Fluid Mechanics of Swimming Propulsion 



K,(s) ■ ■'•'" 



G(s) = . .:%...,: ..M...... :...:, (32b) 



Ko(s) + Kj(s) 



and Kg and Kj are modified Bessel functions of the second kind. 



Finally, we note that after the inverse transform the solution of on the 

 body surface is 0*(x,t) = (p{x,0+,t) = -cp{x,0-,t) = -0"(x,t), 



1/2 J / 2^^''^ 



**(x,t) = iu(.,a.(.)(^) ^'^l[^ ^^^df, (U1<1),33) 



T 



9 ri V(x, t) cos n9 ,^^^ 



\„(r) = - I TTi"'^''' (''=^°^^' n=0. 1, •••) (35) 



-1 (l-x2) 



b+ 103 



G(T) = — I e^^G(s) ds . (36) 



977i J 



2-ni ^ 



b — : 



The coefficient a^it) gives the strength of the leading edge singularity, which 

 is the first term on the right-hand side of (33); the integral term is regular 

 wherever \jj is continuous. The pressure difference (Ap) defined previously is, 

 by (16), 



Ap = 2p4>\x,t) , (|x| < 1) . (37) 



The following cases have been developed earlier: 



Simple Harmonic Time Motion; Constant Swimming Speed 



The motion is prescribed by Eq. (7) (with the third coordinate z omitted, 

 and s given by |x| < 1). It has been shown by Wu (1961) that 







= [(^o + ^i)^(^) - ^J • (^ = ^/U) (38a) 



e(cr) = K^(ja)/[Ko(jc7) + Kj(jcr)] = ?(o-) + j § (a) , (38b) 



in which k^ and A.j are still given by (35), now having the time factor exp(ja;t). 

 C (a) is the Theodorsen Function, 5 and g being its real and imaginary parts, 

 and a is the reduced frequency based on half- chord (which is taken to be unity). 



1153 



