Page 143 -

126 3. Further applications

This, and other aspects of compressible flow past aerofoils, will be put to one side in
the current discussion (but some additional ideas are addressed in Q3.10 and 3.11).
Here, we will work with the mathematical model for the flow in which and
are bounded, so that the only non-uniformity arises as
The solution in the far-field is written in terms of the variables and
with Equation
(3.22) now becomes

where, for simplicity, we have written down only the leading terms on the right-hand
side of the equation. We seek a solution

and then satisfies the nonlinear equation

or

and note that The general solution for is

where H is an arbitrary function; solution (3.39) provides, for general H, an im-
plicit representation only for (which has far-reaching consequences, as we shall see
shortly).
We determine H by matching, and this is most easily done by matching and
—but for this we require the expansion of It is left as an exercise to show that,
from (3.35), we obtain

which gives