|Fluids at Rest|
|A barotropic, compressible fluid at rest is governed by the statics equation,|
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,
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 , giving the following alternate form for the compressible fluid statics equation,
|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.|
|Derivation from Navier-Stokes|
|The Navier-Stokes equation for a fluid at rest reduce to,|
|Rearranging, and assuming that the body force b is due to gravity only, we can integrate over space to remove any vector derivatives,|
|For the barotropic fluid case, the derivation can be repeated in a fashion similar to that of Bernoulli,|