= Minimum shear for capillary Kelvin-Helmholtz instability
{title2=$U_{\min}^2=\frac{2(\rho_1+\rho_2)}{\rho_1\rho_2}\sqrt{(\rho_2-\rho_1)g\gamma}$}
For two infinitely deep inviscid layers with stable density ordering $\rho_2>\rho_1$ and positive <surface tension> $\gamma$, the velocity jump destabilizes wavenumber $k$ precisely when
$$
U^2>\frac{\rho_1+\rho_2}{\rho_1\rho_2}
\left(\frac{(\rho_2-\rho_1)g}{|k|}+\gamma|k|\right).
$$
The threshold diverges at both small and large wavenumber. Minimizing the bracket gives $|k|=\sqrt{(\rho_2-\rho_1)g/\gamma}$ and the stated minimum velocity jump. A larger jump opens a bounded band of unstable wavenumbers, obtained from the two positive roots of $\gamma q^2-[\rho_1\rho_2U^2/(\rho_1+\rho_2)]q+(\rho_2-\rho_1)g=0$. Equality at threshold is marginal rather than exponential growth.
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