derivation of heat equation

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

$$-dt\int_vc\varrho u'_t\,dv.$$ (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

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

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

$$-dt\int_vk\,\nabla\cdot\nabla{u}\,dv \;=\; -dt\int_vk\,\nabla^2u\,dv.$$ (2)

Equating the expressions (1) and (2) and dividing by $dy$ , one obtains

$$\int_vk\,\nabla^2u\,dv \;=\; \int_vc\,\varrho u'_t\,dv.$$

Since this equation is valid for any region $v$ in the matter, we infer that

$$k\,\nabla^2u \;=\; c\,\varrho u'_t.$$

Denoting $$\frac{k}{c\varrho} = \alpha^2$$ , we can write this equation as

$$\alpha^2\nabla^2u \;=\; \frac{\partial u}{\partial t}.$$ (3)

This is the differential equation of heat conduction, first derived by Fourier.

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• pahio

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