? = 



c sin 



4^- 



[61n] 



The volume of fluid that flows per second through unit length of the slot is represented by the 

 increase in i/f from one hyperbolic face to the other. 



Flow through a slot 



If the cylinders are allowed to shrink onto the a;-axis, the flow becomes that through a 

 slot of width 2c in a plane solid sheet extending to infinity on both sides, as illustrated in 

 Figure 94a. The volume passing per second, per unit length of the slot, is then n-. On either 

 face of the solid sheet cos \^=^-\, x--c cosh and, from [61d,e] 



9 = 



c |sinh ^1 



[61o] 



v^^T 



Thus 5' -» oo at the edges of the slot. 



If, for generality, (^ and i/f are replaced in all formulas by (^/k and i/</A;, all velocities 

 and the issuing volume are multiplied by k. 



Circulatory Flow Around an Elliptic Cylinder or Plane Lamina 



For the conjugate flow, with potential (p' = tp and stream function i/f '= -<f>, an ellip- 

 tical cylinder may be inserted along one of the ^' or (f) curves, or a plane lamina of width 2c 

 along the i-axis between - c. The irrotational flow around such a cylinder or lamina is then 

 represented. This is illustrated in Figure 94b, where any one of the curves may represent the 



Figure 94a 



Figure 94b 



Figure 94 — Streamlines for (a) flow through a slot in an infinite plane wall, 

 (b) circulatory flow around a plane lamina. 



139 



