FLUID MECHANICS OF SWIMMING 

 PROPULSION 



T. Yao-tsuWu 

 California Institute of Technology 

 Pasadena, California 



ABSTRACT 



In this paper the fluid mechanics of swimming propulsion of objects of 

 various sizes is discussed for the cases of both large and small values 

 of the Reynolds number. Several problems of current interest will be 

 examined. The content is partly a review of the literature and partly 

 presentation of some new results. 



INTRODUCTION 



Swimming objects propelling themselves in water or in other liquid media 

 span a wide range in their sizes and speeds. Large cetaceans, such as whales 

 and porpoises, have lengths from 6 to 90 ft, and can swim with cruising speeds 

 from 14 to 20 knots. Microscopic organisms as small as turbatrix (vinegar 

 worm) and spermatozoa— ranging from 0.2 to 0.005 cm in length with length- 

 diameter ratio from 20 to 100 — swim with speeds from 0.05 to 0.002 cm/sec. 

 In between these two extremes there are many species of fish of various sizes. 

 If I is some characteristic length of a body moving with velocity u in a liquid 

 of kinematic viscosity v, the Reynolds number R = Ml /v measures the rela- 

 tive magnitude of inertial stress to viscous stress. The value of R is of order 

 10^ for the most rapid cetaceans, 10^ for migrating fishes, 10'* - 10^ for a 

 great variety of fishes, about 10^ for tadpoles, down to about 10"^ for turbatrix 

 and 10" 3 or less for spermatozoa. Thus, the Reynolds number R covers prac- 

 tically the entire range of interest known to hydrodynamicists. Although R may 

 vary greatly from case to case, the swimming motions of these different bio- 

 logical objects have been observed to differ very little from a vibrating motion 

 of the body, in a wave form propagating along the body. In this kind of body 

 motion, the stresses arise from the reaction between the waving surface and 

 the surrounding fluid, and the propulsive thrust is derived from the resultant 

 of these surface forces. 



The hydrodynamics of swimming motion has been recently investigated for 

 both large and small values of the Reynolds number. For the case of large 

 Reynolds number, the swimming propulsion depends primarily on the inertial 

 effect, since the flow outside a thin boundary layer next to the body surface is 

 irrotational. The viscous effect is relevant only in its role of determining the 

 vorticity shed into the wake, and of producing a skin friction at the body sur- 

 face. Although in principle the latter problem can be evaluated separately, it 



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