{a,b) 



Figure 51 — Symbols for source or 

 vortex at (a,b). 



Figure 52 — Streamlines for a superposed 

 line source and vortex: 

 M) = - (Aj - iA2) In z . 



approximate those due to the vortex alone and consist of closed loops surrounding P; as P is 

 approached, these loops approximate circles centered at P. 



To locate the source or vortex at (a,b) instead of at the origin, it is only necessary to 

 replace in the formulas 2 by z-a-ih, hence x by x~a and yhyy-b, and to write 



T = [(x-a)^ + {y-h)'i]^\ e = tan-1 \iy-b)/{x-a)\ 

 Thus Q is measured from a line drawn in the direction of positive x, as illustrated in Figure 51. 



Combined Source and Vortex 



A line source and vortex may be imagined to coexist on the same line. The combined 

 potential and stream function and the resultant velocity may be written 



= - 4j Inr - ^2 (9, xJi = - A^d + A^lnr 



[40n,o] 

 [40p,q,r] 



The corresponding complex potential is w = - (.4, -142)ln 2. The streamlines are equiangular 

 spirals defined by = {A^/A,) Inr + constant, as illustrated in Figure 52. The equipotential 

 curves are a similar set of spirals turning in the opposite direction. 



Rigid walls might be inserted along any one of the spirals on which i// is constant. If 

 walls are inserted along n of them, chosen to be equally spaced about the axis, a first step is 

 taken toward the idealization of a radial centrifugal pump; see Section 97. 



84 



