Figure 158 — Two circular cylinders in a stream. See Section 94. 



The first term represents the flow past each cylinder as if the other were absent; it is con- 

 structed out of Equation [69j], in which, for the term representing the contribution of the 

 second cylinder, z is replaced by 2 + c? in order to displace its axis to (-c?, 0). 

 The forces on the cylinders may be found from the Blasius theorem. Here 



dz 



= fye-'« - Ve' 



{2 + d)' 



2-n 



2+ d 



By proceeding as in Section 42, it is found from Equation [74g] that the force on A has 

 components 



Aj =- pY^V sin a + pi 



sin a + + ^npV 



d^ lud 



7 «'&' 



^ cos 2o( , [94b] 



Y y= pY ^V cos a + pU cos a - inpW 



a"- b' 



sin 2 a . 



[94c] 



The force A',, J', °" ^ ^^^ ^^ found by substituting in these expressions -c? for o? and also 

 interchanging a? with b^ and Fj with F2- All terms containing d thus merely change sign. 



The first terms of X and Y represent the usual lift, and the next terms a possible 

 modification of the lift due to the presence of the other cylinder. The term in F^ F2 represents 

 a first approximation to the circulatory interaction, as found for a simpler case in Equation 

 [42c'] of Section 42. It indicates that like circulations cause repulsion, unlike, attraction. 

 The terms not containing a? or b^ have the same values for two slender cylinders of any 

 cross-sectional shapes. 



233 



