Talk:PlanetPhysics/Reynolds Transport Theorem

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Reynolds' transport theorem %%% Primary Category Code: 51.10.+y %%% Filename: ReynoldsTransportTheorem.tex %%% Version: 4 %%% Owner: PhysBrain %%% Author(s): PhysBrain %%% 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 $F(\mathbf{r},t)$ represent the amount of some physical property of a continuous material medium per unit \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html}. The total amount of this property present in a finite region ${\cal V}$ of the material is obtained through the volume integral. \[ \int_{\cal V} F(\mathbf{r},t) \;dV \]

If this property is being transported by the action of the flow of the material with a \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} $\mathbf{u}(\mathbf{r},t)$, then \emph{Reynolds' transport theorem} states that the rate of change of the total amount of $F$ within the material volume is equal to the volume integral of the instantaneous changes of $F$ occuring within the volume, plus the surface integral of the rate at which $F$ is being transported through the surface ${\cal S}$ (bounding ${\cal V}$) to and from the surrounding region. \[ \frac{d}{d t} \int_{\cal V} F(\mathbf{r},t) \;dV = \int_{\cal V} \frac{\partial F}{\partial t} \;dV + \int_{\cal S} F\mathbf{u} \cdot \mathbf{n} \;dS \] Here, $\mathbf{n}$ is a \htmladdnormallink{unit vector}{http://planetphysics.us/encyclopedia/PureState.html} indicating the normal direction of the surface (oriented to point out of the volume).

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