Page 156 - Aerodynamics for Engineering Students

Potential flow 139

Fig. 3.28 Spherical coordinates

Again the relationship between the stream function and the velocity components
must be such as to satisfy the continuity Eqn (3.61); hence
1 ay 1 a+
qR=-- q, ---- (3.62)
R2sincp a(p '- RsmcpdR

3.4.3 Axisymmetric flow from a point source
(or towards a point sink)
The point source and sink are similar in concept to the line source and sink discussed
in Section 3.3. A close physical analogy can be found if one imagines the flow into or
out of a very (strictly infinitely) thin round pipe - as depicted in Fig. 3.29. As
suggested in this figure the streamlines would be purely radial in direction.
Let us suppose that the flow rate out of the point source is given by Q. Q is usually
referred to as the strength of the point source. Now since the flow is purely radial
away from the source the total flow rate across the surface of any sphere having its
centre at the source will also be Q. (Note that this sphere is purely notional and does
not represent a solid body or in any way hinder the flow.) Thus the radial velocity
component at any radius R is related to Q as follows

Thin pipe

Fig. 3.29