Slender viscous bubble

For a long axisymmetric bubble with slowly varying radius $a(z,t)$, the leading normal-stress balance and internal lubrication flux couple capillary pressure to conservation of the cross-sectional area.

Internal Hagen-Poiseuille flow of viscosity $\lambda \mu$ has axial volume flux $Q=-\pi a^4{P}_z/(8\lambda \mu )$ when the interfacial axial speed is negligible to leading order. Conservation of mass gives $(\pi a^2{)}_t+Q_z=0$. Combining with the exterior radial normal-stress balance gives the displayed implicit evolution equation. The radial viscous normal stress and capillary pressure both matter in its local pinch-off similarity.

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.

Near a finite-time viscous-capillary pinch-off, balancing time dependence, viscous normal stress, capillary pressure, and axial lubrication flow can select self-similar radius and axial scales.

For $\tau =t_{\ast }-t$, balance $\mu a_t/a\sim \gamma /a$ to obtain radius proportional to $\tau$. Axial lubrication conservation then makes the axial length proportional to $\tau$ as well. The displayed variables reduce the lubrication equation for a bubble with viscous exterior to $\zeta A{A}^{\prime }-A^2=\frac{1}{16}\{A^4[(2\zeta A^{\prime }+1)/A{]}^{\prime }{\}}^{\prime }$. The physical interface slope is $\sqrt{\lambda }{A}^{\prime }$, compatible with a slender neck when $\lambda \ll 1$ and the dimensionless profile varies on order-one scales.