An Unsteady Cavity Flow 



To obtain F explicitly we must first obtain F| and Fp. In 

 this paper we are not going to obtain the exact forms of F, and Fp, 

 even though they can be expressed in terms of the solutions of the 

 Mathieu equation given in (57) . We shall, instead, obtain the asymp- 

 totic representations for F, and Fg as w— oo. We should mention 

 here that Langer [ 1934] has developed asymptotic representations 

 for the solutions of the Mathieu equation with at least one parameter 

 large; however, instead of modifying his asymptotic representations 

 to cover the equation shown in (53) with j another imaginary unit, 

 we shall derive the asymptotic representations for F. and F^ 

 below. 



Let us denote 



X(U = i^-^, (60) 



then, the homogeneous equation of (54) can be written as 



dV 



^=jKa)x(UF. (61) 



Due to the symmetric properties of our problem we need only con- 

 siderhcilf of the flow region, say the region bounded by ICAI, which 

 in the C,-plane is a quarter circle as shown in Fig. 2. In this region 

 X has a simple zero at ^ = 1 and a pole of order 3 at t, = 0, If we 

 make a typical change of the independent variable to §, defined by 

 (Jeffreys [1962]) 



|e'''=('^ X'^a)d;, (62) 



I 



and put 



-1/2 



F=(^) U, (63) 



the differential Eq. (61) becomes 



= [jKcae+ r(e)]U, (64) 



2 

 d U 



where 



201 



