Retraction dynamics of highly viscous liquid sheets

Upon rupture, a planar liquid sheet undergoes retraction driven by unbalanced capillary forces. This process is governed by two dimensionless groups: the sheet's aspect ratio and the Reynolds number. In this talk, we will theoretically investigate the retraction dynamics of a two-dimensional viscous sheet in the regime of small aspect ratios and low Reynolds numbers, assuming that the sheet is free at one extremity and fixed at the other. An asymptotic model of the dynamics, derived using matched asymptotic expansions in an appropriate distinguished limit, will be presented. Our analysis reveals that the dynamics is dictated by a remote region where inertial and viscous effects balance. In that region, the flow has a conserved quantity, thereby enabling the reduction of the problem to a one-dimensional diffusion equation for the sheet thickness profile subject to effective boundary conditions at the free end. This reduced description facilitates the identification and analysis of distinct retraction regimes, which are characterized by the time elapsed since rupture and by the relative smallness of the aspect ratio to the Reynolds number.