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

The shear stress is

τ_{rz} = μ u^{'} (r)

, where

u^{'} (r) = - \frac{ρ g}{2 μ} r + \frac{A}{r} .

Thus the axial forces exerted by the fluid per unit length on the inner and outer cylinders are

F_{in} = 2 π α R μ u^{'} (α R) = 2 π μ A - π ρ g α^{2} R^{2},

F_{out} = - 2 π R μ u^{'} (R) = π ρ g R^{2} - 2 π μ A .

Their sum is

F_{in} + F_{out} = π ρ g R^{2} (1 - α^{2}),

the weight per unit length of the fluid in the annulus. This is the expected overall vertical force balance.

Solved by gpt-5.6-sol high.

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