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Seek separable solutions of the form
Substituting into Laplace's equation we obtain
On dividing by
and rearranging into functions of
and
functions of
alone, this gives
 |
(5.14) |
The form of the solution depends on the sign of the separation
constant
.
. The solution is
In general, we need to sum over all values of
to obtain the most
general solution.
. In this case the solution is
. The solution is somewhat simpler in this case, with
linear solutions for both
and
.
Example 5. .1Consider an irrotational, incompressible flow in a channel of width
. The situation is shown in Figure 5.1
Figure 5.1:
An irrotational, incompressible flow in a channel of width
.
|
The velocity components are
The boundary conditions are that there is no normal flow through the
channel walls. Hence,
where
is a specified function of
. The boundary conditions
in the
-direction imply that the solution should be oscillatory
in
while in the
-direction they imply that the variation is exponential
in
. Therefore, we try a solution of the form,
is taken
as positive,
This satisfies the boundary condition at
and the condition for
large positive
. Hence,
Thus, the general solution is
 |
(5.15) |
and
The constants in the summation are given from the condition on
at
by inverting the Fourier
series as
 |
(5.16) |
Next: Analytical Functions of a
Up: Cartesian Coordinates,
Previous: Cartesian Coordinates,
Prof. Alan Hood
2000-11-06