= Uniform-pressure evolution of a slender viscous bubble
{title2=$P=\gamma f/\alpha_0,\quad\dot f=-\gamma g^2/(2\mu\alpha_0),\quad\dot g=\gamma fg/(2\mu\alpha_0)$}
A periodic slender bubble with $a=f+\sqrt2g\sin kz$ and negligible internal pressure variation has exterior radial velocity $u_r=a\dot a/r$ and pressure $P=2\mu\dot a/a+\gamma/a$. Its conserved mean cross-sectional area gives $\alpha_0=f^2+g^2=f(0)^2+g(0)^2$. Comparing the constant and sinusoidal terms gives the displayed equations. The long-wave growth rate about radius $a_0$ is $\gamma/(2\mu a_0)$, and $\dot a_{\min}=-\gamma(g^2+\sqrt2fg)/(2\mu\alpha_0)<0$. Small internal viscosity eventually produces appreciable axial pressure gradients near a neck, so this uniform-pressure reduction cannot persist to pinch-off at fixed positive viscosity ratio.
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