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[parent] derivation of heat equation (Derivation)

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

$\displaystyle -du\;c\,\varrho\,dv \;=\; -u'_t\,dt\,c\,\varrho\,dv$
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
$\displaystyle -dt\int_vc\varrho u'_t\,dv.$ (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

$\displaystyle -k\,\nabla{u}\cdot d\vec{a}\;dt,$
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

$\displaystyle -dt\oint_ak\nabla{u}\cdot d\vec{a},$
which is, by the Gauss's theorem, same as
$\displaystyle -dt\int_vk\,\nabla\cdot\nabla{u}\,dv \;=\; -dt\int_vk\,\nabla^2u\,dv.$ (2)
Equating the expressions (1) and (2) and dividing by $dy$, one obtains

$\displaystyle \int_vk\,\nabla^2u\,dv \;=\; \int_vc\,\varrho u'_t\,dv.$
Since this equation is valid for any region $v$ in the matter, we infer that

$\displaystyle k\,\nabla^2u \;=\; c\,\varrho u'_t.$
Denoting  $\displaystyle\frac{k}{c\varrho} = \alpha^2$,  we can write this equation as
$\displaystyle \alpha^2\nabla^2u \;=\; \frac{\partial u}{\partial t}.$ (3)
This is the differential equation of heat conduction, first derived by Fourier.



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Cross-references: differential equation, theorem, flux, volume, temperature, conduction, heat

This is version 1 of derivation of heat equation, born on 2009-01-20.
Object id is 417, canonical name is DerivationOfHeatEquation.
Accessed 385 times total.

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Physics Classification: 44. (Heat transfer)
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