 Solution

ID: 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

R_{c}

in unbounded two-dimensional incompressible potential flow, a velocity potential is

φ = u (r + R_{c}^{2} / r) cos θ

. It gives

v_{r} = u (1 - R_{c}^{2} / r^{2}) cos θ, v_{θ} = - u (1 + R_{c}^{2} / r^{2}) sin θ .

v_{r} = 0

. The points nearest the banks are the sides perpendicular to the stream,

θ = π /2, 3 π /2

, hence the speed is $2 u$ 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.

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