Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 329 / 3 / ii / Solution

Apply axial momentum balance to the fluid between the remote sections

x_{1}

and

x_{2}

. The pressure forces on the end discs, the integrated tube-wall shear, and the force exerted by the sphere are the only resultant axial forces. The sphere is force-free, so its contribution vanishes and

a \int_{x_{1}}^{x_{2}} \int_{0}^{2 π} σ_{r x} (a, θ, x) d θ d x = π a^{2} [p (x_{2}) - p (x_{1})] .

This is the wall-shear and pressure-drop balance in a tube.

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