40. LOGARITHMIC FLOW 



w = - A In z 

 Let 4 be real. Then, i{ z = x + iy = re^" 



cf, + i ill = - Aln (re'^) = - A\n r - i AS 



4, ^ - Alnr, i', = - Ad 



u / 2 2\''^^ 9 = tan~^ v/x; 



where r = {x^ +y ) ' 



dw 

 dz 



A A 



u = — cos 9, V = — sin 9, <7 = — 



[40 a] 



[40b,c] 



[40d] 

 [40e,f,g] 



The origin, 2 = or a; = y = 0, is a singular point. 



Line Source 



The equipotential curves defined by <;6 = constant are circles about the origin, each 

 defined by a constant value of r or by 



x^ +y2 = e-2<A/^ , ■■ 



The streamlines, defined by i/; = constant, are radial lines from the origin; see Figure 50. 



This is the flow due to a uniform line 

 source, as described in Section 15. The vol- 

 ume emitted by the source per second, per 

 unit of its length, is 2nrq = 27t A. The veloc- 

 ity becomes infinite as the origin is approach- 

 ed and is undefined at the origin itself. This 

 type of flow is physically impossible in in- 

 compressible, indestructible fluid, but it is 

 useful mathematically in building up by 

 super-position the solutions for more com- 

 plicated problems. 



The stream function i// is many-valued. 

 In going around the origin in the positive 

 direction, increases by 2n and ip decreases 

 by 2n A. Thus the volume of fluid emitted 



Figure 50 - Flow net for a line source ^y the source per second and per unit length, 



or vortex: w ^ - A In z 

 (Copied from Reference 7) represented by 2v A, is equal to the decrease 



82 



