An Unsteady Cavity Flow 



where h is regarded as a given function of s and t. Since along the 

 solid body dfg = q^ ds , which is purely real, (13) may be written as 



s 



r 9h 

 Im f , = - \ ^ ds - qgh, (14) 



where we have set Im f| to zero at s = 0, the stagnation point of the 

 basic flow. For the basic flow considered and from the definition 

 adopted for the intrinsic coordinates (s,n), we note that along CB 



and along CA 



qo= Wq, (s,n) = (x.y), (15) 



Wq, (s,n) = - (x,y). (16) 



If we denote the prescribed motion of AB as y = T||(x,t) instead of 

 n = h(s ,t) , with the aid of (15) and (16) , (14) becomes 



Imf, = >! ^ dx- w^Tl,. (17) 



Let us assume that the prescribed motion of AB is given by 



r||(x,t) = e cos cot, (18) 



where e is a very small constant quantity and oi is the frequency of 

 oscillation. For convenience in the following anailysis , let us intro- 

 duce an imaginary unit j = v-1 which is regarded as different and 

 non-interacting with the imaginary unit i used in defining the com- 

 plex variable z = x + iy. If we agreed that only the real part with 

 respect to j of a quantity is meaningful to us, we may write (18) as 



Ti,(x,t) = eeJ*^^ (19) 



To avoid any confusion in the notation, from now on when we mention 

 the real (or imaginary) part of a function 2f» denoted by Re 2? (or 

 Im ?f) as it has been used so far, we mean the real (or imaginary) 

 part of t5 with respect to i, not with respect to j, even if z5 con- 

 tains j. Only when the final result is obtained shall we take the real 

 part with respect to j as our solution. 



If we assume that the disturbance has already been applied for 

 a long tlnne so that the entire flow Is In harmonic oscillation, we may 

 write the complex velocity potential f (z,t) as 



193 



