Liquids at rest
So a liquid is “runny” as we have seen above and can only be at rest in a gravitational field when it is wholly supported by solid boundaries having the shape of a bowl or a bottle. If we observe a liquid in, say a tea cup, (figure 2-1) we know from experience that the cup could be moved to any one of many positions and the liquid would always move until eventually it is at rest occupying whatever constitutes the bottom of the cup. Of course it is not really at rest internally as the ordinary molecular motion continues. However, for most purposes of engineering, we would regard the liquid as not moving and being in equilibrium with the forces exerted on it by the cup. Here we are with Newton’s “forces” all of which are the consequence of the gravitational attraction of the Earth. We shall have to think about how these forces are exerted by the cup and how they are distributed.
How can we proceed from this point? One thing that we can recognise is the fact that the free surface appears to be flat and also horizontal. Of course we know from our experience that the cup is just a small version of an ocean and that, when an ocean is viewed from space its surface is seen to be spherical. The tiny surface of the liquid in the cup must really be aligned with a more or less spherical surface of immense size compared with the cup and merely appears to be flat and horizontal. The gravitational field of the Earth is usually modelled as many concentric spherical surfaces and for each surface the strength of the gravitational field is the same all over it.[1] Our free surface is aligned with one of these spherical surfaces. In the way scientists have of describing fields the surface is said to be all at the same potential. We shall have to give more thought to the meaning of that word in the next chapter.
What else can we find out by reasoning from experience? That nice flat horizontal surface is actually a surface of separation between the water and the air above it. What then can we deduce? First, the water must be in equilibrium under the gravitational force exerted on it, the forces exerted on it by the cup, and the force exerted on it by the air. It is the force on the surface of separation that is interesting. It can only be vertical and there is no influence that can make the force anything other than uniformly distributed all over the surface of separation. Now we have not just a force but a distributed force! We say that it produces a uniform pressure and find the intensity of this pressure by dividing the magnitude of the force by the area.
A further observation is that liquids can resist forces that tend to compress them. This is not a surprise in view of the molecular structure of the liquid with its closely packed molecules. In fact liquids mostly behave as if they are incompressible and only liquids under very high pressure contract measurably. However, even though the liquid is incompressible just like a solid, in one way it quite different to a solid. Most solids can resist the distributed gravitational force that acts on all its distributed mass. It follows that a solid can withstand a system of forces impressed on it without a significant change in shape. This is because it is held together by forces acting between its atoms or molecules that are “rigidly” set in some form of lattice. The internal forces in the solid may be of tension, compression or shear. By contrast the liquid must move and go on moving until the only forces on it are compressive. At rest it is free from tensile forces and from shearing forces as the molecular structure cannot sustain a resistance to such forces.
A cup is a very simple container for a liquid because it has a single free surface. What happens to a liquid in a vessel shaped as shown in figure 2-2? It is clear that there are now two free surfaces and that the liquid is continuous below these surfaces. The two separate surfaces an only be aligned with the same surface of uniform potential so that the surfaces have the same level.
