Introduction

Mechanical engineers are involved with fluids (ie liquids and gases) that flow through, over and round solid boundaries as in pipes, weirs and aerofoils. We have seen how the pressure varies through a liquid when it is at rest and now we want to find a way of deciding how a fluid might behave when it moves and, for preference, find a simple way to make calculations about flow to an accuracy that is sufficient for our needs as engineers.

We would not have to do much to get the water in a plastic cup to start to move, just make a hole in the cup and let gravity do the rest. In many cases the flow of liquid in particular is initiated and sustained by gravity but the flow of liquids and gases can be maintained by a pressure exerted on them in some way. We have to find a way to predict the behaviour of flowing fluids and once more the methods that we use have evolved as a result of contributions by others who have gone before us and which can now be seen to be coherent and mostly adequate for our task. The accumulation of methods that has evolved is based on two simple expressions, the continuity equation and the energy equation and we must come to understand both of these because they will appear in several forms throughout this study.

Let me make a start. I have now used the word energy twice without offering an explanation of it. I find it to be a difficult concept to pin down to a few words for use by "the man in the street", indeed I think that it is used in so many ways in the wake of global warming that everyone ought to stop and decide what we do mean when we use the word. The root of our problem is that energy is a word that has a specific meaning in physics and a much earlier meaning in common parlance, eg a person might be described as being energetic. It is the special meaning that energy has been given by physicists that interests us.

Physicists do not define energy as such but give us a test to decide whether something "contains" energy. The test for energy is that it must be possible to conceive of it being stored by the lifting of a weight in a gravitational field. This leads us directly back to the model of the gravitational field of the Earth referred to in Chapter 2. There we said that the gravitational field could be modelled as many concentric spherical shells each having the same potential all over it. Why do we use the word potential? In order to answer this we have to contemplate some practical tests. Suppose that you hold a mass of perhaps 2 kg in your hand. If you hold it still, you have to exert an upward force of 2.g Newtons, ie 19.62 N. You could lift this mass and, if you do, you do work because the force you exert has moved through a distance. Where has this work gone? There is absolutely no evidence in the mass for the work having been done. Your arm may be tired but that is not relevant. The only idea that fits the facts is that work can be done against the gravitational field. If it can be done in this way the next question must be, can we have it back? The answer is yes. You can lower the mass and work is done on you and this work is exactly equal to the work that you did initially. It seems to be inescapable that work can be stored in a gravitational field and that in order to store work in this way a mass must be lifted[1]. It also seems that a mass in a gravitational field has the potential to do work if it is lowered. Hence the idea of each surface of the model of gravity having the same potential and of course the idea that the higher the level the greater the potential.

This test is sometimes easy to perform but sometimes it is not so easy. Consider a litre of petrol. Everyone would say, in common parlance, that it contains energy but showing that it can be stored in the gravitational field is quite impossible. The energy in the petrol is stored in its chemical structure and can only be released by combustion. It appears as heat and we cannot store heat in the gravitational field and we know of no way to convert all of the heat energy released to work[2].. However by dint of other relationships that have been established we can decide how much work would be done if only we could make the conversion

It is now important to distinguish between energy and power. If something contains energy in a form that can be used to do work it can be made to deliver power. Power is another word that has be taken from long established common parlance and given a special meaning. It simply means the rate of doing work, ie the rate at which energy is used to do work.

With this preamble we can look at a fluid on the move but I shall have to build up the ideas slowly from special cases to get to the general.

[1] This is the underlying principle of the hydroelectric power station. The sun operating through the atmosphere lifts water vapour which may fall as rain on high ground. There it can be stored and, under controlled conditions, the energy stored in the water high up in the gravitational field can be extracted by the turbines to drive alternators and produce power.

[2] Politicians, and others who should know better, make silly mistakes because they do not understand this problem.