For motion at a given speed we assume that the 

 only component of drag that varies between 

 straight-line and circular swimming is the in- 

 duced drag. This is probably an underestimate, 

 because the friction and form may also increase 

 slightly, as mentioned in the previous section. 

 Neglecting these, however, we find that the drag 

 when turning Dt (the drag D, appearing in Equa- 

 tions (l)-(9), is Z)^) is greater than the straight-line 

 drag at the same speed Ds by 



Dt-D, = D.^ 



D 



IS 



(11) 



where D It and D is are the induced drag in turning 

 and in straight-line swimming, respectively. The 

 drag is usually (Hoerner 1965) written as a func- 

 tion of the fin lift force G 



2G' 



D; = 



eirpb^U' 



(12) 



where b is the fin span and e a fin shape factor. For 

 our purposes, Di can be written as 



A = k£, (13) 



as the quantities collected in K are invariant for a 

 given fish. 



The fin span is variable, as many species of 

 interest can adjust the sweepback angle, changing 

 the span. 



Substituting Equation (13) in Equation (11) and 

 applying the geometrical relations of Figure IB 



^t-D, 



K^ 



U^ 



.bt- 



(14) 



where bt and bs are the pectoral fin span in turn- 

 ing and in straight-line swimming, respectively. 

 Dividing hy Ds, 





= 1 + 







(15) 



However, Dig ^ KiL^/U^bg, so that 



D. D. 



' = 1 + _ii 



D. 



= 1 + 



D„ 





(rJ (t)' 



(16) 



In horizontal swimming L = W, so that 



D. 



V. 



Ojf 



V, 



Ojs 



= 1 + 





+ 1 



and as W = (p/ 



(2) 



, (17) 



Pw)gV, by substituting Equation 



v: 



o^t 



V, 



= 1 



OjS 



+ 



D 



Pf + ^Pu. U 



[((^ 



(18) 



and again, as in Equation (10), the oxygen con- 

 sumption is higher when the fish is swimming in a 

 circular path. The two techniques, analyzed above 

 are available to negatively buoyant fish, which 

 can choose between producing the centripetal 

 force by asymmetric thrust or by banking. Neu- 

 trally buoyant fish would have a more difficult 

 time using the banking method [Equation (18)], as 

 the vertical component L would cause upward mo- 

 tion, or complicated and strenuous compensatory 

 motions. 



Returning to the negatively buoyant species, 

 such as skipjack tuna, Katsuwonus pelamis, it is 

 reasonable to assume that they will choose the less 

 costly method, or a combination of the two 

 techniques. This will be studied quantitatively in 

 the next section, but one can see immediately that 

 the banking method is highly dependent on 

 swimming speed. This indicates that there may be 

 a threshold speed for each species above which the 

 banking method is more costly. 



Next, we estimate the influence of swimming in 

 a curved path on the hydrodynamic minimum 

 swimming speed (Magnuson 1970). At the mini- 

 mum swimming speed, the pectorals are already 

 producing the highest possible lift coefficient Cl 

 (which will be designated as CLmax)- Banking the 

 fins reduces the vertical component of the force 

 (Figure IB), since part of the hydrodynamic force 

 is used to turn the fish. But the vertical force is 

 prescribed by the weight, so that the swimming 

 speed must be increased to obtain the required 



173 



