Wu 



depend on the time t, through a fluid which is otherwise at rest. We choose a 

 Cartesian coordinate system (x, y,z) fixed at the body, with the stretched plan 

 form of the body lying in the y = plane and with the free- stream velocity 

 u(t) pointing in the positive x direction. The body motion can be written 

 generally as 



y = h(x,z,t) , (x,z e S) (1) 



where s is the stretched plan form of the body (when h vanishes identically), 

 h is an arbitrary function of x, z, and t, with |Bh/3tl and swimming velocity 

 u assumed to be small (compared with the speed of sound in the fluid) so that 

 the flow may be regarded as incompressible, and with |Bh/Bx| and |Bh/Bz| 

 assumed also small enough to justify the linear theory. 



The Reynolds number r = UP/v, based on the velocity u and body length £ 

 (in the streamwise direction), is taken to be so large that the boundary layer 

 is thin and the inertial effects can be evaluated with the inviscid flow assump- 

 tion. Then the boundary condition requiring the normal component of velocity 

 relative to the solid surface to vanish provides the y component of the flow 

 velocity at the planar surface 



Bh Bh 



v(x,+0,z,t) = V(x,z,t) = — + U , (x.z e S) . (n\ 



Bt Bx ^ ' 



The planar body may admit of sharp leading edges and sharp trailing edges. 

 When the latter kind is present, we shall impose, as usual, at such edges the 

 Kutta condition that the velocity is required to be finite, and hence the pressure 

 continuous at a sharp trailing edge. The following discussion can also be ap- 

 plied to plane flows, say, in the xy plane, in which case the dependence on z 

 simply drops out, and all the quantities will then refer to a unit span in the z 

 direction. 



The thrust (positive when directed in the negative x direction) acting on 

 the body, based on the inviscid linear theory, results from the integration of 

 the pressure component in the forward direction, 



T = Tp + T^ = J (Ap) ^dS + J F^(x,z,t)dz, (3) 



S L. E. 



where (Ap) denotes the pressure difference across the flexible plate, Ap = 

 p(x, -0, z,t) - p(x,+0,z,t), Fg is the singular force per unit arc length along 

 the leading edge due to the leading edge suction, and the last integral is evalu- 

 ated along the leading edge z = b(x). The power required to maintain the mo- 

 tion is equal to the time rate of work done against the reaction of the fluid in 

 the direction of the lift, 



= - (Ap) — dS . 4) 



J 3 1 



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