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Introduction to the Principles of Fluid Mechanics
Little drops of water
Little grains of sand
Make the mighty ocean
And the pleasant land.
R. L. STEVENSON
A Child’s Garden of Verses
Fluids appear everywhere around us in nature. In this section of the book, we shall discuss some of the basic laws which govern the behavior of fluids, and look at the applications of these laws to various physical systems. We shall see that good understandings of the workings of many different types of physical systems can be derived in this way.
Perhaps the most amazing idea that will be developed is that fluid mechanics is not limited in its applications to discussing things like the flow of fluids in laboratories, or the motion of tides on the earth, but that it can successfully be applied to systems as different as the atomic nucleus on the one hand, and the galaxy on the other. Because in dealing with a fluid, we are in reality dealing with a system which has many particles which interact with each other, and because the main utility of fluid mechanics is the ability to develop a formalism which deals solely with a few macroscopic quantities like pressure, ignoring the details of the particle interactions, the techniques of fluid mechanics have often been found useful in making models of systems with complicated structure where interactions (either not known or very difficult to study) take place between the constituents. Thus, the first successful model of the fission of heavy elements was the liquid drop model of the nucleus, which treats the nucleus as a fluid, and thus replaces the problem of calculating the interactions of all of the protons and neutrons with the much simpler problem of calculating the pressures and surface tensions in a fluid. Of course, this treatment gives only a very rough approximation to reality, but it is nonetheless a very useful way of approaching the problem.
A classical fluid is usually defined as a medium which is infinitely divisible. Our modern knowledge of atomic physics tell us, of course, that real fluids are made up of atoms and molecules, and that if we go to small enough scale, the structure of a fluid will not be continuous. Nevertheless, the classical picture will be approximately correct provided that we do not look at the fluid in too fine a detail. This means, for example, when we introduce “infinitesimal” volume elements of the fluid, we do not mean to imply that the volume really tends to zero, but merely that the volume element is very small compared to the overall dimensions of the fluid, but very large compared to the dimensions of the constituent atoms or molecules. So long as we talk about classical macroscopic fluids, there should be no difficulty in making this sort of approximation. Indeed, what is “infinitesimal” is largely a matter of the kind of problem one is working on. It is not at all unusual for cosmologists to consider “infinitesimal” volume elements whose sides are measured in megaparsecs!
A. THE CONVECTIVE DERIVATION
If we are going to describe the motion of fluids, we will have to know how to write Newton’s second law for an element of the fluid. This law takes the form
where m is the mass of the element. We are led naturally, then, to consider total time derivatives of quantities which describe the fluid elements. While this may seem straightforward, the fact that the fluid element is in motion makes it somewhat more complicated than it would seem at first glance. To see why this is so, let us consider some quantity f associated with a fluid element (for definiteness, we could think of pressure or entropy or velocity). Then, if the element is at a position x at a time f, at a time t + Δt it will be at a new position....
