A shallow layer of liquid in a mug has been brought to a steady rigid rotation about…
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow
Fluids and Continuum Physics, Ch Vorticity, circulation and vortex motion
Problem
A shallow layer of liquid in a mug has been brought to a steady rigid rotation about the vertical axis of the mug at angular velocity Ω=3.0rads−1, so that the velocity field is that of solid-body rotation: a fluid particle at distance r from the axis moves in a horizontal circle with speed Ωr. Take a horizontal circular loop of radius r=0.040m centred on the axis, traversed anticlockwise when viewed from above, with z pointing upwards, out of the page.
((a)) Compute the circulation Γ round the loop directly, by adding the velocity component along the loop all the way round it.
((b)) Compute the vertical vorticity ωz of solid-body rotation, and obtain Γ a second way as the vorticity flux ωz times the area enclosed by the loop.