diameters are as large as the distance between their surfaces, the maximum and minimum 

 diameters of each cylinder differ by less than 4 percent; if their diameters are at most half of 

 the distance between them, the difference is less than 1 percent. 



In the case of two cylinders, the forces on them, which are equal and opposite, are 

 easily found from the Blasius theorem. For the cylinder surrounding z = b, write dw/dz thus: 



du, 

 dz 



1 {z - b){z + 36) 



2 



_(2 - bf 46^ 46^ (3+ 6)^ 



Substitution in Equation [74g] and evaluation of the integral from the residues, as in Section 30, 

 then gives for the force X = n p ^i^ /{2b^). The cylinders repel each other, because of lower 

 velocities between them. For the approximately circular cylinder, 1/^2 may be dropped in 

 comparison with 1/t^, and Equation [92g] then gives for its radius a=rj = C/V2 = \Jy./V. 

 Thus in this case the force is, approximately, X = npV a /(26^). 



(For notation and method; see Section 34; Reference Muller 124.) 



93. TWO EQUAL LINE DIPOLES WITH AXES TRANSVERSE; FLOW PAST 

 ONE OR BETWEEN TWO SIMILAR CYLINDERS. 



u, = Vz + p. 



1 



1 



V, fi and b real constants and 6>0; 



'cos d, 



c6 = Ux + fi 



z - ib z + ib 



cos 6^ 



Ux + fix 



[93a] 



[93b] 



sin 0j sin ^^ 

 ifj = Uy - n [ + — ) = y 



where 



1 1 ib^ 



V -u ~ + — - 



Tj = [a;2 + (y - 6)2] ', r^ = [x^ + {y + bf] \ 



'1 '2 



, [93c] 



[93d, e] 



and the significance of 6^ and 6^ is shown in Figure 154. Hence 



''sin 2 6, sin 2 



u = - U + 11 



V - fl 



, [93f,g] 



q^=U^-2iiU 



cos 2 d. cos 2 ' 



+ M 



^ ^ 2 cos (2 6^-2 6^) 



,2 ,2 



228 



