derivation of heat equation
Let us consider the heat conduction in a and specific heat capacity . Denote by the temperature in the point at the time . Let be a surface in the matter and the spatial region by it.
When the growth of the temperature of a volume element in the time is , the element releases the amount
of heat, which is the heat flux through the surface of . Thus if there are no sources and sinks of heat in , the heat flux through the surface in is
(1) |
On the other hand, the flux through in the time must be proportional to , to and to the derivative of the temperature in the direction of the normal line of the surface element , i.e. the flux is
where is a positive (because the heat always from higher temperature to lower one). Consequently, the heat flux through the whole surface is
which is, by the Gauss’s theorem, same as
(2) |
Equating the expressions (1) and (2) and dividing by , one obtains
Since this equation is valid for any region in the matter, we infer that
Denoting , we can write this equation as
(3) |
This is the differential equation of heat conduction, first derived by Fourier.
References
- 1 K. Väisälä: Matematiikka IV. Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
Title | derivation of heat equation |
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Canonical name | DerivationOfHeatEquation |
Date of creation | 2013-03-22 18:45:04 |
Last modified on | 2013-03-22 18:45:04 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 9 |
Author | pahio (2872) |
Entry type | Derivation |
Classification | msc 35K05 |
Classification | msc 35Q99 |
Related topic | DerivationOfWaveEquation |