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A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica and Matlab) do not offer a solution, is to find all the real solutions of a system of N nonlinear equations in a certain finite domain of the N-dimensional space of variables. We present an algorithm of minimum length and computational weight to solve this problem, resembling a graphical tool of edge detection in an image extended to N dimensions. Once the hypersurfaces (edges) defined by each nonlinear equation have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in the vicinity of all the hyperpoints that constitute the solutions makes the final Newton-Raphson step rapidly convergent to all the solutions with the desired degree of accuracy. As long as N remains smaller than about five, which is often the case for physical systems that depend on fewer than five parameters, this approach demonstrates excellent effectiveness., Comment: 10 pages, 3 figures

The average Rayleigh capillary breakup length of a cylindrical Newtonian viscous liquid jet moving with homogeneous velocity $\hat{U}$ (negligible external forces) must be determined by the selection of normal modes with time-independent amplitude and wavelength (invariant modes, IMs). Both positive and negative group velocity IMs exist in ample ranges of the parameter domain (Weber and Ohnesorge numbers), which explains (i) the average breakup length independence on ambient conditions (long-term resonance), and (ii) its proportionality to the inverse of the spatial growth rate of the dominant positive group velocity IM. Published experimental results since Grace (1965, PhD Thesis) confirm our proposal., Comment: 9 pages, 7 figures

A major fraction of the atmospheric aerosols come from the ocean spray originated by the bursting of bubbles from breaking waves. A theoretical framework that incorporates the latest knowledge on film and jet droplets from bubble bursting is proposed. Assuming that their relics constitute the ultimate origin of primary and secondary sea aerosols through a diversity of physicochemical routes, the model can be reduced to a single controlling parameter to predict the global probability density distribution (pdf) of the ocean spray. The bursting and collapse of small bubbles on the sea surface from about 10 to 100 microns produces an extreme energy focusing and the ejection of a rapid liquid spout whose size reaches the free molecular regime of the gaseous environment. In these rarefied conditions, simulations show that this spout yields a jet of sub-micrometer and nanometric scale droplets whose number and speed can be far beyond any previous estimation, overcoming by orders of magnitude alternative mechanisms recently proposed. The one-parameter model fits remarkably well published experimental measurements along five orders of magnitude of spray size, from about 5 nm to about 0.5 mm. According to this proposal, the majority of aerosols determining the life on our planet would have their extremely elusive birth in the uterus-like nano-shape of small bursting bubbles on the ocean surface at the very latest instants of collapse., Comment: 12 pages, 13 figures

Current available data of the worldwide impact of the COVID-19 pandemic has been analyzed using dimensional analysis and self-similarity hypotheses. We show that the time series of infected population and deaths of the most impacted and unprepared countries exhibits an asymptotic power law behavior, compatible with the propagation of a signal in a fractal network. We propose a model which predicts an asymptotically self-similar expansion of deaths in time before containment, and the final death toll under total containment measures, as a function of the delay in taking those measures after the expansion is observed. The physics of the model resembles the expansion of a flame in a homogeneous domain with a fractal dimension 3.75. After containment measures are taken, the natural fractal structure of the network is drastically altered and a secondary evolution is observed. This evolution, akin to the homogeneous combustion in a static isolated enclosure with a final quenching, has a characteristic time of 20.1 days, according to available data of the pandemic behavior in China. The proposed model is remarkably consistent with available data, which supports the simplifying hypotheses made in the model. A universal formulation for a quarantine as a function of that delay is also proposed.

The transient ejection due to a bubble bursting at the interface of a liquid with a gas environment is here described using a dynamical scaling analysis along the process. We show here that the ejection of a liquid microjet requires the backfire of a vortex ring inside the liquid to preserve physical symmetry, which involves a non-trivial scaling. We present the first single uniformly valid expression for the size and speed of ejected droplets for the whole range of the Ohnesorge and Bond numbers where droplet ejection occurs. The evolution of the flow variables, the apparent singularity for a critical Ohnesorge number, and the dispersion of data around this point are explained. Our model generalizes or displaces other recently proposed ones, impacting for instance the statistical description of sea spray., Comment: 10 pages, 3 figures

Dripping, jetting and tip streaming have been studied up to a certain point separately by both fluid mechanics and microfluidics communities, the former focusing on fundamental aspects while the latter on applications. Here, we intend to review this field from a global perspective by considering and linking the two sides of the problem. In the first part, we present the theoretical model used to study interfacial flows arising in droplet-based microfluidics, paying attention to three elements commonly present in applications: viscoelasticity, electric fields and surfactants. We review both classical and current results about the stability of jets affected by these elements. Mechanisms leading to the breakup of jets to produce drops are reviewed as well, including some recent advances in this field. We also consider the relatively scarce theoretical studies on the emergence and stability of tip streaming flows. In the second part of this review, we focus on axisymmetric microfluidic configurations which can operate on the dripping and jetting modes either in a direct (standard) way or via tip streaming. We present the dimensionless parameters characterizing these configurations, the scaling laws which allow predicting the size of the resulting droplets and bubbles, as well as those delimiting the parameter windows where tip streaming can be found. Special attention is paid to electrospray and flow focusing, two of the techniques more frequently used in continuous drop production microfluidics. We aim to connect experimental observations described in this section of topics with fundamental and general aspects described in the first part of the review. This work closes with some prospects at both fundamental and practical levels., Comment: 63 pages, 55 figures

Despite their fundamental and applied importance, a general model to predict the natural breakup length of steady capillary jets has not been proposed yet. In this work, we derive a scaling law with two universal constants to calculate that length as a function of the liquid properties and operating conditions. These constants are determined by fitting the scaling law to a large set of experimental and numerical measurements, including previously published data. Both the experimental and numerical jet lengths conform remarkably well to the proposed scaling law. This law is explained in terms of the growth of perturbations excited by the jet breakup itself., Comment: 6 pages, 7 figures

A general scaling of the evolution of an exploding liquid jet under an ultra short and intense X-ray pulse from a X-ray free electron laser (XFEL) is proposed. A general formulation of the conservation of energy for blasts in vacuum partially against a deformable object leads to a compact expression that governs the evolution of the gap produced by the explosion. The theoretical analysis contemplates two asymptotic stages for small and large times from the initiation of the blast. A complete dimensional analysis of the problem and an optimal collapse of experimental data reveal that the universal approximate analytical solution proposed is in remarkable agreement with experiments.

Jet droplets from bubble bursting are determined by a limited parametrical space: the liquid properties (surface tension, viscosity, and density), mother bubble size and acceleration of gravity. Thus, the two resulting parameters from dimensional analysis (usually, the Ohnesorge and Bond numbers, Oh and Bo) completely define this phenomenon when both the trapped gas in the bubble and the environment gas have negligible density. A detailed physical description of the ejection process to model both the ejected droplet radius and its initial launch speed is provided, leading to a scaling law including both Oh and Bo. Two critical values of Oh determine two limiting situations: one (Oh$_1$=0.038) is the critical value for which the ejected droplet size is minimum and the ejection speed maximum, and the other (Oh$_2$=0.0045) is a new critical value which signals when viscous effects vanish. Gravity effects (Bo) are consistently introduced from energy conservation principles. The proposed scaling laws produce a remarkable collapse of published experimental measurements collected for both the ejected droplet radius and ejection speed., Comment: 14 pages, three figures, published in 2018 in Physical Review Fluids

The liquid cone-jet mode can be produced upon stimulation by a co-flowing gas sheath. Most applications deal with the jet breakup, leading to either of two droplet generation regimes: jetting and dripping. The cone-jet flow pattern is explored by direct axisymmetric VOF numerical simulation; its evolution is studied as the liquid flow-rate is increased around the jetting-dripping transition. As observed in other focused flows such as electrospraying cones upon steady thread emission, the flow displays a strong recirculating pattern within the conical meniscus; it is shown to play a role on the stability of the system, being a precursor to the onset of dripping. Close to the minimum liquid flow rate for steady jetting, the recirculation cell penetrates into the feed tube. Both the jet diameter and the size of the cell are accurately estimated by a simple theoretical model. In addition, the transition from jetting to dripping is numerically analyzed in detail in some illustrative cases, and compared, to good agreement, with a set of experiments., Comment: Submitted to the Physical Review E on December 8th, 2007

Recent experimental studies on the instability appearance of capillary jets have revealed the capabilities of linear spatiotemporal instability analysis to predict the parametrical map where steady jetting or dripping takes place. In this work, we present an extensive analytical, numerical and experimental analysis of confined capillary jets extending previous studies. We propose an extended, accurate analytic model in the limit of low Reynolds flows, and introduce a numerical scheme to predict the system response when the liquid inertia is not negligible. Theoretical predictions show a remarkable accuracy with results from the extensive experimental exploration provided., Comment: Submitted to the Physical Review E (20-March-2008)

Capillary jetting of a fluid dispersed into another immiscible phase is usually limited by a critical Capillary number, a function of the Reynolds number and the fluid properties ratios. Critical conditions are set when the minimum spreading velocity of small perturbations $v^*_-$ along the jet (marginal stability velocity) is zero. Here we identify and describe parametrical regions of high technological relevance, where $v^*_- > 0$ and the jet flow is always supercritical independently of the dispersed liquid flow rate: within these relatively broad regions, the jet does not undergo the usual dripping-jetting transition, so that either the jet can be made arbitrarily thin (yielding droplets of any imaginably small size), or the issued flow rate can be made arbitrarily small. In this work, we provide illustrative analytical studies of asymptotic cases for both negligible and dominant inertia forces. In this latter case, requiring a non-zero jet surface velocity, axisymmetric perturbation waves ``surf'' downstream for all given wave numbers while the liquid bulk can remain static. In the former case (implying small Reynolds flow) we found that the jet profile small slope is limited by a critical value; different published experiments support our predictions., Comment: Submitted first (24-August-2008) to Physics of Fluids, withdrawn from that journal on 6-April-2008, and submitted to Physical Review E the same day