The velocity of a fluid particle on S' relative to the body is 



q + r X ell 



Therefore, the normal distance between S'and S", the amount the control surface is deformed 

 relative to the body in time 5 t, is 



\(q + r X 0) • n\ 8t 



Accordingly, in the expression for S771, we can write for dr 



dr = -iq + r X a) • n8td(7 in F^ 



dT={(\+rxa)'n8tda in ^2 



where da is taken on S'. Then, since the portions of S' which bound F, and F, complement 

 each other, 



(f -f \pq(t,+St)dT =-f 



^(L +8tXq- nj 



+St) 

 + (t X la • n)q (t^ + 8t) 8td(j 



Substituting this in Equation [14] and allowing 5i! to approach zero, we have 

 din^d^ r p^dT - I p(\{(\-n)da- \ p{x x ut • n) c^da 



dt ^^)y' J5' Js' 



We can further reduce the volume integral which appears in [15], since 



r p^dr = -p \ V <^dT = Q $n 



d^ 



V -V - s+S' 



by Gauss* theorem. The unit normal can be written 



dx dy , dz 



I — + J + K 



dn dn dn 



By Green's reciprocal theorem 



f ^d^da= [ x^-^da 



S+5' 



[15] 



[16] 



