Chavaatevistios of Ship Boundary Layers 



C [W, - V X V,(^)] X n, dS, - V, C (n, . n,) • V, (^) dS, 



Pq'-" ^S V^PO 



= 4TrUi X Up . (48) 



The singularity has been removed from the first integral in (48) 

 because a factor proportional to r-- is contained in 



The second integrand is also singularity-free at P since 

 — /1\ n_»r„„ -1 



PQ ^ PQ PQ 



and 



n^ . V, 





Thus we see that (n^ + n^) • Vpii/r^^) is regular at P. 



A procedure for obtaining a numerical solution of the integral 

 equation (48) consists of replacing the integrals by quadrature for- 

 mulas to obtain sets of linear equations. Expressing y in terms of 

 Cj and e„ as in (42) , for each of n points P the quadrature for- 

 mula yields a linear equation in the unknown values of u and v at 

 n points Q_^ This gives n vector equations or, resolving in the 

 directions e^ and e- at P, 2n scalar equations in 2n unknowns. 

 When Q coincides with P, the integrand is set equal to zero. 



_ Taking the scalar products of the members of (48) by e|p and 



Cgp, we obtain the pair of scalar equations 



y^ rv„ - \) X V, (:^) . e„ dS, . u, £ (n, . T^) . V, (^) dS^ 



= 4TrUT • ijp (49) 



^ (V« - Vp) X V, (^) . i,p dS, . vp j"^ (n, . V . V, (^) dS, 



= 4TrUl • egp . (50) 



In applying these equations, one needs to express \q and n^ in 



467 



