Page 146 - Aerodynamics for Engineering Students

Potential flow 129

Now
r; = x2 + y2 - 2xc + c2

and

Therefore
m x2+y2-2xc+c2
$I=-ln
47r x2 + y2 + 2xc + c2

and dividing out

4s x2 + y2 + c2 + 2xc
On expanding,

- ...I
Therefore:

$I=- - 4xc - 1 6x2 c2
47r -[ x2 + y2 + c2 + 2xc 2(x2 + y2 + c2 + 2x42

Since c is very small 2 can be neglected. Therefore, ignoring c? and higher powers of c

m 4xc
$I=--
4s x2 + y2 + 2xc
and as c * 0, and 2mc = p (which is the strength of the doublet) a limiting value of $I
is given by

Therefore
b=-- case (3.38)
27rr

3.3.9 Flow around a circular cylinder given by a doublet
in a uniform horizontal flow
The stream function due to this combination is:

+L sine - Uy (3.39)
2sr