Dduque: Some hydrodynamics ...

2010-02-05T14:32:53Z

Some hydrodynamics ...

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{{Stub-general}}
The '''continuity equation''' expresses the conservation of mass. It is a direct consequence of [[Gauss theorem]].

If the mass enclosed in a region <math>\Omega</math> is <math>M</math>, by definition of mass density <math>\rho</math>:

:<math>M=\int_\Omega \rho dV .</math>

The net loss of matter in this region must be caused by an outward flow <math>\rho \vec{v}</math> across its boundary:

:<math>\frac{\partial M}{\partial t}= - \int_{\partial\Omega} \rho \vec{v} \cdot d\vec{S} .</math>

According to [[Gauss theorem]],

:<math>\int_{\partial\Omega} \rho \vec{v} \cdot d\vec{S} = \int_\Omega \nabla( \rho \vec{v} ) dV .</math>

Since the region is a general one, and it does not change with time, the resulting equation is

:<math> \frac{\partial \rho}{\partial t} + \nabla (\rho \vec{v}) =0 .</math>

As a direct consequence an incompressible fluid, with constant <math>\rho</math>, implies a [[solenoidal]] velocity field:
<math> \nabla \vec{v} =0 </math>.

Continuity equation - Revision history

Dduque: Some hydrodynamics ...