 Solution

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

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

With

U = U_{0}

and

β = 0

, the Orr-Sommerfeld equation factors into constant-coefficient operators:

(D^{2} - α^{2}) (D^{2} - γ^{2}) v = 0, γ^{2} = α^{2} + i R e (α U_{0} - ω) .

For distinct nonzero characteristic roots, the general solution is

v = a_{1} e^{α y} + a_{2} e^{- α y} + a_{3} e^{γ y} + a_{4} e^{- γ y} .

γ

gives the same solution space. The no-slip boundary condition and impermeability impose

v = D v = 0

at each wall.

The displayed exponential basis needs the usual repeated-root qualification. If

γ^{2} = α^{2} \neq = 0

, replace it by

(a_{1} + a_{2} y) e^{α y} + (a_{3} + a_{4} y) e^{- α y}

. If just one root pair is zero, that pair contributes

1, y

; if both pairs are zero, the basis is

1, y, y^{2}, y^{3}

. These are the complete limiting cases, not four independent copies of a repeated exponential.

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