Knowledge card

Energy Equation

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

T⁻¹Θ

Symbol

\nabla^{2} T

The baseline thermal balance: unsteady and convective transport against conduction and thermal radiation, with Joule heating and viscous dissipation explicitly switched off.

Main equation

Level 0 of 12

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 12. 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{\partial T}{\partial t} S2.1 + u \cdot \nabla T) S2.3 = (k \nabla T S2.2 - q_{r}))

Dimension:T⁻¹Θ (s⁻¹ K)Maths:𝔹 equation(·, ·, ·, ·) — A statement that two sides are equal, depending on another quantity, another quantity, another quantity, another quantityDimensionally verifiedMathematically verifiedFully reduced to base quantities

In words:plusequalsminus.

Open full derivation chain: 1 stepsDerivation

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Open Step 1: Given / definition
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.
$((\frac{\partial T}{\partial t} S2.1 + u \cdot \nabla T) S2.3 = (k \nabla T S2.2 - q_{r}))$

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Domain Analysis

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

Step-by-step unit check (1 steps)

Step 1 given
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: ∇²T

“∇²T” 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 (1 steps)

Step 1 — given

Heat Transfer Equation

Essence
S1Unsteady + convective transport balanced by conduction and radiative-flux divergence. [1.1]

Dimension I

Definition

What it is — and what it is not

What it is: S2The baseline unsteady convective energy balance with conduction and linearised Rosseland diffusion.
S2.1essenceThe sum of ∂T/∂t and u·∇T. [1.1]
S2.2essencek∇T reduced by q_r. [1.1]
S2.3essence(∂T/∂t + u·∇T) set equal to (k∇T − q_r). [1.1]
Boundary of meaning: S3Joule heating and viscous dissipation are explicit zero-valued model switches in this baseline; retaining either requires adding its source term before reduction.

Dimension II

In practice

How to deal with it

No practical guidance recorded yet.

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Encyclopedia

Energy Equation

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Domain Analysis

Mathematical Analysis