Figure 224 — Velocity due to rotation. 



The component Q^ normal to the boundary, at a point on the boundary where the direction 

 cosines of its normal are I, m, n, is then 



} = W + mV + nW = oj {mx - ly). 



[135a] 



For rotation at velocity w^ about the a;-axis, or w about the y-axis, similarly, 



}n = '^x (.'^y - '"2), Q„ = W {Iz - nx). 



[135b, c] 



The three types of rotation may be superposed in order to obtain the most general type of 

 rotation about an axis through the origin. 



The normal component of the velocity of the fluid, on the other hand, in terms of its 

 cartesian components u, v, w, is 



a = lu + mv + nw. 

 Equating q^ to Q gives as the boundary condition for the most general case 



lu + mv + nw = d)^ {ny - ms) + a, (Iz - nx) + co^ (mx - ly). [135d] 



If the equation of the surface of the boundary is given as 



/ (x, y, z) = 0, 



[135e] 



339 



