Pressure in compressible fluids

In working out the pressure at a given depth in water we assume that the water is incompressible – that is its density is constant with depth. This is very nearly accurate. However this would not be true of a fluid such as air. In the atmosphere the pressure increases with depth but the density of the air also varies from 1.2 kgm^-3 at sea level to virtually zero at the edge of space.

Let the change in pressure due to a small change of height Dh be DP
Take the density at a height h to be r and the density at the height h+Dh to be r-Dr.
If the temperature T is constant r is directly proportional to P and so
P/P_o = r/r_o where Po and D_o are the pressure and density of air at sea level.
Therefore DP/Dh = - rg = -gr_o[P/P_o]
When this is integrated we have:


[Note: I have had to use r' instead of r_o here]

Substituting for the accepted values of g, P_o and r_o we have:


(N.B – the height h should be given in kilometres here)



The temperature of the atmosphere also decreases with height.
The standard variation being a drop of 6 ^oC for every kilometre above the ground up to 11 km. After that it remains constant at around – 55 ^oC.