where r denotes distance of (x,y,2) from 

 the location of the source, and /I is a 

 constant, positive for an actual source and 

 negative for a sink. The quantity inA 

 represents the volume of fluid emitted by 

 the source per second. The streamlines 

 are radii drawn from the source. 



An axis of symmetry may be drawn 

 through the source in any direction; let it be 

 taken as the axis of polar coordinates 

 T, 6, CO with origin at the source. Let a 

 circle be drawn as before, with the axis of 

 symmetry as its axis, but now consider 

 this circle as the perimeter of a spherical 

 cap C cut out of a sphere centered at the 

 source; see Figure 197 again, where a 



source is now understood to be located at (?, and V = 0. The area of this cap is S = 2rrr^ 

 (1-cos 0), where r and 6 refer to any point on the circle, and the rate of outflow of fluid 

 across it is S = SA/r-^, because of the symmetry. Hence the value of the stream function at 

 P is - SA/t'^ divided by 2u or, after inserting the value of S and dropping the constant term -A, 



Figure 197 — Symbol? for flow symmetric 

 about a line Q Q'. 



t/f = -4 cos 6 



[119e] 



The total range in the values of ip from 6=0to6-nis thus -2A, which equals the 

 volume output from the source per second or inA reversed in sign and divided by 2w. 



If Cartesian axes are introduced and the a;-axis is drawn parallel to the axis of symmetry 

 and toward 6 = 0, and if the source is at (a;^, y^, z^) as in Figure 198, then 



T = [(a;-a!j)2 + (y-y^)^ + (s-2^)2] ' 

 and 1 



i// = A 



[119f] 



For a point dipole, let polar coordinates r, 6, a> be employed for the moment, with the 



origin at the dipole and the axis for 6 lying along the axis of the dipole. Then the potential 



is, as in [12d] 



fi cos 6 

 = [119g] 



300 



