Path instability of an air bubble rising in water.

It has been documented since the Renaissance that an air bubble rising in water will deviate from its straight, steady path to perform a periodic zigzag or spiral motion once the bubble is above a critical size. Yet, unsteady bubble rise has resisted quantitative description, and the physical mechanism remains in dispute. Using a numerical mapping technique, we for the first time find quantitative agreement with high-precision measurements of the instability. Our linear stability analysis shows that the straight path of an air bubble in water becomes unstable to a periodic perturbation (a Hopf bifurcation) above a critical spherical radius of R = 0.926 mm, within 2% of the experimental value. While it was previously believed that the bubble's wake becomes unstable, we now demonstrate a new mechanism, based on the interplay between flow and bubble deformation.

Dynamics of a silicone oil drop submerged in a stratified ethanol-water bath.

We analyze numerically the Marangoni flow around an immiscible droplet submerged in a stably stratified mixture of ethanol and water. The linear stability analysis shows that the base flow undergoes a supercritical Hopf bifurcation that leads to oscillations. The theoretical prediction for the critical droplet radius is consistent with previous experimental results. Ethanol diffusion in water is critical in the flow stability for both low and high droplet viscosity. Direct numerical simulations of the nonlinear oscillatory flow show that the frequency of those oscillations approximately equals that of the critical eigenmode. The nonlinear convective term of the ethanol diffusive-convective transport equation fixes the amplitude of the droplet oscillations. The viscous dissipation associated with the Marangoni flow inside the droplet considerably reduces the oscillation.

Elastic Rayleigh-Plateau instability: dynamical selection of nonlinear states.

A slender thread of elastic hydrogel is susceptible to a surface instability that is reminiscent of the classical Rayleigh-Plateau instability of liquid jets. The final, highly nonlinear states that are observed in experiments arise from a competition between capillarity and large elastic deformations. Combining a slender analysis and fully three-dimensional numerical simulations, we present the phase map of all possible morphologies for an unstable neo-Hookean cylinder subjected to capillary forces. Interestingly, for softer cylinders we find the coexistence of two distinct configurations, namely, cylinders-on-a-string and beads-on-a-string. It is shown that for a given set of parameters, the final pattern is selected via a dynamical evolution. To capture this, we compute the dispersion relation and determine the characteristic wavelength of the dynamically selected profiles. The validity of the "slender" results is confirmed via simulations and these results are consistent with experiments on elastic and viscoelastic threads.

Breakup of an electrified viscoelastic liquid bridge.

We study both numerically and experimentally the breakup of a viscoelastic liquid bridge formed between two parallel electrodes. The polymer solutions and applied voltages are those commonly used in electrospinning and near-field electrospinning. We solve the leaky-dielectric finitely extensible nonlinear elastic-Peterlin (FENE-P) model to describe the dynamical response of the liquid bridge under isothermal conditions. The results show that the surface charge screens the inner electric field perpendicular to the free surface over the entire dynamical process. The liquid bridge deformation produces a normal electric field on the outer side of the free surface that is commensurate with the axial one. The surface conduction does not significantly affect the current intensity in the time interval analyzed in the experiments. The force due to the shear electric stress becomes comparable to both the viscoelastic and surface tension forces in the last stage of the filament. However, it does not alter the elastocapillary balance in the filament. As a consequence, the extensional relaxation times measured from the filament exponential thinning approximately coincides with the stress relaxation time prescribed in the FENE-P model. The above results allow us to interpret correctly the experiments. In the experiments, we measure the filament electrical conductivity and extensional relaxation time for polyethylene oxide (PEO) dissolved in deionized water and in a mixture of water and glycerine. We compare the filament electrical conductivity with the value measured in hydrostatic conditions for the same estimated temperature. Good agreement was found for PEO dissolved in water + glycerine, which indicates that the change in the filament microscopic structure due to the presence of stretched polymeric chains does not significantly alter the ion mobility in the stretching direction. Significant deviations are found for PEO dissolved in deionized water. These deviations may be attributed to the heat transferred to the ambient, which is neglected in the calculation of the filament temperature. We measure the extensional relaxation time from the images acquired during the filament thinning. The relaxation times obtained in the first stage of the exponential thinning hardly depend on the applied voltage. Little but measurable influence of the applied voltage is found in the last phase of the filament thinning.

Controlled cavity collapse: scaling laws of drop formation.

The formation of transient cavities at liquid interfaces occurs in an immense variety of natural processes, among which the bursting of surface bubbles and the impact of a drop on a liquid pool are salient. The collapse of a surface liquid cavity is a well documented natural process that leads to the ejection of a thin and fast jet. Droplets generated through this process can be one order of magnitude smaller than the cavity's aperture, and they are consequently of interest in drop on demand inkjet applications. In this work, the controlled formation and collapse of a liquid cavity is analyzed, and the conditions for minimizing the resulting size and number of ejected drops are determined. The experimental and numerical models are simple and consist of a liquid reservoir, a nozzle plate with the discharge orifice, and a moving piston actuated by single half-sine-shaped pull-mode pulses. The size of the jetted droplet is described by a physical model resulting in a scaling law that is numerically and experimentally validated.

Mean flow produced by small-amplitude vibrations of a liquid bridge with its free surface covered with an insoluble surfactant.

As is well known, confined fluid systems subject to forced vibrations produce mean flows, called in this context streaming flows. These mean flows promote an overall mass transport in the fluid that has consequences in the transport of passive scalars and surfactants, when these are present in a fluid interface. Such transport causes surfactant concentration inhomogeneities that are to be counterbalanced by Marangoni elasticity. Therefore, the interaction of streaming flows and Marangoni convection is expected to produce new flow structures that are different from those resulting when only one of these effects is present. The present paper focuses on this interaction using the liquid bridge geometry as a paradigmatic system for the analysis. Such analysis is based on an appropriate post-processing of the results obtained via direct numerical simulation of the system for moderately small viscosity, a condition consistent with typical experiments of vibrated millimetric liquid bridges. It is seen that the flow patterns show a nonmonotone behavior as the Marangoni number is increased. In addition, the strength of the mean flow at the free surface exhibits two well-defined regimes as the forcing amplitude increases. These regimes show fairly universal power-law behaviors.

Effect of a Surrounding Liquid Environment on the Electrical Disruption of Pendant Droplets.

The effect of a surrounding, dielectric, liquid environment on the dynamics of a suddenly electrified liquid drop is investigated both numerically and experimentally. The onset of stability of the droplet is naturally dictated by a threshold value of the applied electric field. While below that threshold the droplet retains its integrity, reaching to a new equilibrium state through damped oscillations (subcritical regime), above it electrical disruption takes place (supercritical regime). In contrast to the oscillation regime, the dynamics of the electric droplet disruption in the supercritical regime reveals a variety of modes. Depending on the operating parameters and fluid properties, a drop in the supercritical regime may result in the well-known tip streaming mode (with and without whipping instability), in droplet splitting (splitting mode), or in the development of a steep shoulder at the elongating front of the droplet that expands radially in a sort of "splashing" (splashing mode). In both splitting and splashing modes, the sizes of the progeny droplets, generated after the breakup of the mother droplet, are comparable to that of the mother droplet. Furthermore, the development in the emission process of the shoulder leading to the splashing mode is described as a parametrical bifurcation, and the parameter governing that bifurcation has been identified. Physical analysis confirms the unexpected experimental finding that the viscosity of the dynamically active environment is absent in the governing parameter. However, the appearance of the splitting mode is determined by the viscosity of the outer environment, when that viscosity overcomes a certain large value. These facts point to the highly nonlinear character of the drop fission process as a function of the droplet volume, inner and outer liquid viscosities, and applied electric field. These observations may have direct implications in systems where precise control of the droplet size is critical, such as in analytical chemistry and "drop-on-demand" processes driven by electric fields.

Off-axis vortex breakdown in a shallow whirlpool.

The off-axis emergence of vortex breakdown (VB) is revealed. The steady axisymmetric flow in a vertical sealed cylinder, which is partially filled with water and the rest is filled with air, is driven by the rotating bottom disk. The numerical simulations show that VB can emerge away from the rotation axis, interface, and walls. As the rotation intensifies, VB first develops in the water region. If the water height is less (larger) than nearly one half of the cylinder radius, VB emerges off (on) the axis. As the rotation further increases, the off-axis VB ring touches the interface and then a thin countercirculation layer develops in the air flow above the water VB domain. This two-fluid VB ring shrinks (it even disappears in a very shallow whirlpool) as the interface approaches the bottom disk.

Development of colliding swirling counterflows.

This numerical study of the axisymmetric motion of a viscous incompressible fluid in an elongated cylindrical container explains how colliding counterflows develop. Two swirling flows enter the container through peripheral inlets and leave it through central exhausts symmetrically from both ends. Different flow rates, characterized by the Reynolds number Re, are studied for a fixed swirl number. For small Re, the throughflow (TF) is limited to the inlet-exhaust vicinities and a few circulation cells occupy the rest of the interior. As Re grows, (i) the circulation cells disappear while the TF reaches the container's midsection and becomes U shaped, moving near the sidewall inward and going back near the container axis; elongated circulation regions develop separating the TF branches; (ii) the flow convergence to the axis focuses near the container's midsection, resulting in the vortex breakdown development; (iii) the swirl-induced low pressure causes suction of the ambient fluid through the central parts of the exhausts; (iv) the suction flow reaches the container's midsection, turns around, mixes with the driving TF, forms an annular outflow, and leaves through the exhaust periphery. The two factors, (a) swirl decay due to friction at the sidewall and (b) the focused flow convergence to the axis, constitute the physical mechanism of the colliding counterflows. Such flow pattern is favorable for a vortex solid-fuel combustor.

Development of a swirling double counterflow.

This numerical study of an axisymmetric motion of a viscous incompressible fluid in an elongated cylindrical container explains how a swirling inflow develops the global meridional circulation and two U-shaped throughflows (TFs). For moderate values of the Reynolds (Re) number, there is a single U-shaped TF: The fluid moves from the peripheral annular inlet near the sidewall to the dead end, turns around, goes back near the axis, and leaves the container through the central exhaust. As Re increases, vortex breakdown occurs near the dead end. If the exhaust orifice is wide, the ambient fluid is sucked into the container near its axis, reaches the dead-end vicinity, merges with the U-shaped TF, and goes back inside an annular region. Thus, a double counterflow develops, where the fluid moves to the dead end near both the sidewall and the axis and goes back in between. The physical mechanism of the double counterflow is a swirl decay combined with the focused flow convergence near the dead end. This double counterflow is beneficial for combustion applications.

Liquid flow focused by a gas: jetting, dripping, and recirculation.

The liquid cone-jet mode can be produced upon stimulation by a coflowing 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 volume of fluid (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.

Bubbling in unbounded coflowing liquids.

An investigation of the stability of low density and viscosity fluid jets and spouts in unbounded coflowing liquids is presented. A full parametrical analysis from low to high Weber and Reynolds numbers shows that the presence of any fluid of finite density and viscosity inside the hollow jet elicits a transition from an absolute to a convective instability at a finite value of the Weber number, for any value of the Reynolds number. Below that critical value of the Weber number, the absolute character of the instability leads to local breakup, and consequently to local bubbling. Experimental data support our model.