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 Incoming links: Viscous diffusion time

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 16A / c / Solution Created 2026-09-24 Updated 2026-09-25

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When

Δ ≪ 1

, the small-argument expansion

sinh z = z + O (z^{3})

gives

f (y) = \frac{( 1 + i ) Δ y + O ( Δ ^{3} )}{( 1 + i ) Δ + O ( Δ ^{3} )} = y + O (Δ^{2}) .

(1)

Therefore

u (y, t) ≃ \frac{U _{0} y}{L _{0}} cos ω t .

(2)

The viscous diffusion time

L_{0}^{2} / ν

is then short compared with the oscillation period. Viscosity communicates the wall motion across the entire layer almost instantaneously, producing the linear profile of quasi-steady Couette flow.

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