VLYMEN: SWIMMING ENERGETICS OF THE LARVAL ANCHOVY 



Thus, given the time of excursion as t^ we get, 



Wh = 1/ 



2/ (pChAV.^ 



•/o 



dV. 



+ 2{m + M)-^ V,)dt> 



(3) 



The element of work performed by the body 

 contributed by viscous-reactive forces can be ap- 

 proximated using experimental values for the 

 normal and tangential drag coefficient of a 

 smooth plate. The choices made are 



^ V — 3: — 



20.37 



'iV 



Re Re 



(Hoerner, 1965) 



(4) 



VWe VRe 



(Schlicting, 1960) (5) 



where Re is the appropriate Reynolds number re- 

 spectively. Although these are primarily low 

 Reynolds number approximations they are 

 within 107f at Re = 30. ThusFy, the normal force 

 on a plate of frontal area A is given by 



F„= 112 O C^;V' A. 



N 



P ^NV fsl 



Since for any position along the fish body 



Re = 



2Y^h{s) 



V the kinematic viscosity, we get using C^ from 

 Equation (4) the normal force dF^^ on an element 

 of body ribbon as. 



In a similar manner F^^. , the tangential force on a 

 plate of total wetted surface area A, is 



F^ = V2pCjV^ A. 



Since the formula for Cj above uses Re = — 7- 

 where / is distance measured along the body, we 

 get the tangential force dFj on an element of 

 body ribbon as 



dFr, 



2pk, 



\rvZs 



Vj hfs) ds 



= 2pkr\^v1.'^^ds. 



(7) 



Multiplying each element of force above by the 

 element of distance in the direction of that force 

 and summing yields, 



dW 



V.R. 



^k ^, V V^ dsdt 



■N 



N 



+ 2Pk^^/^Vp-^ his) dsdt 



where dW^f^ is the element of viscous reactive 

 work performed by an element of body ribbon. 

 Using the Equations (1') and (2') for Vj^ and Vt 

 and integrating over the excursion time t^ and 

 projected body length excluding the head we 

 get, 



dsdt 



^[-(^J" 



dsdt 



where /ff is thejc-projected length ofthe inert head. 

 Eliminating ds by the relation, 



yields finally, 



/2 



dx 



b^mT 



As will be noted s is present in the second of the 



887 



