All-101 99 



The solution may now bo written as the sum of two parts- 

 '<\>^ given by (^-)(the "interior solution"), iK bein^ sensibly 

 large only near the boundary and negligibly small in the in- 

 terior, (the "boundary layer contribution"). V/e must now try 

 to determine the boundary layer contribution. 



The nature of the total solution itself is the guiding 

 factor in the investigation, V/e have supposed that near the 

 boundaries x = 0,r, i]-!^ has large derivatives with respect to x 

 while t|)j_ is everyv^here smooth and of order unity. Thus, if we 

 write our solution in two parts, i.e., ^. + ip, , the differential 

 equation can be written in the form 



eAA\ljj_ + ^^^ '^13 - "^ix - "^bx "^ ^'^'^^ ^^"^ T:)sin nsy. 



Now the term eAAif) . is of order e , the terms underlined twice 

 arc of order unity and the order of magnitude of the terms 

 underlined once is as yet undetermined. Since the terms in 

 ^■u are to have derivatives with respect to x v/hich are (assumed) 

 large, we have "^-^ » 1. Hence at least one of the terms of 

 eAArjj^ must be as large as ijj^^ in order to balance this term. 

 The equation will then be satisfied approximately if we write 



-lb- = (1+a sin t) sin nsy 

 and 



eAAilj^ - ^bx =^ ° ^^^ 



We must now integrate these equations and then add the two 

 solutions ^j_ and 4) ]3 to form the com.plete solution \|) . 



