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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-1/36a/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 36A / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Integrating the velocity profiles over their respective half-channels gives volume flux per unit width

Q_{-} = \frac{G a ^{3} ( μ _{+} + 7 μ _{-} )}{12 μ _{-} ( μ _{+} + μ _{-} )}, Q_{+} = \frac{G a ^{3} ( 7 μ _{+} + μ _{-} )}{12 μ _{+} ( μ _{+} + μ _{-} )} .

With

r = μ_{+} / μ_{-}

, the requested ratio is

Q_{+} / Q_{-} = \frac{1 + 7 r}{r ( 7 + r )} .

At

r = 1

, both fluxes equal

G a^{3} / (3 μ_{-})

and the ratio is one. For

r \to \infty

,

Q_{-} \to G a^{3} / (12 μ_{-})

,

Q_{+} \sim 7 G a^{3} / (12 μ_{+})

and the ratio is asymptotic to

7/ r

, tending to zero. The lower fluid flows between the wall and an interface which becomes effectively stationary; the upper fluid carries vanishing flux despite finite limiting stress.

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