Knowledge card

Substituted Convective Heat Transport

Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory

T⁻¹Θ

Symbol

(c Δ T_{0} x / (1 - γ t)^{3}) (f' θ - f θ')

The convective heat transport after substitution, with the independent thermal amplitude ΔT₀.

Main equation

Level 0 of 5

Brief — named quantities

About levels

Why the levels load one at a time

Only level 0 travels with this article; the expansion runs to level 5. 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.

((\frac{c Δ T _{0} x}{( 1 - γ t ) ^{3}}) f^{'} θ S2.1 - (\frac{c Δ T _{0} x}{( 1 - γ t ) ^{3}}) f θ^{'})

Dimension:T⁻¹Θ (s⁻¹ K)Maths:ℝ scalar(·, ·, ·) — A single number (real numbers), depending on another quantity, another quantity, another quantityDimensionally verifiedMathematically verifiedFully reduced to base quantities

In words:minus.

Open full derivation chain: 3 stepsDerivation

1
Open Step 1: Given / definition · The source relation this block transforms: the advective transport of heat carried by the flow …
The source relation this block transforms: the advective transport of heat carried by the flow — u∂T/∂x plus v∂T/∂y.Given / definition
Level 0 · more available
Brief — named quantities
About 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.
$((u S3.1 \frac{\partial T}{\partial x}) S3.3 + (v S3.2 \frac{\partial T}{\partial y}))$
2
Open Step 2: Substitution · Insert u=U_wf′, v=−√[cν_f/(1−γt)]f, ∂T/∂x=[ΔT₀/(1−γt)²]θ, and ∂T/∂y=[ΔT₀x/(1−γt)²]√[c/(ν_f(1−γt…
Insert u=U_wf′, v=−√[cν_f/(1−γt)]f, ∂T/∂x=[ΔT₀/(1−γt)²]θ, and ∂T/∂y=[ΔT₀x/(1−γt)²]√[c/(ν_f(1−γt))]θ′.Substitution Substitution · η
Open operand record: Substitution η
Substitution:L0 $η$ L2 $(\frac{c}{ν _{f} ( 1 - γ t )}) \sqrt ? y$
Open term-change ledger: 2 records
Open record 1: ∂T/∂x → U_w f′·(ΔT₀/(1−γt)²)θ
Consumes:L0 $\partial T / \partial x$ Produces:L0 $U_{w} f' \cdot (Δ T_{0} / (1 - γ t)^{2}) θ$ Streamwise advection becomes (U_wf′)[ΔT₀/(1−γt)²]θ and remains nonzero because the reference excess grows with x.U_w f′·(ΔT₀/(1−γt)²)θ
Open record 2: ∂T/∂y → −√(cν_f/(1−γt))f·(ΔT₀x/(1−γt)²)√(c/(ν_f(1−γt)))…
Consumes:L0 $\partial T / \partial y$ Produces:L0 $- \sqrt (c ν_{f} / (1 - γ t)) f \cdot (Δ T_{0} x / (1 - γ t)^{2}) \sqrt (c / (ν_{f} (1 - γ t))) θ'$ Wall-normal advection becomes −√[cν_f/(1−γt)]f·[ΔT₀x/(1−γt)²]√[c/(ν_f(1−γt))]θ′.−√(cν_f/(1−γt))f·(ΔT₀x/(1−γt)²)√(c/(ν_f(1−γt)))θ′
Level 0 · more available
Brief — named quantities
About 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.
$(U_{w} f^{'} \cdot (\frac{Δ T _{0}}{( 1 - γ t ) ^{2}}) θ S1.1 - - \sqrt (\frac{c ν _{f}}{1 - γ t}) f \cdot (\frac{Δ T _{0} x}{( 1 - γ t ) ^{2}}) \sqrt (\frac{c}{ν _{f} ( 1 - γ t )}) θ^{'})$
★
Open Step 3: Simplification · Both products reduce to the same cΔT₀x/(1−γt)³ scale.
Both products reduce to the same cΔT₀x/(1−γt)³ scale.Simplification
Open term-change ledger: 2 records
Open record 1: U_w f′·(ΔT₀/(1−γt)²)θ → (cΔT₀x/(1−γt)³)f′θ
Consumes:L0 $U_{w} f' \cdot (Δ T_{0} / (1 - γ t)^{2}) θ$ Produces:L0 $(c Δ T_{0} x / (1 - γ t)^{3}) f' θ$ U_w[ΔT₀/(1−γt)²]=cΔT₀x/(1−γt)³ leaves the f′θ term.(cΔT₀x/(1−γt)³)f′θ
Open record 2: −√(cν_f/(1−γt))f·(ΔT₀x/(1−γt)²)√(c/(ν_f(1−γt)))… → (cΔT₀x/(1−γt)³)fθ′
Consumes:L0 $- \sqrt (c ν_{f} / (1 - γ t)) f \cdot (Δ T_{0} x / (1 - γ t)^{2}) \sqrt (c / (ν_{f} (1 - γ t))) θ'$ Produces:L0 $(c Δ T_{0} x / (1 - γ t)^{3}) f θ'$ The wall-normal dimensional factors also combine to cΔT₀x/(1−γt)³, leaving fθ′.(cΔT₀x/(1−γt)³)fθ′
Level 0 · more available
Brief — named quantities
About 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.
$((\frac{c Δ T _{0} x}{( 1 - γ t ) ^{3}}) f^{'} θ S2.1 - (\frac{c Δ T _{0} x}{( 1 - γ t ) ^{3}}) f θ^{'})$

Equation Workspace

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Equation workspace

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Domain Analysis

Unit status, dimensional trails, per-step checks, and custom unit expressions.

Step-by-step unit check (3 steps)

Step 1 given
Expand checked this step.
Step 2 substitution
Expand checked this step.
Step 3 simplify
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Reads as: (cΔT₀x/(1−γt)³)(f′θ−fθ′)

“(cΔT₀x/(1−γt)³)(f′θ−fθ′)” 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.

Check one step at a time (3 steps)

Step 1 — given

Step 2 — substitution

Step 3 — simplify

Essence
S1The raw substituted heat-advection products retain ΔT₀ as a thermal amplitude distinct from c.
S1.1essenceU_w f′·(ΔT₀/(1−γt)²)θ reduced by −√(cν_f/(1−γt))f·(ΔT₀x/(1−γt)²)√(c/(ν_f(1−γt)))θ′.
S2The substituted transport is [cΔT₀x/(1−γt)³](f′θ−fθ′).
S2.1essence(cΔT₀x/(1−γt)³)f′θ reduced by (cΔT₀x/(1−γt)³)fθ′.

Dimension I

Definition

What it is — and what it is not

What it is: S3The advective transport of heat carried by the flow: u∂T/∂x + v∂T/∂y.
S3.1essenceThe product of u and ∂T/∂x.
S3.2essenceThe product of v and ∂T/∂y.
S3.3essenceThe sum of (u · ∂T/∂x) and (v · ∂T/∂y).

Dimension II

In practice

How to deal with it

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Encyclopedia

Substituted Convective Heat Transport

Equation Workspace

Domain Analysis

Mathematical Analysis