Equation [76e] for the entrained area S^'then gives 



\'=2n{R) \b(e-^'y + \-&' [76i] 



V 



where iS'is the cross-sectional area of the transformed cylinder. 



(See Reference 74, where a proof not employing complex variables is given by Leathern.) 



AIRFOILS 

 77. THE JOUKOWSKI TRANSFORMATION 



By making a transformation from s to a new variable 2', the flow with circulation around 

 a circular cylinder can be transformed into the flow around a cylinder of a different shape. 

 The equipotential curves and streamlines on the s-plane transform into curves on the s '-plane 

 in association with the same values of (f> and \p. Since the total change in <jS on going around 

 a closed curve thus remains unaltered, the circulation around the transformed cylinder is the 

 same as that around the original circular cylinder. 



It was shown by Joukowski that a first step toward obtaining in this way useful pro- 

 files for airfoils could be taken by using the simple transformation 



3'= 3 + — , [77a] 



2 



where c is a real positive constant, 2'= a;'+ iy' , z = x + iy. 



If c is chosen equal to the radius a of the given cylinder, 3 'becomes simply the complex 

 potential w for the flow past the circular cylinder itself, as obtained in Section 69. Assume, 

 therefore, that c < a. 



The transformation Equation [77a] has a singularity at 3 = 0. Furthermore, since 



dz' c2 1 



— =1 = — (z + c)(z- c), [77b] 



dz „2 2 



the transformation is not conformal at either of the points 3 = ic, or 3'=^ 2c, where 

 dz'/dz = C. In fact, as 3 passes through either of these points along a smooth curve, since 

 dz'/dz changes sign, the motion of 3'reverses, so that the corresponding 3'curve exhibits a 



CVlS"p. 



173 



