We study both numerically and experimentally the steady cone-jet mode of electrospraying close to the stability limit of minimum flow rate. The leaky dielectric model is solved for arbitrary values of the relative permittivity and the electrohydrodynamic Reynolds number. The linear stability analysis of the base flows is conducted by calculating their global eigenmodes. The minimum flow rate is determined as that for which the growth factor of the dominant mode becomes positive. We find a good agreement between this theoretical prediction and experimental values. The analysis of the spatial structure of the dominant perturbation may suggest that instability originates in the cone-jet transition region, which shows the local character of the cone-jet mode. The electric relaxation time is considerably smaller than the residence time of a fluid particle in the cone-jet transition region (defined as the region where the surface and bulk intensities are of the same order of magnitude) except for the high-polarity case, where these characteristic times are commensurate with each other. The superficial charge is not relaxed within the cone-jet transition region except for the high-viscosity case, because significant inner electric fields arise in the cone-jet transition region. However, those electric fields are not large enough to invalidate the scaling laws that do not take them into account. Viscosity and polarization forces compete against the driving electric shear stress in the cone-jet transition region for small Reynolds numbers and large relative permittivities, respectively. Capillary forces may also play a significant role in the minimum flow rate stability limit. The experiments show the noticeable stabilizing effect of the feeding capillary for diameters even two orders of magnitude larger than that of the jet. Stable jets with electrification levels higher than the Rayleigh limit are produced. During the jet break-up, two consecutive liquid blobs may coalesce and form a bigger emitted droplet, probably due to the jet acceleration. The size of droplets exceeds Rayleigh’s prediction owing to the stabilizing effect of both the axial electric field and viscosity.

We examine both theoretically and experimentally the breakup of a pendant drop loaded with an insoluble surfactant. The experiments show that a significant amount of surfactant is trapped in the resulting satellite droplet. This result contradicts previous theoretical predictions, where the effects of surface tension variation were limited to solutocapillarity and Marangoni stresses. We solve numerically the hydrodynamic equations, including not only those effects but also those of surface shear and dilatational viscosities. We show that surface viscosities play a critical role to explain the accumulation of surfactant in the satellite droplet.

In this review, we describe both theoretical and experimental results on the dynamics of liquid bridges under isothermal conditions with fixed triple contact lines. These two major restrictions allow us to focus on a well-defined body of literature, which has not as yet been reviewed in a comprehensive way. Attention is mainly paid to liquid bridges suspended in air, although studies about the liquid–liquid configuration are also taken into account. We travel the path from equilibrium to nonlinear dynamics of both Newtonian liquid bridges and those made of complex fluids. Specifically, we consider equilibrium shapes and their stability, linear dynamics in free and forced oscillations under varied conditions, weakly nonlinear behavior leading to streaming flows, fully nonlinear motion arising during stretching and breakup of liquid bridges, and problems related to rheological effects and the presence of surfactant monolayers. Although attention is mainly paid to fundamental aspects of these problems, some applications derived from the results are also mentioned. In this way, we intend to connect the two approaches to the liquid bridge problem, something that both theoreticians and experimentalists may find interesting.

We study numerically the nonlinear stationary states of a droplet covered with an insoluble surfactant in a uniaxial extensional flow. We calculate both the eigenfunctions to reveal the instability mechanism and the time-dependent states resulting from it, which provides a coherent picture of the phenomenon. The transition is of the saddle-node type, both with and without surfactant. The flow becomes unstable under stationary linear perturbations. Surfactant considerably reduces the interval of stable capillary numbers. Inertia increases the droplet deformation and decreases the critical capillary number. In the presence of the surfactant monolayer, neither the droplet deformation nor the stability is significantly affected by the droplet viscosity. The transient state resulting from instability is fundamentally different for drops with and without surfactant. Tip streaming occurs only in the presence of surfactants. The critical eigenmode leading to tip streaming is qualitatively the same as that yielding the central pinching mode for a clean interface, which indicates that the small local scale characterizing tip streaming is set during the nonlinear droplet deformation. The viscous surface stress does not significantly affect the droplet deformation and the critical capillary number. However, the damping rate of the dominant mode considerably decreases for viscous surfactants. Interestingly, shear viscous surface stress considerably alters the tip streaming arising in the supercritical regime, even for very small surface viscosities. The viscous surface stresses alter the balance of normal interfacial stresses and affect the surfactant transport over the stretched interface.

The production of viscoelastic capillary jets with gaseous flow focusing is studied experimentally. In this technique, the liquid is injected at a constant flow rate through a feeding capillary located in front of the discharge orifice. A gas stream coflows with the jet across the orifice driven by a constant pressure drop. The gas stream sucks and drags the liquid, reducing the jet's diameter well below the orifice diameter. Because of the rheological nature of the liquid, this focusing phenomenon differs from the Newtonian one in several regards. For given values of the polymer concentration, the injected flow rate, and the applied pressure drop, there is an interval of the capillary-to-orifice distance for which the jetting regime is reached. Outside that interval, the jet either suffers from the pull-out instability or breaks up before reaching the discharge orifice. Significant free surface oscillations can be observed in most of the jetting realizations. This oscillatory behavior is caused by a transient die swell effect which continuously appears right at the capillary exit. Ejection interrupts because the jet bulges to such an extent that the free surface pinches. Because of the stabilizing effect of the polymeric contribution to the axial stress, micrometer filaments with lengths up to 1 cm and Weber numbers on the order of 10 −4 can be produced in front of the discharge orifice. The shear viscous stresses exerted on the emitted jet by the gas stream beyond the discharge orifice prevent the macromolecule recoiling. The resulting extensional viscosity inhibits the break-up process, and thus very long jets are produced.