Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 38A / d

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- Solution d

Solution

Integrating the viscous traction

2 ρ ν e_{r θ}

over the side and end surfaces gives

T = - \frac{8 π ρ z _{0} ω _{0} ν ^{2}}{α} [1 - (1 + s_{0}) e^{- s_{0}}] = - A .

It is always negative. In steady state the cylinder stores no angular momentum, so the divergence theorem for angular-momentum balance requires viscous torque to cancel net advective influx.

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