,2,2 ttB nTr, nrr, 



1^2 = ?• cot— 2 



„2 aU a a 



use series [33e] for cot, substitute for r^ in terms of r^, and determine Cj, c. by equa- 

 ting coefficients of like powers. The result is 



-o[-i(?)^^(?r ] 



Thus, when aB/nV is small, so that nf^/a is small, so that nr^/a is small, r^ and r^ 

 agree to the second order, and S closely approximates a circle. Its radius becomes r as 

 aB/V -» 0. Even if r, = a/4, r. exceeds r„ by less than 2 percent, although both are about 

 10 percent smaller than t . 



The kinetic energy of the fluid, when the cylinder moves at velocity V in translation 

 parallel to the walls while the fluid at infinity is at rest, is easily found from Equation [76a] 

 in Section 76. For such motion the complex potential, obtained by dropping again the term 

 f/2, is w as given by [46a]. Let the path of integration be displaced outward from the contour 

 of the cylinder until it becomes a long rectangle with sides lying along the walls and ends at 

 X = - I. Then on the walls dz = dx and ib = 0; hence the walls contribute nothing to (/') <^wdz. 

 On the ends, dz = i dy and <f, -* ^ B sinh (2nl/a)/^osh (277 l/a) + l] = ± B tanh (nl/a) -► ± S as 

 Z -» cx), so that the ends contribute 



■a/2 



2B I dy = 2aB 



-a/2 



J -a/ 

 Thus the kinetic energy of the fluid, per unit length of the cylinder, is, from [76a], 



T^=lpU (2aB - t/S) = Ip U^ ^Trr^^ - Sj [46p] 



where S is the cross-sectional area of the cylinder. If r^/a is small, the radius of the cylinder 

 can be taken to be, from [46m, o], r = r^ [1 - {nt ^/ a)^ / %\ so that r^^ = r [1 + (;7r/a)2/6], approxi- 

 mately; and S = irT^. Then (see Taylor^^) 



^.=i-'^''l-f(f/ ] 



[46q] 



(3) Flow Through a Grating. When the stream is present, a similar surface 

 S surrounds each of the points (0, 0), (0, - a), (0, i 2a) . . . . On the surface surround- 

 ing (0,a), for example, i// = all, and the equation of the surface can be written 



111 



