Math 2.0

Classical thermodynamics, heat transfer Definition

Steady-state heat conduction

Builds on Thermal conductionThermal conductivityFourier's law of heat conductionHeat equationLaplace's equationPoisson's equation

Definition

Let a body with constant thermal conductivity k>0k > 0 occupy an open set Ω⊆ℝn\Omega \subseteq \R^n. Let its temperature TT obey the heat equation with a heat source of density f(x)f(x) per unit volume. The conduction is in steady state when TT does not depend on time. Then T=T(x)T = T(x) satisfies the Poisson equation −k ΔT=fin Ω.-k \, \Delta T = f \quad \text{in } \Omega. Without sources, f=0f = 0, and TT satisfies Laplace's equation ΔT=0\Delta T = 0.

Remarks

Example: in a slab 0<x<L0 < x < L with faces held at T(0)=T1T(0) = T_1 and T(L)=T2T(L) = T_2 and no sources, T′′=0T'' = 0. So T(x)=T1+(T2−T1)x/LT(x) = T_1 + (T_2 - T_1) x / L, and by Fourier's law the flux q=k(T1−T2)/Lq = k (T_1 - T_2) / L is the same at every xx.

Non-example: T(x,t)=e−tsin⁡xT(x, t) = e^{-t} \sin x on 0<x<π0 < x < \pi solves the heat equation, but it is not steady, since ∂tT≠0\partial_t T \ne 0.

A steady state needs boundary data that do not change in time. On a bounded domain with fixed boundary temperatures, solutions of the heat equation tend to the steady state as t→∞t \to \infty.

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To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat equation, concept, no evidence levelHeat equationconceptHeat flux, concept, no evidence levelHeat fluxconceptLaplace's equation, concept, no evidence levelLaplace's equationconceptPoisson's equation, concept, no evidence levelPoisson's equationconceptTemperature, concept, no evidence levelTemperatureconceptThermal conduction, concept, no evidence levelThermal conductionconceptThermal conductivity, concept, no evidence levelThermal conductivityconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptGradient, concept, no evidence levelGradientconceptFourier's law of heat conduction, statement, no evidence levelFourier's law of heat…statementSteady-state heat conduction, definition, no evidence levelSteady-state heat con…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Heat equation concept
  4. Heat flux concept
  5. Laplace's equation concept
  6. Poisson's equation concept
  7. Temperature concept
  8. Thermal conduction concept
  9. Thermal conductivity concept
  10. Total derivative concept
  11. Function of several variables concept
  12. Differentiable function of several variables concept
  13. Partial derivative concept
  14. Gradient concept
  15. Fourier's law of heat conduction statement
  16. Steady-state heat conduction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Heat_equation
About this page
TagKXWB2NQZXP
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MSC80A19, 35J05
DefinesSteady-state heat conduction
UsesThermal conduction, Thermal conductivity, Fourier's law of heat conduction, Heat equation, Laplace's equation, Poisson's equation
Changed3 November 2026

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