surface and may be replaced by a rigid wall. Then half of the field represents flow along a wall 

 and a slender circular cylinder parallel to it. For an exact solution, one of the treatments of 

 two cylinders may be used, or an earlier direct treatment by Riabouchinsky.^^^ 



A first approximation to the force on the cylinder, which is equal and opposite to that 

 on the wall, is given by Equation [94b, c]. 



Put a = 90 deg, so that the 2;-axis is perpendicular to the wall, Fj = - r2 = F where 

 F is the circulation around the cylinder, and d = 2h where h is the distance from the wall to 

 the axis of the cylinder. Then J', = and, writing V instead of U for the velocity of the 

 stream and X for the force on the cylinder, 



^2\ pr2 1 ^^2 "' 



_ pry 1 + — - npv^ — , [94d] 



where a is the radius of the cylinder. Thus the effect of V alone, or of F alone, is an 

 attraction between the wall and the cylinder. The joint effect of circulation and stream is 

 likewise an attraction when the two resulting components of velocity are in the same 

 direction along the wall. 



95. SLENDER CIRCULAR CYLINDERS MOVING INDEPENDENTLY, 

 OR NEAR A WALL 



When cylinders such as those considered in the last section move independently, the 

 method of images can still be used, since a moving cylinder is equivalent to a dipole located 

 on its axis, but the motion cannot be made steady by a suitable choice of the frame of 

 reference. The forces can then be determined either by integration of the pressure or, more 

 conveniently, by first finding the energy and then using the Lagrange equations. The latter 

 method will be used here, to a first approximation only. 



(A) Two Slender Cylinders 



Let two parallel cylinders A and B have radii, respectively, a and b, and let A be 

 moving at velocity F in a direction inclined at an angle a to a line PQ drawn through the 

 axes of the cylinders in the direction from A toward 6, while B is moving with velocity W 

 inclined at the angle /3 to the same line; see Figure 160. Assume that there is no circulation 

 around either cylinder; and for the present let the fluid be at rest at infinity. 



235 



