Fluid Mechanics of Swimming Propulsion 



divided into a "D.C." part f (y) =o(b2) independent of t and x, and a "second- 

 harmonic" part which varies sinusoidally in t and x, and with which we shall 

 not be concerned. 



The solution for the linearized flow is obtained by inspection, with the result 



0^ = -i'hl(l +k) 



1 k 



(' = k2 + 



1 /2 



(80) 



The equation satisfied by the "D.C." second approximation is 



dy^ 



<"l^lx + Vl^ly> • (81) 



where a = k + P , / = F + i' = 2j{(i'). The solution for ^2 which corresponds to 

 a velocity u^, at y = ■ is 



^2 = U^y + i 6Jb2|a|^'K 



L p-ccy _ Ji -yy 



(82) 



The boundary condition to be satisfied on y = is obtained by substitution of 

 the expansion (79) into the boundary conditions (78), resulting in (75) and (76). 



HIGH-FREQUENCY MOTION OF MICROSCOPIC ORGANISMS 



The asymptotic limit of large s has been evaluated by Wu (1966) for the 

 swimming of a slender microscopic body, of length P and a circular cross sec- 

 tion of radius a (a << c )• The body motion is again given by 



y= b(x) ei('^t-k'^) , (0^x<i) . (83) 



We shall assume co to be sufficiently large, and kb(b = max |b(x)l) sufficiently 

 small, that 



a^w > V , kb < 1 . ' ■■''■' • " -■''" ''"'' (84) 



The first condition implies that h = (a), which is small compared to body 

 length P for slender bodies; the second condition means that the flow field does 

 not vary rapidly with respect to x. As the first approximation, we may there- 

 fore regard the flow as consisting of two components: One is the cross flow due 

 to the lateral oscillations, and the other is the longitudinal flow along the mean 

 (stretched straight) position of the body. The transverse component gives rise 

 to lateral force, thrust, etc., and the longitudinal component produces the friction 

 drag. 



We first evaluate the cross flow by using a slender-body approximation. At 

 a station x(0 < x < f) the cross section of the body is taken to be fixed at the 



1197 



