Talk:PlanetPhysics/Derivation of Heat Equation

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: derivation of heat equation %%% Primary Category Code: 44. %%% Filename: DerivationOfHeatEquation.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Let us consider the \htmladdnormallink{heat}{http://planetphysics.us/encyclopedia/Heat.html} \htmladdnormallink{conduction}{http://planetphysics.us/encyclopedia/Conduction.html} in a homogeneous matter with density $\varrho$ and specific heat capacity $c$.\, Denote by \,$u(x,\,y,\,z,\,t)$\, the \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html} 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 \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html} 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 \htmladdnormallink{flux}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} 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{align} -dt\int_vc\varrho u'_t\,dv. \end{align} 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 \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}, same as \begin{align} -dt\int_vk\,\nabla\cdot\nabla{u}\,dv \;=\; -dt\int_vk\,\nabla^2u\,dv. \end{align} 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\, $\displaystyle\frac{k}{c\varrho} = \alpha^2$,\, we can write this equation as \begin{align} \alpha^2\nabla^2u \;=\; \frac{\partial u}{\partial t}. \end{align} This is the \htmladdnormallink{differential equation}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} of heat conduction, first derived by Fourier.

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