All-101 ^6 



The term of highest order derivative in ^ is matched 

 with the remaining largest term in the equation. Hence, we 

 formally match ^'^'^^lum ^i*h V,^. Then k = I/3* and the 

 equation becomes 



^lb£E^ " ^Ib "" acos T sin nsy[(re"^^^-2e"^'^3)cos(-LL3) + 



+ ^■^—^ sin(-^) ] e'^^^ + 0(£^/^). (33) 

 /3 2 



The term V can now be expanded in an asymptotic series 



in e and only the first terras will be kept» Since the inhomo- 



geneous term of (33) contains only exponential and trigonometric 



functions, let us try a solution of the form 



V^, = a cos T sin nsy e " \ V° cos(-2_L£)+V^ sin(-li-3) re 

 " 1^ 2 ^ 2 J (3^) 



where V? and V? are the first terms of asymptotic expansions 

 and are to be determined. 



If V-,, as given by (3*+) be substituted into (33) and 



sin r \/o 



if coefficients of ^os (-5^) be equated, two simultaneous differ- 



ential equations with constant coefficients result. 

 -Kc -i^?^^ ^^X«, -^ V°^ ^3^^?« - - -^/3 



(35) 



3 /3 yo _ 3 \/3 yo _ 3 yo ^ 3 yo , yo _ _ r 

 -2— ^^" 2 'U^ 21^ 2^U^^^UC -^' 



* The fact that k = I/3 indicates that the thickness of the 

 boundary layer is of the same order of magnitude in the zero 

 and first order solution, as was anticipated. 



