If t/ = 0, a little reflection shows that the force on the cylinder is directed more or 

 less towards the source or vortex. 



The torque L on the cylinder per unit of its length is given by Equation [74h]. In 

 evaluating this integral around a, the factor z in the integrand can be written as (z-a) + a. 

 It is found that 



Write b. - \b {e'l^^i. Then, for a line source at s = a, 



L = p[A {2nav^^ - r)- 2;rf/|6j| sin (6^ - y)], [75f] 



or for a line vortex, 



L =p[ar^u^^ -27rV\b^\ sin(,6j - y)]. [75g] 



To use these results, u , and v , must be known. 



^ ac ac 



If several sources or vortices are present, the values of X, Y, and L due to all of 

 them are simply added to obtain the total force and torque, as is easily verified. The 

 principle of the superposition of flows will not hold, however, since in each formula the 

 values of u and v are influenced indirectly by all sources or vortices. The sources 

 or vortices may be fictitious, introduced to represent the effect of another cylinder, with 

 or without circulation around it; in this way it may be possible to calculate the interaction 

 of two or more cylinders. 



(See Reference 2, Section 8.63, 9.53, 13.62; also Section 8.83 for an extension to 

 dipoles.) 



76. KINETIC ENERGY IN TRANSLATIONAL MOTION 



By using the same stratagem as in Section 74, some useful formulas can be derived 

 for the kinetic energy of the fluid surrounding a cylinder that is moving in translation per- 

 pendicularly to its generators. Let the velocity of the cylinder be V with the fluid at rest 

 at infinity and with no circulation around the cylinder, and first, for simplicity, let it be 

 moving toward positive x. Then the common normal velocity of the surface of the cylinder 

 and the fluid, at a point where the x direction cosine of the normal to the surface is I, is 

 q^ where q = W, and the kinetic energy of the fluid per unit length of the cylinder is T^ 

 where, from Equation [17d], 



170 



