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Home » Gate Study Material » Civil Engineering » Hydrostatic Forces

Hydrostatic Forces

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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).


Pressure at any point is the same in all directions.

 

This is known as Pascal's Law and applies to fluids at rest.

4. Variation Of Pressure Vertically In A Fluid Under Gravity

 

 

Vertical elemental cylinder of fluid

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.

 

 

Horizontal elemental cylinder of fluid

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

 



 

Pressure in the horizontal direction is constant.

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.

 

 

Two tanks of different cross-section connected by a pipe

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.

 

 

A cylindrical element of fluid at an arbitrary 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)

 

 

Fluid head measurement in a tank.

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.


 

Note: If pressure is quoted in head, the density of the fluid must also be given.

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, .

 

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