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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 39D / iv / Solution

For a Newtonian fluid,

σ_{ij} = - p δ_{ij} + μ (\frac{\partial u _{i}}{\partial x _{j}} + \frac{\partial u _{j}}{\partial x _{i}}) .

On

r = a

, with

n = x / a

,

A^{'} = - \frac{3}{2 a}, B = 0, B^{'} = \frac{3}{2 a ^{3}}, p = \frac{3 μ}{2 a} (U \cdot n) .

The surface velocity gradient is therefore

\frac{\partial u _{j}}{\partial x _{i}} = \frac{3}{2 a} n_{i} ((U \cdot n) n_{j} - U_{j}) .

Contracting the stress with

n_{j}

, the normal part of the viscous stress cancels the pressure contribution, leaving the uniform traction

(σ \cdot n)_{i} = - \frac{3 μ}{2 a} U_{i} .

Solved by gpt-5.6-sol high.

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