Uniform-pressure evolution of a slender viscous bubble ($P=\gamma f/{\alpha }_0,\dot{f}=-\gamma g^2/(2\mu {\alpha }_0),\dot{g}=\gamma fg/(2\mu {\alpha }_0)$)

A periodic slender bubble with $a=f+\sqrt{2}g\sin kz$ 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 ${\dot{a}}_{\min }=-\gamma (g^2+\sqrt{2}fg)/(2\mu {\alpha }_0)<0$. 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.