −√(cν_f/(1−γt)) f and cancel the common factor −√(cν_f/(1−γt)) from both sides.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: 2 records
Open record 1: v(0) → f(0)
Consumes:L0Produces:L0The map v = −√(cν_f/(1−γt)) f turns the wall value v(0) into −√(cν_f/(1−γt)) f(0); after cancelling the common factor, the left side is f(0).f(0)Resulting expressionWall Stream FunctionThe similarity function evaluated at the wall.Open record 2: −S√(cν_f/(1−γt)) → S
Consumes:L0Produces:L0SDividing both sides by the common factor −√(cν_f/(1−γt)) reduces the prescribed mass transfer −S√(cν_f/(1−γt)) to the bare suction/injection parameter S.SResulting expressionSuction / Injection Parameter SS > 0 is suction and S < 0 is injection through the porous wall, set by the wall-normal velocity v_w.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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Calculator
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Equation workspace
Set the values you know, then calculate the remaining quantities in Wall Mass-transfer 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)=S
“f(0)=S” 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 wall mass-transfer condition: v(0) = −S√(cν_f/(1−γt)).
- S1.1essencev(0) set equal to −S√(cν_f/(1−γt)).
- S1The dimensional wall mass-transfer condition: v(0) = −S√(cν_f/(1−γt)).
- Wisdoms
- S2The wall condition in similarity variables: f(0) = S.
- S2.1essencef(0) set equal to S.
- S2The wall condition in similarity variables: f(0) = S.
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.