On the breakup of viscous liquid threads

A one-dimensional model evolution equation is used to describe the nonlinear dynamics that can lead to the breakup of a cylindrical thread of Newtonian fluid when capillary forces drive the motion. The model is derived from the Stokes equations by use of rational asymptotic expansions and under a slender jet approximation. The equations are solved numerically and the jet radius is found to vanish after a finite time yielding breakup. The slender jet approximation is valid throughout the evolution leading to pinching. The model admits self-similar pinching solutions that yield symmetric shapes at breakup. These solutions are shown to be the ones selected by the initial boundary value problem, for general initial conditions. Furthermore, the terminal state of the model equation is shown to be identical to that predicted by a theory which looks for singular pinching solutions directly from the Stokes equations without invoking the slender jet approximation throughout the evolution. It is shown quantitatively, therefore, that the one-dimensional model gives a consistent terminal state, with the jet shape being locally symmetric at breakup. The asymptotic expansion scheme is also extended to include unsteady and inertial forces in the momentum equations to derive an evolution system modeling the breakup of Navier-Stokes jets. The model is employed in extensive simulations to compute breakup times for different initial conditions; satellite drop formation is also supported by the model and the dependence of satellite drop volumes on initial conditions is studied.