Knowledge card
Thermal Diffusion Term
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The conduction-plus-radiation term — the effective coefficient times the temperature curvature θ″.
About levels
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In words:times.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · The source relation this block transforms: the substituted effective diffusion — conduction and…
The source relation this block transforms: the substituted effective diffusion — conduction and radiative conduction, both proportional to θ″.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: Parameter definition · Divide by the thermal scale and identify the coefficients by the definitions of Pr and Rd: cond…
Divide by the thermal scale and identify the coefficients by the definitions of Pr and Rd: conduction and radiation GATHER into the single verified coefficient k₁ = (1/Pr)(1/φ_Cp)(φ_k + 4Rd/3), times θ″.Parameter definition Defining ruleΠ ≔ (grouped physical quantities)Naming a recurring group of quantities as a single parameter (A ≔ γ/c, M ≔ σB₀²/(ρc), …), replacing the group by its name.Open term-change ledger: 1 records
Open record 1: (k/(ρCp))_hnf(ΔT₀x/(1−γt)²)(c/(ν_f(1−γt)))θ″ + … → k₁
Consumes:L0L0Produces:L0k₁Dividing by the thermal scale and applying the definitions of Pr = μCp/k and Rd = 4σ*T∞³/(k*k): conduction contributes (1/Pr)(φ_k/φ_Cp) and radiation (1/Pr)(1/φ_Cp)(4Rd/3) — BOTH stored against (ρCp)_hnf, so 1/φ_Cp divides the WHOLE group (the verified grouping): k₁ = (1/Pr)(1/φ_Cp)(φ_k + 4Rd/3); the θ″ factor is carried explicitly.k₁Resulting expressionEffective Thermal Diffusivity CoefficientThe leading coefficient of the energy ODE — (1/Pr)(1/φ_Cp)(φ_k + 4Rd/3). Verified grouping: 1/φ_Cp divides the WHOLE diffusive group, because conduction AND radiation are both stored against the mixture’s volumetric heat capacity.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
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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 parameter_definition
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: k₁θ″
“k₁θ″” 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 — parameter definition
Essence
S1The effective thermal-diffusion term of the reduced ODE: (1/Pr)(1/φ_Cp)(φ_k + 4Rd/3)θ″.
- S1.1essenceThe product of k₁ and θ″.
Dimension I
Definition
What it is — and what it is not
- What it is
- S2The substituted effective diffusion: the conductive and radiative-conductive fluxes, each the diffusive coefficient times θ″.
- S2.1essenceThe sum of (k/(ρCp))_hnf(ΔT₀x/(1−γt)²)(c/(ν_f(1−γt)))θ″ and (1/(ρCp)_hnf)(16σ*T∞³/3k*)(ΔT₀x/(1−γt)²)(c/(ν_f(1−γt)))θ″.
- S2The substituted effective diffusion: the conductive and radiative-conductive fluxes, each the diffusive coefficient times θ″.
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.