 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/20/a/solution

x = ρ cos ϕ

y = ρ sin ϕ

v = α z ρ ϕ, v_{ρ} = v_{z} = 0.

For each fixed height, particles follow concentric circular streamlines around the vertical axis. At

z > 0

the motion is anticlockwise viewed from above, while at

z < 0

it is clockwise. Within a horizontal plane the angular velocity is

α z

, independent of radius, and the speed is

α ∣ z ∣ ρ

. Both the plane

z = 0

and the vertical axis are stationary. The rotation rate varies with height, so the entire liquid is not one rigidly rotating body.

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