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 Incoming links: Dissipation identity for a homogeneous Squire mode

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 79 / 5 / a / iii / Solution Created 2026-10-03 Updated 2026-10-14

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Multiply the homogeneous Squire equation by

\overset{η}{ˉ}

and integrate over the channel. The zero boundary values remove the integration by parts boundary term, giving

- iω \int_{- 1}^{1} ∣ η ∣^{2} d y + ik \int_{- 1}^{1} U ∣ η ∣^{2} d y + \frac{1}{R} \int_{- 1}^{1} (∣ η^{'} ∣^{2} + q^{2} ∣ η ∣^{2}) d y = 0.

(1)

For real

U

and

k

, the middle term is purely imaginary. Taking real parts yields the dissipation identity for a homogeneous Squire mode:

Im ω = - \frac{\int _{- 1}^{1} ( ∣ η ^{'} ∣ ^{2} + q ^{2} ∣ η ∣ ^{2} ) d y}{R \int _{- 1}^{1} ∣ η ∣ ^{2} d y} < 0.

(2)

The strict inequality holds for a nonzero mode with zero endpoint values and positive Reynolds number. Hence all homogeneous Squire modes are temporally stable. In the complete shear-flow perturbation system, the Squire variable can also be forced by the wall-normal velocity; this homogeneous-mode result does not rule out instability of that coupled system or its transient growth.

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