PlanetPhysics/Derivation of Heat Equation

Let us consider the [[../Heat/|heat]] [[../Conduction/|conduction]] in a homogeneous matter with density $\varrho$ and specific heat capacity $c$.\, Denote by \,$u(x,y,z,t)$\, the [[../BoltzmannConstant/|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/|volume]] element $dv$ in the time $dt$ is $du$, the element releases the amount $-duc\varrho dv=-u{\text{'}}_{t}dtc\varrho dv$ of heat, which is the heat [[../AbsoluteMagnitude/|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

$\begin{array}{c}-dt\underset{v}{\int }c\varrho u{\text{'}}_{t}dv.\end{array}$

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 $-k\mathrm{\nabla }u\cdot d\overrightarrow{a}dt,$ 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 $-dt{\oint }_{a}k\mathrm{\nabla }u\cdot d\overrightarrow{a},$ which is, by the Gauss's [[../Formula/|theorem]], same as

$\begin{array}{c}-dt\underset{v}{\int }k\mathrm{\nabla }\cdot \mathrm{\nabla }udv=-dt\underset{v}{\int }k{\mathrm{\nabla }}^{2}udv.\end{array}$

Equating the expressions (1) and (2) and dividing by $dy$, one obtains $\underset{v}{\int }k{\mathrm{\nabla }}^{2}udv=\underset{v}{\int }c\varrho u{\text{'}}_{t}dv.$ Since this equation is valid for any region $v$ in the matter, we infer that $k{\mathrm{\nabla }}^{2}u=c\varrho u{\text{'}}_t.$ Denoting\, $\frac{k}{c\varrho }={\alpha }^2$,\, we can write this equation as

$\begin{array}{c}{\alpha }^2{\mathrm{\nabla }}^{2}u=\frac{\partial u}{\partial t}.\end{array}$

This is the [[../DifferentialEquations/|differential equation]] of heat conduction, first derived by Fourier.