Page 140 - Aerodynamics for Engineering Students

Potential flow 123

To find the shape of the streamlines associated with this combination it is neces-
sary to investigate Eqn (3.28). Rearranging:
2cy

or

or

2Ir$
x2 + y2 - 2ccot-y - c2 = 0
m
which is the equation of a circle of radius cdcot2 (27r$/m) + 1, and centre
c cot (21r$/m).
Therefore streamlines for this combination consist of a series of circles with centres
on the Oy axis and intersecting in the source and sink, the flow being from the source
to the sink (Fig. 3.16).
Consider the velocity potential at any point P(r, O)(x, y).*

(3.29)

6 = (x - c)2 + y2 = 2 + y2 + 2 - 2xc

r; = (x+c)~ + y2 = 2 + y2 + 2 +2xc

Fig. 3.16 Streamlines due to a source and sink pair

*Note that here ro is the radius of the equipotential Q = 0 for the isolated source and the isolated sink, but
not for the combination.