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Pressure

Pressure gradient


In order to determine the pressure gradient dP/dy as a function of the depth y in a liquid, consider the situation as shown in the figure below.

The adjacent figure shows the surface of a liquid in contact with air. Liquid and air are in a static equilibrium. Apart from motion at microscopic distances, at macroscopic distances everything is in perfect rest.
The dark rectangle represents an imaginary box below the surface of the liquid with a horizontal upper surface of area S and an equally large horizontal bottom surface. The bottom surface of the imaginary box is at height y in some system of coordinates, whereas the top surface of the imaginary box is at height y+Δy. Hence, the imaginary box is Δy high and encloses a volume V=S×Δy of liquid. The pressure, which is due to the movement of the molecules, at the bottom of the box is given by P(y), at the top by P(y+Δy).


The mass of the amount of liquid inside the imaginary box is given by

M = V ρfluid = S Δy ρfluid


Since, moreover, the liquid is in rest, the weight of the amount of liquid inside the imaginary box Mg must be compensated by the sum of the forces on the bottom SP(y) and on the top -SP(y+Δy) of the box. Hence,

S Δy ρfluid g = Mg = -SP(y+Δy) + SP(y)


After dividing out SΔy, we obtain

{P(y+Δy) - P(y)}/Δy = -ρfluid g


The limit Δy→0 of the above relation gives the pressure gradient dP/dy as a function of the height y in the liquid.

dP/dy = -ρfluid g


For an incompressible liquid, which is a good approximation for most liquids, the density is constant and given by ρliquid. The above differential equation is then readily solved by

P(y) = P(y=0) - ρliquid g y


If we take y=0 for the surface of the liquid where the pressure is given by the air pressure Pair, we end up with

P(y) = Pair - ρliquid g y


Notice here that y is negative below the surface of the liquid and, consequently, the pressure grows as one sinks deeper under the surface of the liquid.
For air, which is compressible, things are slightly more complicated.


fluid pressure