Semi-analytical model for the rising sheet generated by droplet-pair impact

Abstract

When two low-Ohnesorge-number drops impact a dry substrate simultaneously, their spreading lamellae collide and lift a free-standing vertical sheet. The sheet grows by inertial feeding from the spreading drops and is pulled back by capillary retraction at its rim. We develop a semi-analytical model for this rising sheet by extending the single-drop impact description of~\citet{Gordillo2019} to the two-drop geometry. The thin-film flow in the sheet is coupled at its base to the colliding lamellae and at its apex to a capillary-retarded rim. The sheet interior is then solved along ballistic characteristics in two stages: a lamella-fed stage, for which the velocity and thickness fields can be obtained in closed form, and a post-lamella stage, for which the inlet conditions are taken from simulations. The resulting framework gives the three-dimensional velocity and thickness fields and therefore the full sheet shape. On the centreline, the apex height and local thickness are obtained explicitly, showing that the different Weber-number exponents reported in the literature arise from a crossover rather than from a single universal scaling law. At sufficiently large Weber number, the apex pinches off. A linear Rayleigh--Plateau analysis, using the time-dependent jet diameter and deceleration predicted by the model, then bounds the maximum attainable height and closes the description of the pinch-off regime.