Fluid Mechanics of Swimming Propulsion 



In this inviscid flow the mechanical energy imparted to the fluid per unit time 

 is equal to the time rate of work done by the pressure over the body surface, or 



•/ 



(Ap) V(x,z,t) dS - T^U . • (5) 



These quantities, of course, satisfy the principle of conservation of energy, 

 which asserts that the power input P is equal to the rate of work done by the 

 thrust TU, plus the kinetic energy w lost to the fluid in unit time, that is, 



P = TU + W . (6) 



K the viscous effects are further taken into accovint, then the thrust T must 

 include the viscous drag due to skin friction and the energy loss must contain 

 the viscous dissipation. - ■ 



On physical grounds, it can be inferred that w is non-negative in several 

 cases of broad interest. One of such cases is the periodic body movement with 

 constant forward velocity, 



U = const. , h(x,z, t) :; Re [hj(x,z) eJ'^*] , (x,z e S) (7) 



where j = v^ is the imaginary unit for the periodic time motion, hi(x,z) may 

 generally be complex with respect to j , and Re denotes the real part. After the 

 transient stage is over, it is clear that the kinetic energy imparted to the fluid 

 is largely confined in the wake which contains the trailing vortex sheet and is 

 lengthening at the rate u. Therefore, w cannot be negative. [A mathematical 

 proof of this statement has been given for the case of plane flows, see Eq. (39).] 

 Another example is when the body starts to swim from a state of rest, 



U = U(t) , h = h(x,z,t) , (t>0) (8) 



while u, h, and (u, v,w) are all zero for t < 0, where u, v, and w are the com- 

 ponents of the perturbation velocity. In this case any disturbance generated in 

 the flow must correspond to a gain of kinetic energy of the fluid. 



The following discussion will be based on the presumption w > 0. Under 

 this condition we have, by (6), 



P > TU i f W > . (9) 



P , however, may not be positive definite. When P is negative, energy is trans- 

 ferred out of the fluid (such as by a turbine). In such case T < 0, indicating that 

 there must be an inertial drag acting on the body. Forward swimming is possible 

 only when the thrust T > 0, large enough to overcome the viscous drag; then 

 P > 0, and hence a power is required to maintain the motion. Now, from (3) it 

 is seen that a positive thrust is assured if Ap and Bh/3x are everywhere of the 

 same sign, since the suction force F^ depends only on the instantaneous local 

 condition and is never negative. In view of the inequality (9) and the expression 



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