Figure 59 — Streamlines between two 



rigid cylindrical surfaces, centered 



at a;,, Xj. See Section 42(C). 



Figure 60 - Representation of a line 



vortex within a cylindrical shell. 



See Section 42(D). 



Here the distance D between the axes of the cylinders equals a;. - 2; , but c is found to be 

 given by [42n] as before, and A = r/2fr. The value of x or Xr^ locates the origin of coordi- 

 nates. In the region between the cylinders 1// and A have opposite signs. 



The formulas represent circulating flow in the space between the two cylinders. A 

 case is illustrated in Figure 59. 



D. Line Vortex Inside a Cylinder 



In the case just described, if the inner cylinder is omitted and tlie formulas are contin- 

 ued down to the point (c,0), the ideal flow is represented around a line vortex inside of a rigid 

 cylindrical shell. The vortex is at (c,0) and there is circulation F = 'in A about any closed curve 

 lying inside the shell and encircling it once in the positive direction. If R is written for the 

 radius of the shell, lA instead of i/f^ for the value of i/f on it, and A for the distance of the 

 vortex from the axis of the shell, which is at (a; ,0), then h = x^-c and from [42u] and [42w], 



R = -c csch (-!io\ h = y^27^ -c,c^ (R^-h^)/2h [42x,y,z] 



which fixes c when R and h are given. The origin lies outside of the shell, at a distance 

 h + c toward negative x from its axis, and inside the sliell lA has the opposite sign to A; see 

 Figure 60. 



If the vortex is assumed to move with the fluid, it revolves about the axis of the cylin- 

 drical shell at velocity A/2c, as in Case B; but here the direction of revolution is the same 

 as that suggested by the circulation around the vortex. If it is located on the axis, the vortex 

 is stationary. 



95 



