An Unsteady Cavity Flow 



as ^ — * CD, and if the distance between the boundary lines of D 

 does not tend to zero as | 5 | -* oo in any subdomain of D, then, 

 a uniformly valid expansion of U in terms of Airy functions can 

 be obtained. Olver [ 1957] later extends the result to the case -when 

 \i.^ is a large complex parameter. For our problem, all the above 

 requirements for a uniformly valid expansion in terms of Airy 

 functions are satisfied except that the parameter \y. in our case is 

 jKw, where j is an imaginary unit Independent of the imaginary unit 

 i used in the complex variable ^. Since i and j do not interact 

 and j may be regarded as a real quantity so far as the imaginary 

 unit of i is concerned, we assume that Olver's result is applicable 

 here and write the two solutions of (64) as 



U, = Ai[(jKw)'^e][l + 0(1)] (70) 



and 



U2= Ai[(jKco)'^e"^'"^^e][l +o(^)] as co— 00, (71) 



where Ai(X) is the Airy function with argument X, which may be 

 expressed either as the sum of two converging series or as an 

 Integral given in the following (Jeffreys & Jeffreys [ 1956]) 



2Tri/3 



Ai(x) =-1- \ e 3 ds. (72) 



ooe 



Substituting x =(jKu)) ^ into (72) and manipulating the result, we 

 can show that 



Ai[(jKa))'/'e] = i{(i-ij)Ai[(iKa))'^'e] + (1 +U) Ai[(-iKco)'^^e] } . (73) 



, /.vl/3 . , .J/3 ,„ , ^ , 7r/6 , -iri/6 



where (1) and (-1) will be taken as e and e respec- 



tively. Therefore, from (63) 



r,=( 



jt -1/2 



- 1/2 

 ~i(4|-) {(l-ij)Ai[(lKco)'^^e] +(l+ij)Ai[(-lKa))'^e]}. (74) 



Similarly , 



203 



