Figure 57 — Diagram to illustrate the transformation of the interior of a 

 circle into an infinite strip, by Equation [41n]. 



c - 2 = r^ e~*^i and if, = A (d{ + 6^). 



The ar-axis between - c now transforms into the entire real axis or xlr = 0. The trans- 

 formation can be visualized by imagining the interior of the circle to be drawn out into a strip 

 as the points c and - c recede in opposite directions to infinity. 



(For notation and method; see Section 34; Reference 1, Article 64, 155; Reference 2, 

 Sections 6.50, 8.22, 13.30.) 



42. CIRCULATING FLOW: CYLINDERS, VORTICES, A WALL 



The flow due to a vortex pair, as described in the last section, maybe used to represent 

 a circulating flow around parallel cylinders, 'iere the potential and stream function are <^' 

 = ill, lb' = - (f), but it will be convenient to drop the primes. Then the new (^ and i/? may be 

 derived from the modified transformation 



w = 4) + i tij = lA [In (z -c) - In (2 + c)] 

 whence, in the notation of the last section, 



(A = - A{9. -61,), i/, = - 4 ln-2 



iAcxy 



fs\n d^ sin ^2\ 4-4c; 



/cos 0, cos d^\ x^-y^ 

 V = A *■ ^ = 2Ac r— 



[42 a] 



[42b,c] 

 [42d] 



[42e] 



91 



