Wu 



While these results have put the whole picture in a much improved aspect, a 

 closer examination of the experimental data as shown in Fig. 5 indicates that 

 there are still cases in which the laminar flow was maintained over a consider- 

 ably greater percent point of the porpoise skin than for an equivalent rigid body. 

 A number of hypotheses have been proposed in an effort to explain the observed 

 low drag. One of the likely effects is attributed to a favorable pressure gradient 

 over a well- shaped streamline body, as indicated by van Driest and Blumer 

 (1963) for laminar flows up to R = 10^. Another possibility is by means of 

 boundary- layer control, such as the compliant skin discovered by Kramer (1960). 

 Subsequent theoretical studies of this effect by Betchov (1959), Benjamin (1960), 

 and Landahl (1961) have indicated that the increases in critical Reynolds num- 

 ber obtainable with passive flexible surfaces are too modest to support this ef- 

 fect on the basis of simple stability theory alone. Even though the possibility of 

 activated flexible surfaces have been proposed, the structural complexity of 

 such skin seems to be biologically infeasible. 



A fairly certain explanation for low drag on fish is the effect on the boundary 

 layer produced by the addition of long-chain molecules, as reported by Fabula, 

 Hoyt, and Crawford (1963). The mucous exuded by fish is composed of a similar 

 type of long- chain molecules and has been found by Hoyt (private communication) 

 to bear significantly the same effect. Still another possible explanation, which 

 seems to be really the principal one to this author, is the unsteady flow effects, 

 due to body undulations, on the hydrodynamic stability. 



SELF- PROPULSION IN A PERFECT FLUID 



The previous theories are concerned with the swimming of bodies in fluids 

 of small, but not zero viscosity. Recently, Saffman (1967) raised the interesting 

 question: can a fish swim in a perfect fluid whose viscosity is identically zero 

 (as in a superfluid) ? It has been shown that the classical paradox of D'Alembert 

 for steady flows of a perfect fluid does not apply to the general unsteady flows 

 past a deformable body and that a fish could indeed swim in a perfect fluid. 



The momentum equation for the rectilinear motion of a deformable body in 

 a perfect fluid can be written 



[M + m(t)] W = -MU(t) - Io(t) , (69) 



where M is the mass and m(t) the virtual mass of the body, w(t) is the velocity 

 of the geometric centroid, u(t) the velocity of the center of mass of the body, 

 and Id is the component of the fluid impulse due to the change in body shape 

 relative to an instantaneously identical rigid body moving with velocity w. The 

 quantities m, u, and l^ are functions only of the shape and structure of the 

 deformable body and are independent of w. It is clear that an arbitrary displace- 

 ment can be effected without a permanent or net deformation of the body if m, u, 

 and Id can be made to vary periodically with t in such a way that w has a non- 

 zero time average 



1192 



