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derivation of heat equation
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(Derivation)
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Let us consider the heat conduction in a homogeneous matter with density $\varrho$ and specific heat capacity $c$ . Denote by $u(x,\,y,\,z,\,t)$ the temperature in the point $(x,\,y,\,z)$ at the time $t$ . Let $a$ be a simple closed surface in the matter and $v$ the spatial region restricted by it.
When the growth of the temperature of a volume element $dv$ in the time $dt$ is $du$ , the element releases the amount $$-du\;c\,\varrho\,dv \;=\; -u'_t\,dt\,c\,\varrho\,dv$$ of heat, which is the heat flux through the surface of $dv$ . Thus if there are no sources and sinks of
heat in $v$ , the heat flux through the surface $a$ in $dt$ is
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(1) |
On the other hand, the flux through $da$ in the time $dt$ must be proportional to $a$ , to $dt$ and to the derivative of the temperature in the direction of the normal line of the surface element $da$ , i.e. the flux is $$-k\,\nabla{u}\cdot d\vec{a}\;dt,$$ where $k$ is a positive constant (because the heat flows always from higher temperature to lower one). Consequently, the heat flux through the whole surface $a$ is $$-dt\oint_ak\nabla{u}\cdot d\vec{a},$$ which
is, by the Gauss's theorem, same as
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(2) |
Equating the expressions (1) and (2) and dividing by $dt$ , one obtains $$\int_vk\,\nabla^2u\,dv \;=\; \int_vc\,\varrho u'_t\,dv.$$ Since this equation is valid for any region $v$ in the matter, we infer that $$k\,\nabla^2u \;=\; c\,\varrho u'_t.$$ Denoting $\displaystyle\frac{k}{c\varrho} = \alpha^2$ , we can write this equation as
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(3) |
This is the differential equation of heat conduction, first derived by Fourier.
- 1
- K. V¨AISÄLÄ: Matematiikka IV. Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
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Cross-references: differential equation, valid, equation, expressions, Gauss's theorem, positive, normal line, derivative, sinks, sources, flux, element, volume element, growth, region, surface, point, capacity
There is 1 reference to this entry.
This is version 6 of derivation of heat equation, born on 2009-01-20, modified 2009-02-04.
Object id is 11528, canonical name is DerivationOfHeatEquation.
Accessed 836 times total.
Classification:
| AMS MSC: | 35K05 (Partial differential equations :: Parabolic equations and systems :: Heat equation) | | | 35Q99 (Partial differential equations :: Equations of mathematical physics and other areas of application :: Miscellaneous) |
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Pending Errata and Addenda
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