Fluid Mechanics of Swimming Propulsion 



approximation. The body, when stretched straight, lies in between x = and 

 X = c ; its cross section is small in dimension compared to ? . The free stream 

 has a constant velocity u in the x direction. The motion of the curve passing 

 through the centroids of the cross section of the body remains in the xz plane, 

 and is prescribed as 



z = h(x,t) (0 <x < O (47) 



with the same qualification for h as before. '■ ... y; 



The flow has two components: One is the steady flow around the stretched 

 straight body, which gives no resultant force or moment for symmetric bodies, 

 and the other is the cross flow due to the displacement h(x, t), which has, in 

 the cross-flow plane, the velocity v(x,t) as given by (18). This latter compo- 

 nent alone determines the lift, moment, thrust, and other relevant quantities. 

 The cross-flow momentum is /^A(x)v(x,t ), where aA(x) is the virtual mass 

 corresponding to the transverse unsteady flow and A(x) can be readily deter- 

 mined for given cross sections. The instantaneous lift acting on a section of 

 length dx at x is equal and opposite to the rate of change of momentum in 

 cross flow (or equivalently, it can be obtained by integrating p over the bound- 

 ary of the body cross- section), that is, . ^ ; 



L(x,t) dx = -p (— + U —\ [A(x)V(x,t)] dx . (48) 



The rate of work done by the body in making the displacement h in the direction 

 of lift is therefore 



r hX(x,t) dx = p — r f AVh, - - AV^") dx + pU [Ah.V] 



(49) 



Alternatively, this expression is obtained by replacing (Ap) dS in (4) by L (x,t)dx 

 over < X < P . The kinetic energy imparted to the fluid due to the lateral mo- 

 tion in unit time is 



W = ( (A + u AWl pAyA d^ = I p A I AV2 dx + i pU [AV2] (50) 



{ \Bt Bx/\2 / 2 Bt -{) 2 x=e 



The time average of P and w are clearly 



P = pUA(0 [htV] = pUA(0 [ht +UhJ] , ,-. ^- (51) 



W=-pUA(e)V2 = - pUA(£)(h, + Uhj' . (52) 



The physical significance of these results is clear. P is equal to the average 

 of the product of the lateral velocity h^ and the rate of shedding lateral 



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