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,
, giving the following alternate form
for the compressible fluid statics equation,
