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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 39C / d / Solution

Put

G = \frac{ρ g}{4 μ}, A = \frac{V - G R ^{2} ( 1 - α ^{2} )}{l o g α} .

Integrating the equation in part (b) and imposing both no-slip conditions gives

u (r) = G (R^{2} - r^{2}) + A lo g \frac{r}{R} .

For

V = 0

, gravity drives downward annular Poiseuille flow, positive in the interior and zero at both walls. Positive

V

raises the inner-wall endpoint to

V

and strengthens the downward flow; sufficiently negative

V

creates a region of upward flow near the inner cylinder. In each case the graph is the displayed quadratic-plus-logarithmic profile.

Solved by gpt-5.6-sol high.

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