The transformation Equation [97a] can be arrived at by first transforming the grating 



into a cylinder represented by the unit circle on the ^-plane, using 2 = (a/u) In [{t+l/t)/2], 



and then assuming w(t) to represent circulatory flow around this cylinder. This is done by 

 Prandtl.^23 



A similar infinite grating with plates of finite width and flow past both sides was 

 treated by Kutta,^^ Grammel,^^'* and Engel.^^^ Other cases are considered by Steuding/'^^ 

 Ringleb, Konig, and, with direct reference to turbine theory, by Busemann and 

 Weinel;^'"^ see also Sedov, Reference 141. 



Circular Gratings 



Gratings may also be constructed with circular symmetry, so as to have the property 

 that the grating coincides with itself after a certain integral fraction of a turn about the axis 

 of symmetry. The flow through such gratings has been studied as a basis for applications 

 in the theory of turbines and centrifugal pumps. For this purpose a line source and vortex 

 may be assumed to exist on the axis, also circulation about the exterior of the grating, and 

 perhaps other line vortices disposed with the symmetry of the grating. References: 

 Spannhake, Schulz, '* Florin, ^^'* and other references there given. 



98. VORTICES NEAR CYLINDERS OR WALLS 



In addition to the simple cases treated in Section 42, many other cases have been 

 studied of a line vortex in the presence of a rigid cylinder or wall. Usually the center of 

 interest has lain in the motion of the vortex itself, which is assumed to move with the fluid. 

 The following may be noted: 



Vortex near a lamina: Cisotti,^'*^ Paul,"*^' ''^ and Caldonazzo.^''^ 



Vortex near a slit in an infinite plane: Paul. 



Vortices near a broken wall or in a channel of varied width: Mazet, Miiller, ' * 



Miyadzu,^"*^ and Zeuli.^^° 

 Vortex near a semicircular lamina or near a wall with a semicircular boss: De. 

 Vortex inside a semicircle: Cisotti. 



Vortex inside a rectangular cylinder: Jaffe',^° Miiller, ^"^^ and Seth.^^^ 

 Vortex inside a curvilinear rectangle: Greenhill,^^ and Kondo. 

 Vortex near an elliptic cylinder or inside an elliptic shell: Coates, and 



Rosenhead;^^^ also Caldonazzo,^^ Poggi,^^^ Sanuki and Arakawa,^^® and 



Tomotika.^^^ 

 Vortex near a parabolic cylinder: Masotti. 

 Vortex near a cylinder of certain other shapes: Caldonazzo,^ a cardioid,^ ^ where 



the force is questionable; Morris. ^^ 



244 



