Here we extend the results in Gordillo et al. (J. Fluid Mech., vol. 866, 2019, pp. 298–315), where the spreading of drops impacting perpendicularly a solid wall was analysed, to predict the time-varying flow field and the thickness of the liquid film created when a spherical drop of a low viscosity fluid, like water or ethanol, spreads over a smooth dry surface at arbitrary values of the angle formed between the drop impact direction and the substrate. Our theoretical results accurately predict the time evolving asymmetric shape of the border of the thin liquid film extending over the substrate during the initial instants of the drop spreading process. In addition, the particularization of the ordinary differential equations governing the unsteady flow when the rim velocity vanishes provides an algebraic equation for the asymmetric final shapes of the liquid stains remaining after the impact, valid for low values of the inclination angle. For larger values of the inclination angle, the final shape of the drop can be approximated by an ellipse whose major and minor semiaxes can also be calculated by making use of the present theory. The predicted final shapes agree with the observed remaining stains, excluding the fact that a liquid rivulet develops from the bottom part of the drop. The limitations of the present theory to describe the emergence of the rivulet are also discussed.

When a drop of a low-viscosity liquid of radius $R$ impacts against an inclined smooth solid substrate at a velocity  $V$ , a liquid sheet of thickness $H_{t}\ll R$ is expelled at a velocity $V_{t}\gg V$ . If the impact velocity is such that $V>V^{\ast }$ , with $V^{\ast }$ the critical velocity for splashing, the edge of the expanding liquid sheet lifts off from the wall as a consequence of the gas lubrication force at the wedge region created between the advancing liquid front and the substrate. Here we show that the magnitude of the gas lubrication force is limited by the values of the slip length $\ell _{\unicode[STIX]{x1D707}}$ at the gas–liquid interface and of the slip length $\ell _{g}\propto \unicode[STIX]{x1D706}$ at the solid, with $\unicode[STIX]{x1D706}$ the mean free path of gas molecules. We demonstrate that the splashing regime changes depending on the value of the ratio $\ell _{\unicode[STIX]{x1D707}}/\ell _{g}$ – a fact explaining the spreading–splashing–spreading–splashing transition for a fixed (low) value of the gas pressure as the drop impact velocity increases (Xu et al., Phys. Rev. Lett., vol. 94, 2005, 184505; Hao et al., Phys. Rev. Lett., vol. 122, 2019, 054501). We also provide an expression for $V^{\ast }$ as a function of the inclination angle of the substrate, the drop radius  $R$ , the material properties of the liquid and the gas, and the mean free path  $\unicode[STIX]{x1D706}$ , in very good agreement with experiments.

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 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.

At room temperature, when a drop impacts against a smooth solid surface at a velocity above the so-called critical velocity for splashing, the drop loses its integrity and fragments into tiny droplets violently ejected radially outwards. Below this critical velocity, the drop simply spreads over the substrate. Splashing is also reported to occur for solid substrate temperatures above the Leidenfrost temperature, $T_{L}$ , for which a vapour layer prevents the drop from touching the solid. In this case, the splashing morphology differs from the one reported at room temperature because, thanks to the presence of the gas layer, the shear stresses acting on the liquid can be neglected. Our purpose here is to predict, for wall temperatures above $T_{L}$ , the critical Weber number for splashing as well as the maximum spreading radius. First, making use of boundary integral simulations, we calculate both the time evolution of the liquid velocity as well as the height of the sheet which is ejected tangentially to the substrate. These results are then used as boundary conditions for the one-dimensional mass and momentum equations describing the dynamics of the rim limiting the expanding liquid sheet. Our predictions for both the maximum spreading radius and for the critical Weber number for splashing are in good agreement with experimental observations.

We experimentally determine the phase diagram for impacting ethanol droplets on a smooth, sapphire surface in the parameter space of Weber number $\mathit{We}$ versus surface temperature $T$ . We observe two transitions, namely the one towards splashing (disintegration of the droplet) with increasing $\mathit{We}$ , and the one towards the Leidenfrost state (no contact between the droplet and the plate due to a lasting vapour film) with increasing $T$ . Consequently, there are four regimes: contact and no splashing (deposition regime), contact and splashing (contact–splash regime), neither contact nor splashing (bounce regime), and finally no contact, but splashing (film–splash regime). While the transition temperature $T_{L}$ to the Leidenfrost state depends weakly, at most, on $\mathit{We}$ in the parameter regime of the present study, the transition Weber number $\mathit{We}_{C}$ towards splashing shows a strong dependence on $T$ and a discontinuity at $T_{L}$ . We quantitatively explain the splashing transition for $T<T_{L}$ by incorporating the temperature dependence of the physical properties in the theory by Riboux & Gordillo (Phys. Rev. Lett., vol. 113(2), 2014, 024507; J. Fluid Mech., vol. 772, 2015, pp. 630–648).

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. Here, we make use of the model of Riboux & Gordillo (Phys. Rev. Lett., vol. 113, 2014, 024507), together with a one-dimensional approximation describing the flow in the ejected liquid sheet and of balances of mass and momentum at the border of the sheet, to calculate mean sizes and velocities of the ejected drops. The predictions of the model are in good agreement with experiments.

Navier–Stokes simulations of the agitation generated by a homogeneous swarm of high-Reynolds-number rising bubbles are performed. The bubbles are modelled by fixed momentum sources of finite size randomly distributed in a uniform flow. The mesh grid is regular with a spacing close to the bubble size. This allows us to simulate a swarm of a few thousand bubbles in a computational domain of a hundred bubble diameters, which corresponds to a gas volume fraction $\alpha $ from 0.6 % to 4 %. The small-scale disturbances close to the bubbles are not resolved but the wakes are correctly described from a distance of a few diameters. This simple model reproduces well all the statistical properties of the vertical velocity fluctuations measured in previous experiments: scaling as ${\alpha }^{0. 4} $ , self-similar probability density functions and power spectral density including a subrange evolving as the power $- 3$ of the wavenumber $k$ . It can therefore be concluded that bubble-induced agitation mainly results from wake interactions. Considering the flow in a frame that is fixed relative to the bubbles, the combined use of both time and spatial averaging makes it possible to distinguish two contributions to the liquid fluctuations. The first is the spatial fluctuations that are the consequence of the bubble mean wakes. The second corresponds to the temporal fluctuations that are the result of the development of a flow instability. Note that the latter is not due to the destabilization of individual bubble wakes, since a computation with a single bubble leads to a steady flow. It is a collective instability of the randomly distributed bubble wakes. The spectrum of the time fluctuations shows a peak around a frequency ${f}_{cwi} $ , which is independent of $\alpha $ . From the present results it is possible to determine the origin of the overall properties of the total fluctuations observed in the experiments. The scaling of the velocity fluctuation as ${\alpha }^{0. 4} $ is a combination of the scalings of the spatial and temporal fluctuations, which are different from each other. As the time fluctuations are symmetric in the vertical direction, the asymmetry of the probability density function of the vertical velocity comes from that of the spatial fluctuations. Both contributions exhibit a ${k}^{- 3} $ spectral behaviour around the same range of wavenumbers, which explains why it is observed regardless of the nature of the dominant contribution.

The charged liquid micro-jet issued from a Taylor cone may develop a special type of non-axisymmetric instability, usually referred to in the literature as a whipping mode. This instability usually manifests itself as a series of fast and violent lashes of the charged jet, which makes its characterization in the laboratory difficult. Recently, we have found that this instability may also develop when the host medium surrounding the Taylor cone and the jet is a dielectric liquid instead of air. When the oscillations of the jet occur inside a dielectric liquid, their frequency and amplitude are much lower than those of the oscillations taking place in air. Taking advantage of this fact, we have performed a detailed experimental characterization of the whipping instability of a charged micro-jet within a dielectric liquid by recording the jet motion with a high-speed camera. Appropriate image processing yields the frequency and wavelength, among the other important characteristics, of the jet whipping as a function of the governing parameters of the experimental set-up (flow rate and applied electric field) and liquid properties. Alternatively, the results can be also written as a function of three dimensionless numbers: the capillary and electrical Bond numbers and the ratio between an electrical relaxation and residence time.

Capillary liquid flows have shown their ability to generate micro and nano-structures which can be used to synthesize material in the micro or nanometric size range. For instance, electrified capillary liquid jets issued from a Taylor are broadly used to spin micro and nanofibers when the liquid consists of a polymer solution or melt, a process termed electrospinning. In this process, the electrified capillary jet may develop a nonaxisymmetric instability, usually referred to as whipping instability, which very efficiently transforms electric energy into stretching energy, thus leading to the formation of extremely thin polymer fibers. Even though non axysimmetric instabilities of electrified jets were first investigated some decades ago, the existing theoretical models provide a qualitative understanding of the phenomenon but none of them is accurate enough when compared with experimental results. This whipping instability usually manifests itself as fast and violent lateral motion of the charged jet, which makes it difficult its characterization in the laboratory. However, this instability also develops when electrospinning is performed within a liquid bath instead of air. Although it is essentially the same phenomenon, the frequency of the whipping oscillations is much slower in the former case than in the latter, thus allowing detailed experimental characterization of the whipping instability. Furthermore, since the outer fluid is a liquid, its density and viscosity may now be used to influence the dynamics of the electrified capillary jet. In this work we present and rationalize the experimental data collecting the influence of the main parameters on the whipping characteristics of the electrified jet (frequency, amplitude, etc.).

An experimental investigation of the flow generated by a homogeneous population of bubbles rising in water is reported for three different bubble diameters (d = 1.6, 2.1 and 2.5 mm) and moderate gas volume fractions (0.005 ≤ α ≤ 0.1). The Reynolds numbers, Re = V0d/ν, based on the rise velocity V0 of a single bubble range between 500 and 800. Velocity statistics of both the bubbles and the liquid phase are determined within the homogeneous bubble swarm by means of optical probes and laser Doppler anemometry. Also, the decaying agitation that takes place in the liquid just after the passage of the bubble swarm is investigated from high-speed particle image velocimetry measurements. Concerning the bubbles, the average velocity is found to evolve as V0α−0.1 whereas the velocity fluctuations are observed to be almost independent of α. Concerning the liquid fluctuations, the probability density functions adopt a self-similar behaviour when the gas volume fraction is varied, the characteristic velocity scaling as V0α0.4. The spectra of horizontal and vertical liquid velocity fluctuations are obtained with a resolution of 0.6 mm. The integral length scale Λ is found to be proportional to V02/g or equivalently to d/Cd0, where g is the gravity acceleration and Cd0 the drag coefficient of a single rising bubble. Normalized by using Λ, the spectra are independent on both the bubble diameter and the volume fraction. At large scales, the spectral energy density evolves as the power −3 of the wavenumber. This range starts approximately from Λ and is followed for scales smaller than Λ/4 by a classic −5/3 power law. Although the Kolmogorov microscale is smaller than the measurement resolution, the dissipation rate is however obtained from the decay of the kinetic energy after the passage of the bubbles. It is found to scale as α0.9 V03/Λ. The major characteristics of the agitation are thus expressed as functions of the characteristics of a single rising bubble. Altogether, these results provide a rather complete description of the bubble-induced turbulence.