Fluid Mechanics of Swimming Propulsion 



div u = , . , ;!.; .;v - . (72) 



where u is the perturbation flow velocity, u is the free- stream velocity di- 

 rected along the x axis, and F stands for the external force and may include 

 surface force acting on the fluid by the moving body. The conditions necessary 

 for this linearization to be valid have been well understood for the case of recti- 

 linear and steady motions. The corresponding conditions for oscillatory mo- 

 tions can be examined as follows. 



When a body, either rigid or flexible, of a characteristic length £ , under- 

 goes an oscillation with frequency w and amplitude a, the flow motion is charac- 

 terized by the following dimensionless parameters: 



R=UCA', s = (-) = e2^/v, X = a/e . (73) 



The rectilinear Reynolds number R measures the relative importance of the 

 translational inertial force and the viscous stresses; the oscillatory Reynolds 

 number S gives the ratio of the body length i to the depth of penetration of the 

 vorticity, s = {v/w) ^^^. The relative magnitudes of these parameters give rise 

 to the following principal regimes of interest: 



(i) R << 1, s « 1, X arbitrary. This is the case of low-frequency 

 oscillation with the amplitude not necessarily small. Consequently, the 

 flow field varies only slowly with time, and the problem may be treated 

 as quasisteady, such that the terms on the left-hand side of (71) can be 

 neglected. 



(ii) S » 1, k « 1, R arbitrary. This is the case of rapid oscilla- 

 tion with amplitude small compared with the body dimension, and 

 hence the unsteady and viscous effects are of equal importance. The 

 depth of penetration of the vorticity is now small compared with the 

 body length; consequently, there exists an unsteady boundary layer, 

 outside of which the flow is inviscid and irrotational. The nonlinear 

 effect is still unimportant in this case, since the amplitude a is small. 

 The Reynolds number R, however, need not be small. 



(iii) s >> 1, \ = o(l). This case represents rapid oscillations 

 with amplitude comparable to, or larger than the body dimension. The 

 effects of unsteadiness, viscosity, and nonlinearity are now of equal 

 importance, consequently the nonlinear terms of the Navier-Stokes 

 equations must be restored, which will give rise to the phenomenon of 

 nonlinear streaming. Two boundary layers are therefore anticipated 

 in the motion, one due to the unsteady effect and the other due to the 

 nonlinearity. 



SWIMMING OF A WAVING PLATE IN A VISCOUS FLUID 



In order to investigate the mechanism of swimming of microorganisms. 

 Sir Geoffrey Taylor (1951) took as his first model a doubly -infinite sheet, 



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