b( b- 



z = z' + — + ^ [72e] 



Then, substituting in (72d), using the binomial theorem and also the series for In (1 + x) after 

 writing In s = In 2' + In [1 + {z - z')/ z'^, 



+0. + c In z' + a^z'. [72f] 



Thus the equivalent dipole at infinity is in general changed, in correspondence with the 

 change in the shape and size of the cylinder. The circulation and source strength remain un- 

 changed. 



73. LIFT ON A CYLINDER; THE KUTTA-JOUKOWSKI THEOREM 



In Section 70 it was shown that a circular cylinder moving through fluid otherwise at 

 rest experiences no drag or force opposing its motion, but, if circulation is present, there is 

 a transverse force or lift of magnitude 



L = pTV [73a] 



per unit of length of the cylinder. Here p is the density of the fluid, V is the velocity of 

 translation of the cylinder perpendicularly to its length, and F is the circulation in the fluid 

 around any closed path encircling it once. The fluid is assumed, as usual, to be incompress- 

 ible and devoid of viscosity. The direction of the lift can be found by rotating the direction 

 of motion of the cylinder through 90 deg in the direction of rotation suggested by F, as illus- 

 trated in Figure 110. 



Figure 110 — Illustrating direction of the lift on a cylinder. 



It was shown by Kutta and independently by Joukowski that the same statements are 

 true for a cylinder of any form. This may be shown by considering the changes in the' momen- 

 tum of the fluid as the cylinder passes. 



Let a frame of reference be used relative to which the cylinder is at rest while the 

 fluid at infinity is moving with velocity -V. Consider the mass of fluid that lies, at any time 

 <j, between two planes drawn perpendicular to the axis of the cylinder and unit distance 

 apart, and also between two other planes PP,, P'P/, drawn perpendicular to the direction of 



162 



