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/s^{2}; 32.2 ft/s^{2}). 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, |