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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-331/3/d/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 3 / d / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

Take

k^{*} > 0

and positive shear scale

S^{*}

, with kinematic viscosity

ν^{*}

. Choose

y = k^{*} y^{*}, x = k^{*} x^{*}, t = S^{*} t^{*}, U = \frac{k ^{*} U ^{*}}{S ^{*}} = y, c = \frac{k ^{*} c ^{*}}{S ^{*}}, R e = \frac{S ^{*}}{ν ^{*} ( k ^{*} ) ^{2}} .

(1)

The velocity scale is

S^{*} / k^{*}

. A reversed shear can be treated by reversing the corresponding coordinate orientation and mode convention. The dimensional Orr-Sommerfeld equation, with

U^{*''} = 0

, is

(U^{*} - c^{*}) (D_{*}^{2} - (k^{*})^{2}) v^{*} = \frac{ν ^{*}}{i k ^{*}} (D_{*}^{2} - (k^{*})^{2})^{2} v^{*} .

(2)

Substitution of the scales and cancellation gives

(y - c) (D^{2} - 1) v = - \frac{i}{R e} (D^{2} - 1)^{2} v .

(3)

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