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Fluid Mechanics: Fluid Statics
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| Fluids at Rest |
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| A barotropic, compressible fluid at rest is governed by the statics equation, |
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| where z is the height above an arbitrary
datum, and g is the gravity acceleration constant (9.81
m/s2; 32.2 ft/s2). This equation describes the
pressure profile of the atmosphere, for example.
For an incompressible fluid, the statics equation simplifies to, |
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| This equation describes the pressure profile in
a body of water, or in a manometer.
If the fluid is compressible but barotropic,
then the density and the pressure can be integrated into the "pressure
per density" function |
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| Note that the equation at the top of the page can still be applied though, as it makes no assumption on the fluid's equation of state. |
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| Derivation from Navier-Stokes |
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| The Navier-Stokes equation for a fluid at rest reduce to, |
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| Rearranging, and assuming that the body force b is due to gravity only, we can integrate over space to remove any vector derivatives, |
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| For the barotropic fluid case, the derivation can be repeated in a fashion similar to that of Bernoulli, |
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, giving the following alternate form for
the compressible fluid statics equation, 
