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148 3. Further applications

Mathieu equation (such as Ince, 1956). A lot of detail, with analytical and numerical
results, including many applications, can be found in McLachlan (1964).
The method of strained coordinates is described quite extensively in van Dyke
(1964, 1975), Nayfeh (1973), Hinch (1991) and, with only slightly less emphasis, in
Kevorkian & Cole (1981, 1996).

EXERCISES
Q3.1 Flow past a distorted circle. Find the third term in the asymptotic expan-
sion, for of the problem described by

with and for

Hence write down the asymptotic solution to this order and observe that,
formally at least, there is a breakdown where Deduce that the
solution in the new scaled region is identical (to the appropriate order) to that
obtained for r = O(1), the only adjustment being the order in which the terms
appear in the asymptotic expansion (and so the expansion can be regarded as
regular). Use your results to find an approximation to the velocity components
on the surface of the distorted circle.
Q3.2 Weak shear flow past a circle. Cf. Q3.1; now we consider a flow with constant,
small vorticity past a circle. Let the flow at infinity be
which has the constant vorticity (in the the problem
is therefore to solve

with as and
Seek a solution find the first two terms and,
on the basis of this evidence, show that this constitutes a two-term, uniformly
valid asymptotic expansion. Indeed, show that your two-term expansion is the
exact solution of the problem.
Q3.3 Potential function outside a distorted circle. (This is equivalent to finding the potential
outside a nearly circular, infinite cylinder.) We seek a solution, of the
problem

with