Hydrostatic Forces

Considering the prismatic element again, is the pressure on a plane at any angle , the x, y and z directions could be any orientation. The element is so small that it can be considered a point so the derived expression . indicates that pressure at any point is the same in all directions.
(The proof may be extended to include the z axis).

4. Variation Of Pressure Vertically In A Fluid Under Gravity

In the above figure we can see an element of fluid which is a vertical column of constant cross sectional area, A, surrounded by the same fluid of mass density . The pressure at the bottom of the cylinder is at level , and at the top is at level . The fluid is at rest and in equilibrium so all the forces in the vertical direction sum to zero. i.e. we have

Taking upward as positive, in equilibrium we have

Thus in a fluid under gravity, pressure decreases with increase in height .

5. Equality Of Pressure At The Same Level In A Static Fluid

Consider the horizontal cylindrical element of fluid in the figure below, with cross-sectional area A, in a fluid of density , pressure at the left hand end and pressure at the right hand end.

The fluid is at equilibrium so the sum of the forces acting in the x direction is zero.

This result is the same for any continuous fluid. It is still true for two connected tanks which appear not to have any direct connection, for example consider the tank in the figure below.

We have shown above that and from the equation for a vertical pressure change we have

and

so

This shows that the pressures at the two equal levels, P and Q are the same.

6. General Equation For Variation Of Pressure In A Static Fluid

Here we show how the above observations for vertical and horizontal elements of fluids can be generalised for an element of any orientation.

Consider the cylindrical element of fluid in the figure above, inclined at an angle to the vertical, length , cross-sectional area A in a static fluid of mass density . The pressure at the end with height is and at the end of height is.

The forces acting on the element are

There are also forces from the surrounding fluid acting normal to these sides of the element.

For equilibrium of the element the resultant of forces in any direction is zero.

Resolving the forces in the direction along the central axis gives

Or in the differential form

 

If then s is in the x or y directions, (i.e. horizontal),so

Confirming that pressure on any horizontal plane is zero.

If then s is in the z direction (vertical) so

Confirming the result

7. Pressure And Head

In a static fluid of constant density we have the relationship , as shown above. This can be integrated to give

In a liquid with a free surface the pressure at any depth z measured from the free surface so that z = -h (see the figure below)

This gives the pressure

At the surface of fluids we are normally concerned with, the pressure is the atmospheric pressure, . So

As we live constantly under the pressure of the atmosphere, and everything else exists under this pressure, it is convenient (and often done) to take atmospheric pressure as the datum. So we quote pressure as above or below atmospheric.

Pressure quoted in this way is known as gauge pressure i.e.

Gauge pressure is

The lower limit of any pressure is zero - that is the pressure in a perfect vacuum. Pressure measured above this datum is known as absolute pressure i.e.

Absolute pressure is

As g is (approximately) constant, the gauge pressure can be given by stating the vertical height of any fluid of density which is equal to this pressure.

This vertical height is known as head of fluid.

Example:

We can quote a pressure of in terms of the height of a column of water of density, . Using ,

And in terms of Mercury with density, .