http://www.sklogwiki.org/SklogWiki/index.php?title=Continuity_equation&feed=atom&action=historyContinuity equation - Revision history2024-03-29T12:56:45ZRevision history for this page on the wikiMediaWiki 1.41.0http://www.sklogwiki.org/SklogWiki/index.php?title=Continuity_equation&diff=9565&oldid=prevDduque: Some hydrodynamics ...2010-02-05T14:32:53Z<p>Some hydrodynamics ...</p>
<p><b>New page</b></p><div>{{Stub-general}}<br />
The '''continuity equation''' expresses the conservation of mass. It is a direct consequence of [[Gauss theorem]].<br />
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If the mass enclosed in a region <math>\Omega</math> is <math>M</math>, by definition of mass density <math>\rho</math>:<br />
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:<math>M=\int_\Omega \rho dV .</math><br />
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The net loss of matter in this region must be caused by an outward flow <math>\rho \vec{v}</math> across its boundary:<br />
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:<math>\frac{\partial M}{\partial t}= - \int_{\partial\Omega} \rho \vec{v} \cdot d\vec{S} .</math><br />
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According to [[Gauss theorem]],<br />
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:<math>\int_{\partial\Omega} \rho \vec{v} \cdot d\vec{S} = \int_\Omega \nabla( \rho \vec{v} ) dV .</math><br />
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Since the region is a general one, and it does not change with time, the resulting equation is<br />
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:<math> \frac{\partial \rho}{\partial t} + \nabla (\rho \vec{v}) =0 .</math><br />
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As a direct consequence an incompressible fluid, with constant <math>\rho</math>, implies a [[solenoidal]] velocity field: <br />
<math> \nabla \vec{v} =0 </math>.</div>Dduque