Charaateristics of Ship Boundary Layers 



-qT- = curl (v X oj) - V curl curl w (l4) 



a form from which the pressure gradient has been eliminated. The 

 velocity, however, still appears. 



In rectangular coordinates we would have 



curl curl u = V X (V X oi) = VV • w - V^ = - V^ 

 since V • <jj = 0, and (14) could be written in the form 



9^ t 9v , ^ 9v , . 9v 8ti dT] dr] , ,, 9^ ,, ex 



9t=^9;^^'l97^^9^-^8^-^9^-^-9i + ^9^ ^^^^ 



9; > 9w , „ 9w , „ 9w 9C a; 9; , ,^ d\ 



9t=^9^^^9^'"^3^-^9;^-^9^-^9^^^-9^ 



We wish to obtain the equivalent set of equations for a three -dinnen- 

 sional boundary layer, employing a triply orthogonal coordinate 

 system (Qr,p,\), where h da and h dP are elements of arc along the 

 lines of principal curvature on S, and y is distance along the nor- 

 mal, with Y = on S. 



Let e , e., e, denote unit vectors in the directions of in- 

 creasing a, p, Y» Put 



V = "ijU + e^v + e^w, (^ = e^ ^ + e^r) + e^C,, 



From OJ = curl v we have in this system of coordinates, with h = 1, 



P - A-[^ 9(h^v) -| _ J_ _9w 9v V 9h2 

 ^ " h2L9p " 9y J hg ap '"§^ " h^ 9y 



_ J_r9_ ,, > 8w1 _ 9u 1 9w , u 9h^ 

 "^ ~ h, L8y ^ i"^' " 9^] 9y " h, da h^ 9y 



„ 1 r 9 ,, . ^ tu \^ i 9v 1 9u 



^ ^hTrLg^^V) - ap (h,u)j= __ - __ 



, 1 / 8ho ah,\ 



"^hXV^-a^- "'■do') 



V-2 - 9P 



461 



