Capillary breakup of a viscous thread surrounded by another viscous fluid

Previous long-wavelength analyses of capillary breakup of a viscous fluid thread in a perfectly inviscid environment show that the asymptotic self-similar regime immediately prior to breakup is given by a balance between surface tension, inertia, and extensional viscous stresses in the thread. In contrast, it is shown here that if viscosity in the external fluid, however small, is included then the asymptotic balance is between surface tension and viscous stresses in the two fluids while inertia is negligible. Scaling estimates for this new balance suggest that both axial and radial scales decrease linearly with time to breakup, so that the aspect ratio remains $\text{O(1)}$ with time but scales with viscosity ratio like ${\mathrm{(\mu }}_{\mathrm{int}}{\mathrm{/\mu }}_{\mathrm{ext}}{)}^{\text{1/2}}$ for ${\mu }_{\mathrm{int}}{\mathrm{\gg \mu }}_{\mathrm{ext}},$ where ${\mu }_{\mathrm{int}}$ and ${\mu }_{\mathrm{ext}}$ are the internal and external viscosities. Numerical solutions to the full Stokes equations for ${\mu }_{\mathrm{int}}{\mathrm{=\mu }}_{\mathrm{ext}}$ confirm the scalings with time and give self-similar behavior near pinching. However, the self-similar pinching region is embedded in a logarithmically large axial advection driven by the increasing range of scales intermediate between that of the pinching region and that of the macroscopic drop. The interfacial shape in the intermediate region is conical with angles of about 6° on one side and 78° on the other.