Thinning and rupture of a viscous filament in a Hele-Shaw cell

The simplest geometry for a thin viscous filament in a Hele-Shaw cell can be modelled by a version of the so-called “thin film equation”. This is a nonlinear fourth-order diffusion model. If fluid is sucked out of this filament, then it will rupture in finite time. For more interesting geometries where the centreline of the filament is free to evolve in time, the model is much more complicated. In particular, I will discuss applications where the evolution of the filament is driven by a pressure difference across the filament, a situation which is commonplace in Hele-Shaw experiments. The discussion will involve some modelling, numerics, linear stability, and the question of whether/how the filament ruptures in this more complicated model.