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.

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.