Flow Parallel to the Line of Axes. Equation [89p] gives similarly in the limit, with 

 V replaced by V , 



I ''A "^ 



w = - inaV sinh — , [90j] 



for flow at infinity at velocity V toward negative y and hence parallel to the line of axes of 

 the touching cylinders. Then 



naV nax nay ndV nax nay 



(b = cosh sin , i/« = - sinh cos [90k, 1] 



G .2 ,2 G ,2 2 



_ n^ a^ \V\ 

 r^G 



1 / 'i.nax 2nay 

 — (cosh + cos 



2 .2 2 



[90m] 



As the origin is approached along or between the cylinders, cosh {2n a x/r') increases without 

 limit and q-*Q. 



The kinetic energy T. of the fluid near unit length of the double cylinder, when moving 

 at velocity U through fluid that is at rest at infinity, can be found conveniently by substituting 

 an for c/m in Equations [89n] and [89s] and then letting c-»0. This gives, since S = 2na^: 

 for motion perpendicular to the line of axes, 



T^= — pi- ij 2na^ U^ ^ - p (4.580 rra^) (y2. [90n] 



for motion parallel to the line of axes, 



Ti= — P i- l) 2na'^ u^ ^ — p (1.290 vra^) (/2. [90o] 



In the case of motion perpendicular to the line of axes, a wall can be inserted, as 

 before; then half of T^ is the kinetic energy of the fluid near unit length of a cylinder that is 

 sliding along a wall. 



(For notation and method; see Section 34; Reference 2, Section 6.52.) 



221 



