91. CYLINDERS OF OTHER FORMS 



Aside from airfoil shapes, cylinders of the following additional cross-sectional shapes 

 have been studied, the cylinder being either stationary in a stream or moving through quiescent 

 fluid. The kinetic energy per unit length of fluid of density p surrounding the cylinder, when 

 it is moving at velocity V with the fluid quiescent at infinity, is denoted below by T^; 

 see also Section 34. 



(a) A hypocycloid, by Agostinelli^°^ and Sestini. °^ 



(b) Rectangular, by Riabouchinsky^^° and J.L. Taylor.^^ For kinetic energy, see 

 Chapter V. 



(c) Equal-sided quadrilateral of side s, moving parallel to a diagonal bisecting an internal 

 angle of d radians, by J.L. Taylor.^^ Here 2 = /[^^/(w;^ - i)]^/(2"-) ^w^ xhe area is 'S^ 



sin 6 and 



r, = — p 

 ^ 2 



2r (3/2) 



r 1 r , 



2n \2n 2, 



s^ V 



[91a] 



where T {x) denotes the gamma function of x. 



(d) Two parabolic arcs meeting at right angles, by J.L. Taylor. ^^ If h is the length of 

 the chord Joining the edges, the area is h /3, and 



(1) For motion parallel to the chord 



6 , , 1/9 1 A^ UK* \ , 1 1 r 



3 = - [Rw^ - l)~^*dw]^, T^=- p — - 1 U^ = — p (0.178) h^V^; [91b, c] 



(2) For motion perpendicular to the chord. 



JW-i 



¥1 



du, 



2 1 A2 (%K* \ 9 1 O r 



^ f^^- p—- [__ _ 1) a^ = — (0.683) A2{y2.[91d,e] 



2 3 \^^3 / 2 



Here 6 is a constant and K is the complete elliptic integral of modulus V 1/2 or 



ff/2 



''I 



1 Sin' 



2 



-Vi 



= 1.8541. 



(e) Four equal semicircles, on the sides of an inscribed square whose diagonal is of 

 length D, by J.L. Taylor.^^ The area is (2 + w) D^/i and 



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