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When a viscous fluid partially fills a Hele--Shaw channel, and is pushed by apressure difference, the fluid interface is unstable due to the Saffman--Taylorinstability. We consider the evolution of a fluid region of finite extent,bounded between two interfaces, in the limit the interfaces are close, that is,when the fluid region is a thin liquid filament separating two gases ofdifferent pressure. In this limit, we derive a second-order `thin filament'model that describes the normal velocity of the filament centreline, andevolution of the filament thickness, as functions of the thickness, centrelinecurvature and their derivatives. We show that the second-order terms in thismodel, that include the effect of transverse flow along the filament, arenecessary to regularise the instability. Numerical simulation of the thinfilament model is shown to be in accordance with level-set computations of thecomplete two-interface model. Solutions ultimately evolve to form a bubble ofrapidly increasing radius and decreasing thickness.

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