Viscous-capillary pinch-off similarity

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.