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116 3. Further applications

3.1 A REGULAR PROBLEM
A simple, classical problem in elementary fluid mechanics is that of uniform flow of an
incompressible, inviscid fluid past a circle. (This is taken as a two-dimensional model
for a circular cylinder placed in the uniform flow.) Represented as a complex potential
(w), the solution of this problem can be written as

where is the velocity potential, the stream function, U the constant speed of the
uniform flow (moving parallel to the x-axis) and a is the radius of the circle, centred
at the origin. In terms of complex potentials, this solution is constructed from the
potential for the uniform flow (Uz) and that for a dipole at the origin the
complex variable is Both and satisfy Laplace’s equation in two
dimensions:

and, if we elect to work with the stream function (as is usual), then we have

expressed in plane polar coordinates. We now formulate a variant of this problem:
uniform flow past a slightly distorted circle.
Let the distorted circle be represented by

where will be our small parameter; in terms of the problem is to solve

(where subscripts denote partial derivatives) with

and

The condition (3.4) ensures that there is the prescribed uniform flow at infinity i.e.
as (see (3.1)), and (3.5) states that the surface of the distorted
circle is a streamline (designated of the flow. We see, immediately, that there