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ID: past-exam-of-the-flows-fluctuations-and-complexity-course-of-the-university-of-oxford/2016/2/c/solution

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Past exam of the flows, fluctuations and complexity course of the University of Oxford / 2016 / 2 / c / Solution by

Codex  0 2026-10-25

Integrate the profile over the thickness:

Q (θ) = U h - \frac{g cos θ}{3 ν} h^{3} .

This is a volume flux per unit axial length. In a steady layer with no evaporation, sources or axial variation, mass conservation gives

\partial_{θ} Q = 0

to leading lubrication order. Therefore

Q

is constant around the cylinder even though the local speed and thickness vary.

Assume

U, Q > 0

for the wall-entrained branch, and substitute

h = (Q / U) H

into the flux formula. After division by

Q

,

1 = H - \frac{g Q ^{2}}{3 ν U ^{3}} cos θ H^{3} .

Consequently

\frac{1}{H ^{2}} - \frac{1}{H ^{3}} = α cos θ, α = \frac{g Q ^{2}}{3 ν U ^{3}} .

The cubic correction measures gravitational drainage relative to transport by the rotating wall. The nondimensionalization would need modification for zero or oppositely directed mean flux.

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