Dimple drainage before the coalescence of a droplet with a smooth substrate

C is a major process in multiphase flows since it controls, with the break-up dynamics, the global evolution of the number of droplets and bubbles in the flow. It is therefore crucial for spray cooling and coating (1), emulsion and foams (2, 3), or droplet transmission of diseases (4, 5) for instance. Coalescence is in fact a complex phenomenon, the effective contact between two liquid bodies being not at all automatic as the interfaces approach each other, since surface interactions and viscous forces can in particular contribute to separate the interfaces (6–9). These forces can lead to the striking “floating drops” phenomena, as already mentioned by Lord Rayleigh (10) and Osborne Reynolds (11): for instance, it can observed in an everyday life “experiment” by watching small coffee drops dancing in the mug above the liquid surface before sinking(12), or when a drop is deposited on a vibrating liquid bath(13, 14). The coalescence dynamics is controlled at first by the drainage of the thin fluid film separating the two liquid bodies, whether we consider drop-drop, drop-film or drop-substrate coalescences. The drainage of this separating film determines the time for coalescence that can be defined as the time between the beginning of the drainage regime (where no motion is present but the drainage) to the time where the film ruptures so that the two liquid bodies enter in contact, or the liquid starts to wet the substrate. Such thin film drainage follows a lubrication dynamics (15–21), where the viscosity of the surrounding fluid induces high pressure in the film that leads to the interface deformation and sometimes to the entrapment of an air bubble at coalescence as observed in drop impacts for instance (22, 23). However, the lubrication dynamics prevents mathematically the rupture of the film in finite time (at least when surface tension is present(24)) even when the film is squeezed, so that the final stage of coalescence has to involve additional physical mechanisms, dominant at microscopic or even nanoscopic scales. Usual suspects are van der Waals interactions (25), surface roughness, thermal fluctuations, Marangoni currents and non-continuum effects (26), although their precise implications remain largely an open question. Even if this final stage of the coalescence dynamics can exhibit thus high fluctuations, it can usually be described as a thickness cut-off below which the film rupture happens rapidly, so that the lubrication dynamics can be taken as the dominant mechanism to determine the coalescence time. Eventually, even the drainage time can exhibit large variations in experiments and models, mostly poorly understood, so that it is crucial to have a better comprehension of the lubrication dynamics of drainage in the different configurations encountered (9, 27). Drop coalescence (28–30) can be obtained through mainly two different practical configurations: drop collision (to an other drop, a bath, a substrate in particular) where the drop velocity is the main control parameter (31); drop deposition, where the drop is smoothly deposited on the substrate, the weight of the drop leading to its coalescence (27). While most of the experimental, numerical and theoretical studies have focused on the drop/drop collision, only little attention has been paid eventually to the case of the deposition of a drop on a smooth liquid substrate, where the coalescence dynamics is driven by the weight of the drop that squeezes the air gap between the drop and the wetted substrate. Recently, an experimental investigation of the coalescence of a millimetric drop gently deposited on a thin viscous film (27) has revealed unexpected drainage dynamics: the drainage time is seen much shorter in the presence of a very thin viscous film than the time of touch down estimated through simple lubrication arguments. The authors show experimentally that tangential flows at both interfaces between the gas and the liquids cannot be neglected as it is often done in theory, allowing a more rapid drainage of the gas layer (32). These tangential flows are created by the viscous entrainment of the liquids by the gas layer that is squeezed, a mechanism also observed recently in Leidenfrost configuration (33). In order to elucidate the