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242 5. Some worked examples arising from physical problems

and thus (5.108a) gives directly

and then (5.108b) becomes

The appropriate solution of this equation, which satisfies both the matching condition
and (5.106d), is

Higher-order terms, in both the outer and boundary-layer solutions, can be found
altogether routinely (although the calculations are rather tedious).

This completes our few examples in this group; we now turn to one of the areas where
singular perturbation theory has played a very significant rôle.

5.5 FLUID MECHANICS
The study of fluid mechanics is broad and deep and it often has far-reaching conse-
quences. Many of the classical techniques of singular perturbation theory were first
developed in order to tackle particular difficulties that were encountered in this field.
Examples that are available are numerous, and any number could have been selected for
discussion here (and some have already appeared as examples in earlier chapters). We
will content ourselves with just four more very different problems that give a flavour
of what is possible, but these are all fairly classical examples of their type. Many others
can be found in most of the texts already cited earlier. We will discuss: E5.19 Viscous
boundary layer on a flat plate; E5.20 Very viscous flow past a sphere; E5.21 A piston
problem; E5.22 A variable-depth Korteweg-de Vries equation for water waves.

E5.19 Viscous boundary layer on a flat plate
The solution of this problem (about 1905), with Poincaré’s work on celestial mechanics,
together laid the foundations for singular perturbation theory. In this example, we
consider an incompressible, viscous fluid (in y > 0) flowing over a flat plate,
the flow direction at infinity being parallel to the plate. The governing equations are the
Navier-Stokes equation (in the absence of gravity) and the equation of mass conservation: