Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 1 / c / iv

Table of contents
- Solution iv

Solution

Use superscripts

T

and

D

for the thin-upper-layer and two-deep-layer cases. The cutoff is identical, while

\frac{k _{m}^{T}}{k _{m}^{D}} \sim \frac{3}{2}, \frac{σ _{m}^{T}}{σ _{m}^{D}} \sim [\frac{3 3 ( ρ _{1} + ρ _{2} )}{8 ρ _{1}} k_{c} L_{1}]^{1/2} ≪ 1.

Thus confinement suppresses the Rayleigh-Taylor instability growth strongly, but moves the most amplified disturbance to a somewhat larger wavenumber. The cutoff depends on the gravity-capillary balance, whereas growth depends also on the inertia of the moving layers.

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