45. ALTERNATING VORTICES OR SOURCES; VORTEX MIDWAY 

 BETWEEN WALLS 



w = lE In tanh ££ , a and T real [45a] 



2/7 2a 



n r ffV nx~\ r «osh -^ + cos — 



(j) =--L. tan~^ sin — /sinh — , ,A = In 2 «_ [45b, c] 



^^ La a J 4;r „^„. nx _ ^^^ „y 



cosh -— - cos 



a a 



u= - -^ cosh ^ sin — , v = -^ sinh 22 cos — [45d,e] 



2aG a a 2aG a a 



r ^ nx - ny 



? =-i-, C = sinh2— +sin2— [45f,g] 



2aG a a 



For the interpretation of tan~^, see Section 44. 



(' nx \ 



2a sinh — 1 [45h] 



On the y-axis,t) =0, u = - T/ ( 2a sin — 1 [45i] 



On the lines y = ± a/2, v = Q, w = + r/(2a cosh — j [45j] 



The tan~^ is to be defined so as to vary continuously with x and y; let one of its values on the 

 positive avaxis be zero. 



There is circulation T around the origin, since tan~^ increases by 2n and </> decreases 

 by r as the origin is encircled. Furthermore, if y is increased by a, all functions merely 

 change sign; hence there is circulation - F about (0,a). Finally, everything repeats when y 

 is changed by 2a. Thus the formulas represent the flow due to a row of vortices spaced a 

 apart along the y-axis, all of equal strength but alternating in direction of rotation. Part of 

 the flow net is shown in Figure 70. 



The conjugate flow with (/>'= 0, iA'= - represents a row of alternate sources and 

 sinks of equal strength, spaced a apart along the y-axis. 



Parallel plane walls may be inserted along the lines y = a/2, on which ilj = 0. Stream- 

 lines for this case are shown by the central section of Figure 70. 



The substitution of tan (^ 2/2 a) for tanh (772/20) rotates everything through 90 degrees, 

 with the sources or vortices on the a>axis. (For notation and method; see Section 34; Jaffe'^°, 

 Reference 32.) 



lor 



