Here we provide a self-consistent analytical solution describing the unsteady flow in the slender thin film which is expelled radially outwards when a drop hits a dry solid wall. Thanks to the fact that the fluxes of mass and momentum entering into the toroidal rim bordering the expanding liquid sheet are calculated analytically, we show here that our theoretical results closely follow the measured time-varying position of the rim with independence of the wetting properties of the substrate. The particularization of the equations describing the rim dynamics at the instant the drop reaches its maximal extension which, in analogy with the case of Savart sheets, is characterized by a value of the local Weber number equal to one, provides an algebraic equation for the maximum spreading radius also in excellent agreement with experiments. The self-consistent theory presented here, which does not make use of energetic arguments to predict the maximum spreading diameter of impacting drops, provides us with the time evolution of the thickness and of the velocity of the rim bordering the expanding sheet. This information is crucial in the calculation of the diameters and of the velocities of the droplets ejected radially outwards for drop impact velocities above the splashing threshold.

Abstract

A drop of radius $R$ of a liquid of density $\unicode[STIX]{x1D70C}$, viscosity $\unicode[STIX]{x1D707}$ and interfacial tension coefficient $\unicode[STIX]{x1D70E}$ impacting a superhydrophobic substrate at a velocity $V$ keeps its integrity and spreads over the solid for $V<V_{c}$ or splashes, disintegrating into tiny droplets violently ejected radially outwards for $V\geqslant V_{c}$, with $V_{c}$ the critical velocity for splashing. In contrast with the case of drop impact onto a partially wetting substrate, Riboux & Gordillo (Phys. Rev. Lett., vol. 113, 2014, 024507), our experiments reveal that the critical condition for the splashing of water droplets impacting a superhydrophobic substrate at normal atmospheric conditions is characterized by a value of the critical Weber number, $We_{c}=\unicode[STIX]{x1D70C}\,V_{c}^{2}\,R/\unicode[STIX]{x1D70E}\sim O(100)$, which hardly depends on the Ohnesorge number $Oh=\unicode[STIX]{x1D707}/\sqrt{\unicode[STIX]{x1D70C}\,R\,\unicode[STIX]{x1D70E}}$ and is noticeably smaller than the corresponding value for the case of partially wetting substrates. Here we present a self-consistent model, in very good agreement with experiments, capable of predicting $We_{c}$ as well as the full dynamics of the drop expansion and disintegration for $We\geqslant We_{c}$. In particular, our model is able to accurately predict the time evolution of the position of the rim bordering the expanding lamella for $We\gtrsim 20$ as well as the diameters and velocities of the small and fast droplets ejected when $We\geqslant We_{c}$.

Here we report the production of monodisperse microbubbles by taking advantage of the large values of both the pressure gradients and of the local velocities existing at the leading edge of airfoils in relative motion with a liquid. It is shown here that the scaling laws for the bubbling frequencies and the bubble diameters are identical to those found in microfluidics. Therefore, the metre-sized geometry presented here is a feasible candidate to circumvent the inherent problems of using micron-sized geometries in real applications – namely, wettability, the low productivity and the clogging of the microchannels by particles or other impurities.

Here we present a method for producing noncoalescing monodisperse microbubbles in an efficient, massive and controlled way. Since our prototype is easily scalable, it can be straightforwardly adapted to satisfy the specific gas injection demands required by the different applications where it can be used, like bioreactors or water treatment or purification plants. The main feature of the bubbling device described here consists of injecting the gas at the leading edge of a wing in relative motion with respect to a liquid. The reasons for this particular design relies on the smallness of the drag coefficient of streamlined bodies and also on the fact that the strongest favorable pressure gradients and the minimum values of the liquid pressures are located at the leading edge of the airfoils composing the wing.