A First Course in Fluid Dynamics

Background knowledge

Quite a lot of mathematics is needed for this text, and most of it must be assumed to have been met elsewhere. However, it would be unfair to assume that you have already learned every mathematical technique, so some new methods will be introduced in sufficient detail as they become necessary. If you need to revise any of the topics mentioned below, do it now, before your lack of knowledge interferes with the fluid dynamics that is being expounded.

(a) Vectors

Fluid dynamics is about the motion of fluids, so velocities must come in; that is, the whole subject will be full of work with vectors. You must be confident in your use of the two products

A · B and A × B,

and in the use of components and unit vectors. These components may be those appropriate to cartesian axes or to polar directions; the important material about polar coordinates and directions is summarised below.

(b) Functions of several variables

The velocities in fluid dynamics will in general depend on position and time. e.g.

v(x, y, z, t).

For an example of this you can think of weather maps, which usually show wind velocities at the surface. These velocities are different for different parts of the country and change from day to day. So we shall certainly need to consider functions of several variables; whenever possible we shall reduce from the generality of four variables down to special cases involving only two, but we cannot often use only one variable.

(c) Vector calculus

Since the velocities change in space and time, we shall need vector calculus. For example, the divergence of v

∇·v or div v

and the curl of v

∇ × v or curl v

are most important. This whole area is vital, and so it is summarised below with special results that are needed in fluid dynamics.