An Unsteady Cavity Flow 



S(r) =4^ U(^lJ^!I^-^[j(Kco)^M,(r)+(Kco)2MjT) 



irl (^2_ ^)i/2 



2/2 ,d/2 



- EttjKcot + b,T^(T'' - ir'^] . (83) 



In (83) we note that although bj is purely real with respect to I, 

 it may be complex with respect to j; we also note that the term 

 ZirjKcoT is extremely small as compared to the term j(Kto) M (t) as 

 00 — ^ oo , and since both of them are dependent on j, so we may 

 neglect ZttjKooT from (83) and write 



, . VTe [i(T +Vt^TT) _ 1] 3-- , . 



g^"^) -^ i ui U(Kco) M (t) 



+ {Ku>fM^{T) + b,TV^- 1)'^^] . (84) 



The asymptotic form of F can now be expressed as 



- Fji Ut)] C F, [t, ((t)] g((r)e^*^''^'d(r } . (85) 



where <r is the integration variable in the T-plane and F|(t,), 

 ^^{Q, g(T) and W(F,,F2) are given in (74), (75), (84) and (77). 



Substituting all the necessary results obtained above into (20), 

 noting the relations 



(i±ij)^ = 2(1 ±ij) 



(86) 



(l+ij)(i-ij) = 



when we are taking the real part of (20) with respect to j, we obtain 

 the complex velocity potential f| as 



' (Kco)'^ (t-Vt2-1 -~) 



x{r.iA^**'^''"'"^'^ {Ai[(iK.)'^e]Al[(iK.)'/^e-2-/^] - 



205 



