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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 38C / Solution

The axial equation is

μ r^{- 1} (r u^{'})^{'} = - G

. Regularity at zero, no slip

u_{a} (a) = 0

, continuity of velocity and shear at

c

give

u_{a} (r) = \frac{G ( a ^{2} - r ^{2} )}{4 μ _{a}}, u_{c} (r) = \frac{G ( c ^{2} - r ^{2} )}{4 μ _{c}} + \frac{G ( a ^{2} - c ^{2} )}{4 μ _{a}} .

The annulus is controlled only by its own viscosity and the outer no-slip condition, explaining the absence of

μ_{c}

. For the annular control volume, kinetic-energy rates vanish; pressure work plus interfacial traction work equals

\int_{V} 2 μ_{a} e_{ij} e_{ij} d V

. Substituting

u_{a}^{'} = - G r / (2 μ_{a})

evaluates both sides to the same boundary expression, verifying the mechanical-energy equation.

Solved by gpt-5.6-sol high.

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