An Unsteady Cavity Flow 



MAT) = 27rT(6T2-5) + j^^^^T^-i) - 2(t^-1)'/2 [ 4G ^ (s+^^j^-Z] 



+ t(4-5t2)(t2-1)'/2 In^ +(2-3t2)(t2-1)'/2 ^(t) ^ (50) 



G = Catalan's constant = 0. 915965594, (51) 



and 



-I 

 -0-2)1/2 (0--T) 



MT)=y ^^_IX/,L ^\ do- - Tr< cos'' o-< 0, (52) 



which cannot be expressed in terms of elementary functions. 



It is not difficult to see from (48) that as | t | — 00 , the 

 dominating term is the one containing b|, which is of the order \t\' 

 Since the remaining terms in (48) are obtained from the integral 

 shown in (42), they are, therefore, of the order |t| . This indi- 

 cates that the b| term is the most important term for the flow field 

 near the point at infinity. 



With H given by (48), (22) may be regarded as a linear, 

 second order, ordinary differential equation for f. 



If we transform the independent variable from fg to t, and 

 make the following change of dependent variable 



f=F(;)(^) e-^""°. (53) 



The differential equation (22) is readily reduced to 



^-ijKa.i^F = G(;) , (54) 



where 



fo and Wq as functions of t, are given by (4) and (5) , and H is 

 given by (48) with T as a function of t, given by (35). To help us 

 to understand the properties of the Eq. (54), let us matke the follow- 

 ing change of variables 



199 



