Page 144 - Aerodynamics for Engineering Students

Potential flow 127

Fig. 3.19

The streamlines due to a source and sink combination are circles each intersecting
in the source and sink. As the source and sink approach, the points of intersection
also approach until in the limit, when separated by an infinitesimal distance, the
circles are all touching (intersecting) at one point - the doublet. This can be shown as
follows. For the source and sink:
$ = (rn/2n)P from Eqn (3.26)
By constructing the perpendicular of length p from the source to the line joining the
sink and P it can be seen that as the source and sink approach (Fig. 3.19),
p -+ 2csinO and also p + rp
Therefore in the limit

2c sin e = rp or p=- 2c sin 8
r

rn2c .
$=-- sin 8
2n r
and putting p = 2cm = strength of the doublet:
$=- sine (3.36)
2nr
On converting to Cartesian coordinates where

and rearranging gives
(X* + y2> - -Y = 0
P
2~
which, when $ is a constant, is the equation of a circle.
Therefore, lines of constant $ are circles of radius p/(4n$) and centres (0, p/(4n$))
(Fig. 3.20), Le. circles, with centres lying on the Oy axis, passing through the origin as
deduced above.