The interaction effect may be stated as follows, in first approximation: 



(1) Acceleration of either cylinder evokes a force on the other whose magnitude equals 

 2np a^b^/r'^ multiplied by the acceleration, and whose direction is inclined to a line drawn 

 through the axes of the cylinders at the same angle as is the acceleration but on the opposite 

 side of that line; compare Figure 161. The force and acceleration have the same direction 

 when they lie along the line of axes, but opposite when they are transverse. For, x^ and y^ 

 are the components of the acceleration of A, and the ratio of the terms containing them in 

 X,D and y, D gives -y./x^ as the tangent of the inclination of the force to the a;-axis, which 

 is now assumed to be the line of axes. 



Figure 161 — Illustrating direction of force due to acceleration of 

 another slender cylinder. See Section 95(a). 



(2) Motion of either cylinder at velocity V relative to the fluid at infinity, in a direction 

 inclined at an angle a to the line of axes drawn toward the other cylinder, evokes a force 

 upon the other whose magnitude is 47rpa^ b^ V^/r^. The direction of this force makes an 

 angle 2 a with the line of axes drawn as described, but it lies on the opposite side of this 

 line from, the direction of the velocity V. This force is shown as Fy in Figure 160. For, 

 when = 0, x^ ~ y^ = F^ cos 2 a , 2 ij y^ = F^ sin 2a. 



In particular, a slender cylinder moving toward or away from another (a = or n-), or 

 stationary but immersed in a stream that is uniform at infinity and directed parallel to the 

 line of axes, repels the other cylinder, whereas, if the motion of the cylinder or of the stream 

 is transverse to the line of axes, the force is attractive. 



If more than two cylinders are present, the forces due to the motion of each are simply 

 additive, in this approximation. 



(B) Slender Cylinder and a Wall 



If 6 = a, if the fluid is at rest at infinity, and if at all times /3 = w - a and ^^' = V, 

 then it is obvious from symmetry that the plane bisecting the line joining the axes of the 

 two cylinders remains a stream surface. A rigid wall may be inserted along this plane, and 

 cylinder A and the fluid on its side of the wall may be discarded. 



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