Knowledge card
Substituted Buoyancy
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The buoyancy after substituting T−T∞=(T_r−T∞)θ; its reference excess is a temperature, so the term remains an acceleration.
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In words:timestimestimestimes.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · The source relation this block transforms: the thermal buoyancy along the tilted plate — gβ_hnf…
The source relation this block transforms: the thermal buoyancy along the tilted plate — gβ_hnf(T − T∞)cos α.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: Substitution · Substitute T−T∞=(T_r−T∞)θ with T_r−T∞=ΔT₀x/(1−γt)². The reference excess remains a temperature …
Substitute T−T∞=(T_r−T∞)θ with T_r−T∞=ΔT₀x/(1−γt)². The reference excess remains a temperature and is not the velocity scale.Substitution Defining rulex = a ⟹ f(x) = f(a)Replacing a symbol or expression by an equal expression; equality is preserved because equals substitute for equals.Substitution · θ Operand nodeDimensionless Temperature θ(η)The scaled temperature (one at the wall scale, zero in the free stream) — the unknown of the energy ODE; its definition is composed on demand.Substitution: The operand (a map or model) replaces the source directly.Open operand record: Substitution θ
Substitution:L0L1Open term-change ledger: 1 records
Open record 1: T−T∞ → Tr−T∞
Consumes:L0Produces:L0The θ-map rewrites the fluid excess as the prescribed reference excess (T_r−T∞) [Θ] times θ.Tr−T∞Resulting expressionReference Temperature ExcessThe prescribed heating-fluid/reference excess above ambient that normalises θ; the actual surface excess is this value times θ(0).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
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Calculator
Equation Workspace
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Equation workspace
Set the values you know, then calculate the remaining quantities in Substituted Buoyancy.
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 substitution
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: gβ_hnf(Tr−T∞)θcosα
“gβ_hnf(Tr−T∞)θcosα” 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 — substitution
Essence
S1The mapped buoyancy gβ_hnf(T_r−T∞)θcosα remains an acceleration.
- S1.1essenceThe product of g and β_hnf.
- S1.2essenceThe product of (g · β_hnf) and Tr−T∞.
- S1.3essenceThe product of ((g · β_hnf) · Tr−T∞) and θ.
- S1.4essenceThe product of (((g · β_hnf) · Tr−T∞) · θ) and cos α.
Dimension I
Definition
What it is — and what it is not
- What it is
- S2The thermal buoyancy driving the flow along the tilted plate: gβ_hnf(T − T∞)cos α.
- S2.1essenceThe product of g and β_hnf.
- S2.2essenceThe product of (g · β_hnf) and T−T∞.
- S2.3essenceThe product of ((g · β_hnf) · T−T∞) and cos α.
- S2The thermal buoyancy driving the flow along the tilted plate: gβ_hnf(T − T∞)cos α.
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.