 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/39a/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Write

f (r) = A + \frac{B}{r ^{3}}, w = Ω \times x, u = f w .

The velocity gradient of the solid-rotation part is antisymmetric, so it drops out of the Newtonian fluid stress tensor. Since

\partial_{i} f = - \frac{3 B}{r ^{5}} x_{i},

and

p = 0

, the stress is

σ_{ij} = - \frac{3 μ B}{r ^{5}} [x_{i} (Ω \times x)_{j} + x_{j} (Ω \times x)_{i}] .

Equivalently, with

n = x / r

σ_{ij} = - \frac{3 μ B}{r ^{3}} [n_{i} (Ω \times n)_{j} + n_{j} (Ω \times n)_{i}] .

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