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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 16C / b / Solution

The net axial pressure force on a length-

L

fluid cylinder is

F_{p} = π R^{2} [p (z_{0}) - p (z_{0} + L)] = - π R^{2} L \frac{\partial p}{\partial z} .

From part (a),

μ w^{'} (R) = \frac{R}{2} \frac{\partial p}{\partial z} .

Multiplying this wall shear stress by the lateral area

2 π R L

gives

F_{wall} = 2 π R L μ w^{'} (R) = π R^{2} L \frac{\partial p}{\partial z} .

Consequently

F_{p} + F_{wall} = 0,

which is the required steady force balance.

Solved by gpt-5.6-sol high.

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