Fluid Mechanics of Swimming Propulsion 



where h* is the complex conjugate of h (with respect to j ). This result is 

 readily extended to the integral form when g, h are expressed by integrals 

 over a continuous spectrum. 



Returning to the waving motion, we consider the fundamental form 



y = Re [hj(x,z)eJ(^*"'^'')] , (x,zeS) (12) 



which represents a simple wave propagating along the planar body in the 

 streamwise direction with phase velocity c = co/k and amplitude hj(x,z). Sub- 

 stituting (12) in (3) and (4), and taking the time average, we obtain 



fp . ^ Re I (Apj) ^jhj + ^ ^) ei^- dS , ;, .. (13a) 



P = - Re r (Apj) (jhi) eJ'^'' dS , (13b) 



s 



where (ApJ is the time-independent part of (Ap), Ap = ( ApJ exp(ja)t) as a re- 

 sult of the linearized theory. Since the thrust T3 due to the leading-edge suc- 

 tion is always non-negative, it follows from the inequality (9) that 



P > u T > u Tp (14) 



provided w > 0. Consequently, if Bh/Bx = (the amplitude h^ is independent 

 of x), or when |Bhj/3x| « Ikhjl , then from (13) and (14) we immediately have 



c = wA > u . (15) 



This result shows that not only is a progressive wave desirable, but also its 

 phase velocity must be greater than u (under the stated conditions) in order to 

 achieve a given swimming velocity u . This qualitative feature remains true for 

 a wide class of amplitude function hi(x, z), particularly when additional thrust 

 is required to overcome the viscous drag. 



SWIMMING OF A TWO-DIMENSIONAL WAVING PLATE 



Although the flow around swimming fish is certainly three-dimensional, the 

 theory of two-dimensional swimming motion has received more attention, partly 

 because the analysis is relatively less complicated. We review in the following 

 the main features of swimming in plane flows. 



Here we consider the incompressible plane flow of an inviscid fluid past a 

 flexible plate of zero thickness, spanning from x= -Ito x= 1, and perform- 

 ing a waving motion of the general form 



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