Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 38B (Fluid Dynamics II)

Let the disturbed sheet be $y=h+\eta$ and take a normal mode proportional to $e^{i(kx-\omega t)}$, with $K=|k|$. Irrotational flow and the rigid-wall condition at $y=0$ give perturbation velocity potentialsThe two linearized kinematic boundary conditions at $y=h$ areThe Unsteady Bernoulli equation gives $p_{-}^{\prime }=i\rho (\omega -kU)A\cosh (Kh)$ and $p_{+}^{\prime }=i\rho \omega B$. Continuity of pressure therefore yields the finite-depth vortex-sheet dispersion relationor, with $T=\tanh (Kh)$,For $k>0$ its two roots areOne root has positive temporal growth for every $k\ne 0$, so every finite wavelength is subject to the Kelvin-Helmholtz instability.

The unstable branch has temporal growth rate and phase velocityFor short waves, $|k|h\gg 1$, soFor long waves, $|k|h\ll 1$, soThus the growth-rate curve starts at the origin with zero slope, rises like $|k{|}^{3/2}$, and approaches the straight-line asymptote $|k|U/2$. The inviscid zero-thickness model consequently has no short-wave cutoff.

If the lower density ${\rho }_{-}$ exceeds the upper density ${\rho }_{+}$, stable density stratification adds a gravitational restoring term. Including surface tension $\gamma$, the same calculation has the schematic dispersion relationThe gravity term dominates the shear term as $K\to 0$ and therefore suppresses sufficiently long waves. The capillary term dominates as $K\to \infty$ and suppresses sufficiently short waves. Together they reduce the Kelvin-Helmholtz instability growth rate and can leave only a finite unstable band, or eliminate instability if the restoring effects are strong enough.