Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / Section A / A10 / Solution

For a cylinder of radius $R_c$ in unbounded two-dimensional incompressible potential flow, a velocity potential is $\phi =u(r+R_c^2/r)\cos \theta$. It givesOn the surface $v_r=0$. The points nearest the banks are the sides perpendicular to the stream, $\theta =\pi /2,3\pi /2$, hence the speed is $2u$ at each. This is the inviscid outer-flow speed; a real no-slip boundary layer has zero velocity at the material surface. The small-radius/wide-river assumption neglects bank effects.