Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 39A / b / iii

Table of contents
- Solution iii

Solution

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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