Long-wave equations and direct simulations for the breakup of a viscous fluid thread surrounded by an immiscible viscous fluid

We consider capillary driven breakup of a viscous liquid thread in a second immiscible viscous liquid of infinite extent and arbitrary viscosity. Postulating the existence of long-wave dynamics, we use the slenderness parameter ε ≪ 1 (proportional to the interfacial slope, for example) to construct consistent asymptotic theories using matched asymptotic expansions at arbitrary viscosity ratios. Three canonical models are found, two of which hold for asymptotically small values of the inner to outer viscosity ratio λ ∼ ε² and λ ∼ 1/ln(1/ε), respectively, and the third valid for large λ ∼ 1/(ε² ln(1/ε)). The smallest and largest λ produce appropriate limits of models described in the literature when the inner or outer fluid is air, respectively. The intermediate λ model is found to be ill-posed and a technique is described to regularize it by considering terms arising from the asymptotic forms of the λ ∼ ε² model as its scaled viscosity ratio becomes large, and from the asymptotic forms of the λ ∼ 1/(ε² ln(1/ε)) as its scaled viscosity becomes small. Time-dependent direct numerical simulations based on boundary integral methods are also used to predict the dynamics for a large range of viscosity ratios (0.001 ≤ λ ≤ 20). Intermediate values of λ indicate that the dynamics is not long-wave, consistent with the asymptotic analysis, but are self-similar in a pinch-off region with order one aspect ratio. Simulations at large and small values of λ produce intricate dynamics near pinching and in particular the formation of necks, threads and bulges. The direct simulations and the asymptotic models valid at λ ∼ ε² and λ ∼ 1/(ε² ln(1/ε)) complement each other and the former confirm the validity of the long-wave assumptions.