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solutions of the form

and

the values of the constants, and are obtained from the numerical solution as
From the behaviour as we see that the solution outside
the boundary layer must now match to

which shows that we require a term in the asymptotic expansion valid in
the outer region.
Thus we seek a solution of the set (5.109) in the form

where q represents each of u, v and p. The problem for the second terms in this region
therefore becomes the set

with

and, in terms of the stream function

This is a classical problem in inviscid flow theory, where the exterior flow is distorted
by the presence of a parabolic surface—the effect of the boundary layer which grows
on the plate. The exact solution can be expressed in terms of the complex variable
(and denotes the real part):

which gives

(It can be shown that, in order to match, the boundary-layer solution must now contain
a term not as might have been expected. For more general surfaces
than a flat plate, the next term in the boundary-layer expansion is indeed