# Drop splashing at smooth dry surfaces

When a drop impacts a smooth, dry surface at a velocity above the so-called critical speed for drop splashing, the initial liquid volume loses its integrity, fragmenting into tiny droplets that are violently ejected radially outwards. Making use of experiments, potential flow and lubrication theories and of numerical simulations, we first develop a model to predict the critical velocity for splashing. We find that dewetting is a necessary but not sufficient condition for splashing. Splashing only occurs when the drop velocity is such that, the much faster and thinner liquid sheet which is expelled in the direction tangent to the solid as a consequence of the impact, is accelerated vertically up to velocities larger than those caused by the capillary retraction of the sheet. The vertical accelerations are imparted to the edge of the spreading liquid sheet by the aerodynamic lift force exerted by the surrounding gas on the edge of the radially expanding lamella. The lift force per unit length results as the addition of: i) the classical term used in aerodynamics (which depends on the product of gas density, the squared relative velocity and the thickness of the lamella) plus ii) a term that depends on gas viscosity as the product of four terms: gas viscosity, the velocity of the liquid sheet edge relative to that of the ambient gas, a constant which is a function of the angle the advancing lamella forms with the substrate and a logarithm which incorporates the ratio between the thickness of the lamella and the mean free path of molecules. The contribution of the logarithm, and as a result, of the mean free path, is essential since, otherwise, these “viscous lift forces” would tend to infinite. The wedge angle is not the static contact angle with the substrate, but it is the angle the advancing liquid front forms with the solid substrate. Our splash criterion can be summarized as follows: if the vertical velocities imparted to the edge of the sheet by the aerodynamic lift forces are larger than the radial velocity at which the edge of the sheet grows by capillary retraction, the edge of the sheet “takes off” and the toroidal rim bordering the sheet is prone to develop Savart-Plateau-Rayleigh capillary instabilities. If the growth time of capillary instabilities is sufficiently small when compared with the time of growth of the toroidal rim, the edge of the lamella breaks into very tiny drops, of sizes comparable to that of the thickness of the rim. For impact velocities larger than the critical splash velocity and making use of a onedimensional approximation describing the ﬂow in the ejected liquid sheet and of balances of mass and momentum at the border of the sheet, we predict the mean sizes and velocities of the ejected drops. The predictions of the model are in good agreement with experiments.

This talk is part of the Institute for Energy and Environmental Flows (IEEF) series.