An Unsteady Cavity Flow 



(-1,0) 



T -plane 



1.0) 



B 



Fig. 3 A conformal mapping plane of the basic flow 



- TT ^ cos' T < 0. 



(37) 



We note that y, shown in (36) , has simple poles at r = and ± 1 , 

 and therefore, it obviously does not satisfy the Holder condition on 

 AB, where -1 < t < 1. In order to find H, which is regarded as 

 a function of t now, let us continue it analytically into the lower- 

 half T -plane by 



H(T) = - H(T) , 



(38) 



and let us define another analytic function S2(t) by 



a{r) = T{T^- 1)'/2h(t), 



(39) 



where the branch cut for (t - 1) ^"^ has been defined after (35). If 

 we denote the limiting values of Q> as Im t — ± by Qi±, then, 

 from (39), (21) and (24), 



n^ - fi. = 



= - 2t(i - T^y/s 



|Re T I > 1 

 iRe T I < 1 



(40) 



With Qi^ - J2_ given by (40), the function ^(t) may be determined 

 (Muskhelishvili [1946]) 



"<^' = MI,^ 



2J/2 , . 



-q- ) 7(0-) 



do- + > b 

 n = 



"'"}' 



(41) 



where bp are arbitrary constants and N is an arbitrary integer. 

 From (39), H(t) may be written as 



197 



