Charaateristios of Ship Boundary Layers 



Also the condition that ds be normal to VF is 



ds • VF = (5) 



Equations (4) and (5) are the differential equations of the lines of 

 principal curvature. 



In terms of their components, (5) becomes 



F dx + F dy + F dz = (6) 



X y ' z 



and from (4) we obtain 



(F F - F F )(dx)^ + (F F - F F ){dyf + (F F - F F )(dz)^ 

 \ y X2 z xy'^ ' ^ z xy x yz ' * x yz y xz 



+ (FF -FF +FF -FF)dydz 

 * x yy x zz z xz y xy -^ 



+ (FF -FF +FF -FF)dzdx 

 ^ y zz y XX X xy z yz 



+ (F F -FF +FF -FF ) dx dy = 0. (7) 



z XX z yy y yz x xz ^ 



Because of the quadratic nature of (7), the sinnultaneous solution of 



(6) and (7) yields a pair of solutions for (dx, dy , dz) , which can be 

 shown to be orthogonal. Thus, from an initial point P, one can 

 calculate the lines of principal curvature in step-by- step fashion. 



If the equation of the surface is given in the form 



y = f(x,z) (8) 



where x is directed from bow to stem, z is positive upwards, 

 ajnd the plane y = is the vertical plane of symmetry, then (6) and 



(7) can be comiljined into the differential equation of the projection of 

 the lines of principal curvature on the plane of symmetry, 



[pqt- s(l+q^)](||)' +[(l+p2)t- (l+q2)r] ^ +[(l+p2)s - pqr] =0 



(9) 



where 



p = f^, q=f^, r = f,,, s=f„, t = f^, (10) 



and the principal radii of curvature p are given by 



459 



