Salve sen 



The flow will be treated as steady in a coordinate system moving with the 

 cylinder. A two-dimensional coordinate system will be used, with the y axis 

 vertical, and the x axis a distance b below the undisturbed free surface. The 

 direction of decreasing x coincides with the direction of motion of the cylinder. 

 The fluid will be assumed to be inviscid, incompressible, and without surface 

 tension. It will also be assumed that the flow is irrotational and that the dis- 

 turbances at the free surface are all small. 



The assumption of irrotationality ensures the existence of a velocity poten- 

 tial 4»(x,y) from which the velocity field can be derived, i.e. 



V = grad <t) = $^ i + <t'y j • (1) 



In view of the equation of continuity, the velocity potential <> is a solution of the 

 Laplace equation 



V2<t> = <I) + = . (2) 



XX yy ^ ' 



Another significant consequence of the irrotational character is obtained by 

 integrating the equation of motion, which leads to the Bernoulli equation: 



- I grad <D|' + -+ gy = Ci , (3) 



where p is the pressure, g is the gravitational acceleration, and p is the density. 



There are two boundary conditions to be satisfied on the free surface. The 

 kinematic condition states that any particle which is on the surface remains 

 there. Thus if y = b+ 77(x) is the equation of the free surface and WCx.y) the 

 stream function, then 



'I'[x, b +T7(x)] = constant = bU (4) 



on the free surface. The dynamical condition requires that the pressure above 

 the free surface is constant, here set equal to zero. By Eq. (3), the Bernoulli 

 equation, it therefore follows that 



— I grad $1 + gy = constant on y = b + •'7(x) . (5) 



The nonlinear conditions above are often referred to as the "exact"* free- 

 surface conditions. 



The boundary condition to be satisfied at the wall of the cylinder is clearly 



^=0. (6) 



on 



*In this work, "exact" in quotation marks refers to exact within the potential 

 flow theory. 



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