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

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

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.



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by pahio.