Variation of pressure with depth in a liquid at rest

 

Before we can tackle this we have to look at a liquid at rest more carefully. Suppose that you could make a bag from thin polythene, fill it with, say, water, and then seal it. It would have a shape that was arbitrary in the sense that it is not a regular shape like a sphere. All that has been done is that a fixed quantity of water has been prevented from mixing when it is released in an expanse of water that can come to rest and, if the polythene has a density equal to that of water, the bag could end up anywhere because wherever it goes it is in equilibrium with the water surrounding it.

 

Figure 2-3 shows the plastic bag filled with water The horizontal line with two shorter lines indicates the  free surface. The water in the bag is in equilibrium under the attraction exerted on it by the Earth and the distributed compressive force exerted on it by the liquid surrounding it. I have tried to represent this distributed force as being made up of lots of small forces acting on the surface each shown by an arrow. These forces must, because there can be no shearing forces, be normal to the parts of the surface on which they act. They must also be, in every respect, equal and opposite to the internal forces in the element and they must combine to produce a force which is equal and opposite to the weight of the element. Furthermore this force must act vertically through the centre of mass of the element. But small forces acting on small areas produce pressures and so this enclosed liquid is subject to pressure over its whole surface but not to a uniform pressure. This enclosure containing the liquid could be of any shape and any size so it is possible to imagine it becoming smaller and smaller until the elemental forces are acting effectively at a point. Then the forces causing the pressure still act in all directions but in the end the pressures tend to become the same in all directions. It is what everyone expects anyway.

 

With these observations and deductions we can find an expression relating pressure and depth in a liquid at rest. Figure 2-4 shows a part of an extensive liquid at rest, with a vertical prism of liquid identified within it. The prism is of square cross-section, (although it could have any cross-section), with one end surface, of area A, in the free surface and the other at a depth h. The other surfaces are vertical. There is a downward vertical force on the top face equal to  where  is the atmospheric pressure.

 

We must now examine the lower face to make certain that it is not subject to any net force tending to make it move. If there were to be a force on it, the force could have vertical and horizontal components. If a horizontal component were to be in existence it could only be the result of a shearing action in the liquid and this we know to be impossible. Most liquids are quite mobile and come to rest quickly with no residual shearing stress. It follows that the horizontal component of any net force on the lower face must be zero. If there were to be a vertical component to this force it would have to be resisted either by shearing forces or by an acceleration impressed on the liquid. As the liquid is at rest we can discount both of these and must conclude that the lower face is in equilibrium under vertical forces.

 

It is easy to see that, in the absence of shearing forces, the downward force is due to gravity acting on the mass of liquid contained in the prism. If we use the concept of density (mass/unit volume), and take the density of a liquid to be independent of pressure, this force is equal to :-

                                               where  is the density of the liquid.

 

Recalling the cup with its free surface, it would be inconsistent with our concept of gravity to suppose that the pressure on this face was other than uniform. If this is accepted the net upward force on the prism is equal to :-

                                      where   is the absolute pressure at depth h in the liquid.

 

Then                              

                                           or

                                                                               

In practice pressures are most frequently measured above and below atmospheric pressure and called gauge pressures, and denoted . Then the gauge pressure  at depth  in a liquid of density  is given by:-

                                                                        

This simple expression opens the way to a whole range of applications that, at one time, depended on trigonometry and integration to find forces on immersed surfaces. The trigonometry is still with us but the integration is no longer a problem with mathematics packages. We need some applications.