A First Course in Fluid Dynamics

Basic ideas and equations of state

(a) Functions of state

In our discussions so far we have introduced five functions to describe the motion of a fluid:

So far there are only four equations in sight for these functions, the equation of mass-conservation, and the equation of momentum change (three components). We clearly need something like an energy equation, which must include energy of compression of a gas; and it is well known that compressing a gas heats it (try pumping a bicycle tyre), so that temperature will come in as well, as a related quantity to heat energy. In order to discuss these ideas clearly, we must set up some, but not too much, of the theory of thermodynamics. By the end of the chapter we will have three standard mathematical models to work with, one for liquids and two for gases under reasonable conditions; but we should also have a good idea of when and why these models are adequate – a model that is not understood is a model that will be used in the wrong way.

A new area of physical theory will have new observables, and the mathematical theory will bring in new functions which are not observable and yet which are the best ones for framing the theory. Our early stages in fluid dynamics are helped by the fact that density and velocity are common measurable concepts in other forms of mechanics; and the idea of a stress tensor, which though not directly observable is extremely useful, is not too unlikely a generalisation from a force vector. But thermodynamics brings in ideas which may be quite new to those without much background in physics, so the early stage of thermodynamics needs careful attention, to note where axioms based on experiments are being brought in, and where new definitions are being made.