Wu 



t 



I 



W(t')dt'^Wt as t^oo(W?iO). /-jqn 



Saffman described two different ways in which this can be accomplished, one for 

 heterogeneous and the other for a homogeneous body. 



For a heterogeneous body, we can have u = 0, and it is simplest to sup- 

 pose that the surface deformation has fore and aft symmetry so that I^ = 0. 

 Then w is positive if u(t) and m(t) - m (0) oscillate periodically in phase or 

 with an in-phase component. The physical explanation of the propulsion mecha- 

 nism in this case is clear. There is a hydrodynamic force on a body whenever 

 the body accelerates, which is described by the virtual mass. Now, if the center 

 of mass is moved backwards, the recoil will send the shell forward. K then the 

 resistance or virtual mass is less when the shell goes forward than it is when 

 the reverse recoil is moving the shell backwards, the distance covered during 

 the forward motion exceeds that covered during the backwards motion and 

 there is a net forward displacement during each cycle. Note that there is no 

 continuing transfer of momentum between the body and the fluid; the momentum 

 of the body oscillates about a nonzero mean while the oscillating deformation 

 continues. There is of course a transfer of energy between body and fluid, but 

 this is loss-free. 



II. SWIMMNG MOTION AT SMALL REYNOLDS NUMBERS 



Propulsion of microscopic organisms always corresponds to a small Reyn- 

 olds number and depends almost entirely on the viscous stresses. Although the 

 body motions of some minute biological creatures bear a close resemblance to 

 those of fish, in that they also send waves of lateral displacement from its head 

 down a thin, long tail (or flagellum), the mechanics of the fluid is however 

 greatly different from the case of large R. The effect of viscous stresses in 

 steady flows at small R extends over a wide range, such that the body tends to 

 drag along a very large volume of the surrounding fluid. The vorticity in steady 

 flows is well diffused, leaving practically no wake near the body. Oscillation of 

 the body reduces the amount of fluid moving with the body with increasing fre- 

 quency, as was discussed by Stokes (1851). The mechanics of swimming in this 

 case obeys, nevertheless, the basic principle of action and reaction, so that the 

 total time rate of production of momentum is zero for a self-propelling body 

 at a constant forward speed. 



IMPORTANT FLOW PARAMETERS 



A wide class of unsteady flows of an incompressible, viscous fluid past an 

 oscillating body of arbitrary shape can be adequately described by the linearized 

 Navier-Stokes equations, or Oseen's equations, 



— + U — = Vp+ vV^u + F , ^71 \ 



1194 



