Two-interface and thin-filament approximation in Hele-Shaw channel flow (Q6552462)

A thin liquid filament separating two gases of different pressure is considered in a Hele-Shaw cell. A highly accurate second-order lubrication flow model is used to describe this phenomenon. The new element, compared with the classical Saffman-Taylor results, is the appearance of a (near circular) bubble which is rapidly expanding and may intersect the channel walls at finite time. Numerical results are obtained by using a finite-difference method and the MATLAB's ode15s algorithm for time stepping. A lot of very useful figures illustrates the bubble evolution. It is important to note that as the bubble size increases, the thickness of the initial fluid filament decreases. The surface tension has a very important role. An useful comparison, between the interfaces obtained with ``filament model'' and the solution of the full problem obtained by using the level-set method, is carefully explained in section 5.2. As the authors emphasized, ``the specifics of the late stages of evolution of the filament...are not (yet) resolved''; they consider the case when ``the thickness does not go to zero at an isolated finite point in space and time''. Moreover, they suggest that considering fully three-dimensional effects could overcome these difficulties. This very interesting paper explains important new phenomena related to the Hele-Shaw flows.