 Lubrication equation for a bubble with viscous exterior

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.