Fluid Mechanics of Swimming Propulsion 



Here, F and G are, respectively, the real and imaginary parts of the function 

 w(ct). F(a) is always > 0. For a « 1, or as oj— '0, the asymptotic value of 

 L is 



1 , / 4V \ 77 ."' 



- log - 7 + - 1 



2 Ua^/ 4 



47T/xV 



(95) 



where 7 = 0.5772; and for a >> 1 



L ~ Trpa^ — + 77apV( 2aJi^)^^^ . /9g\ 



The first term in this expression is due to the apparent mass in potential flow 

 past a cylinder, while the second gives the limit of the dissipative force. 



It can be shown that favorable values of thrust can be achieved if 



c = a; A > U , b = const. (97) 



Under this condition we obtain . ^ 



f = 7T/Li.ecT2F(o-) (c-U)(kb)2 , /;.,:. .,•: (98) 



W = (c-U) T , .. •„ ,•;: . J ■■ (99) 



so that '/'-., . ' .J -.. .■: . . :'■■':'' l 



T7 = U/c , (for c>U) •■ ' >■- •. (100) 



which is simply the ratio of the swimming speed to wave speed. v ;v 



REFERENCES T: v^: 



Benjamin, T.B. (1960) J. Fluid Mech. 9, 513 - 532 



Betchov, R. (1959) Douglas Aircraft Co. Rept. ES-29174 



Fabula, A.G., Hoyt, J.W., and Crawford, H.R. (1963) Am. Phy. Soc. Meeting, 

 Buffalo, N.Y. 



Gray, J. (1948) Nature, London, 161, 199-200 



Gray, J. (1949) Nature, London, 164, 1073 - 1075 



Gray, J., and Hancock, G.J. (1955) J. Exp. Biol. 32, 802 - 814 



Hancock, G.J (1954) Proc. Roy. Soc. A 217, 96 - 121 



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