Figure 61 — Streamlines between a 



circular cylinder and a wall. 



See Section 42 (E). 



E. Cylinder and Plane Wall 



By expanding the outer circle in Case C until it coincides with the y-axis, circulating 

 flow is represented between a circular cylinder and a plane boundary or wall. Writing /? in- 

 stead of R for the radius of the cylinder, H for the distance of its axis from the wall, and ip 

 instead of ih^ for the value of t/i on it, it is found that H = x. and 



/? = - c csch (-A, H = yc2 + ff2, c = Jh^-R^ 



[42a', b', c'] 



which fixes c when R and H are given. The circulation around any closed curve encircling the 

 cylinder positively, once but not crossing the wall is F or 2nA. 



Since i^= on the wall, - il/ represents the volume of fluid that passes between unit 

 length of the cylinder and the wall per second, taken positive in the direction of counterclock- 

 wise motion around the cylinder. See Figure 61. 



F. Line Vortex and Rigid Wall 



If, in Case E, R is allowed to shrink to zero, the flow is represented due to an ideal 

 line vortex parallel to a rigid wall and distant H or c from it. The wall is at a; = 0; the vortex 

 is at (//,0), and the circulation around it is F or inA. The terms in <}> and xp that involve refer- 

 ence to (-//,0) can be regarded as arising from an image vortex at (-H ,0). 



The velocity at the wall, from [42e], in which now 



is _- 



2AH F /; 



y2 + ;/2 ^ y2^;^2 



If the vortex is again assumed to move with the fluid, it moves parallel to the wall with 

 the velocity due to the image vortex, which is A/2H or F/(4ffW), toward negative y if ^ > 0. 

 The axes must be assumed to move with the vortex. 



96 



