Capillary Time Scale Research Articles

Droplet–jet collision following the monodispersedly dripping of coaxial binary droplets above a pool surface

In the present study, the collision between a falling droplet and a rising Worthington jet was experimentally studied. The event is followed by the monodispersedly dripping of coaxial binary droplets into a quiescent pool of glycerol solution. Different concentrations of the solution are considered. Unique droplet–jet collision characteristics are observed when the dripping flow rate is manipulated to release binary droplets. When the first droplet impacts the pool, a significant disturbance is imposed onto the pool, forming a deep crater followed by a Worthington jet. The second droplet is timed to collide with the rising jet to create a unique mushroom-shaped droplet–jet collision. Two jet pinch-off modes (tip pinch-off and no pinch-off) and four distinct collision regimes (partial rebounding, end-pinching, elongated, and clotted central jet collision) are recognized. Liquid viscosity and jetting mode significantly influence the collision dynamics and splattering characteristics. To achieve partial rebounding collision at low Weber number, a high-impact coefficient incorporating characteristic dimensions of the droplets and the Worthington jet is required, whereas a low-impact coefficient is required at high Weber number to attain clotted jet collision. The overall end-pinching phenomenon occurs due to the interaction between liquid flow toward the jet tip and the retraction of the tip, which causes the jet neck diameter to decrease on a capillary timescale. As the impact parameter decreases, the Worthington jet is inhibited, and the mushroom-shaped collision splash spreading is suppressed.

Coalescence of immiscible sessile droplets on a partial wetting surface

Droplet coalescence is a common phenomenon and plays an important role in multidisciplinary applications. Previous studies mainly consider the coalescence of miscible liquids, even though the coalescence of immiscible droplets on a solid surface is a common process. In this study, we explore the coalescence of two immiscible droplets on a partial wetting surface experimentally and theoretically. We find that the coalescence process can be divided into three stages based on the time scales and force interactions involved, namely (I) the growth of a liquid bridge, (II) the oscillation of the coalescing sessile droplet and (III) the formation of a partially engulfed compound sessile droplet and the subsequent retraction. In stage I, the immiscible interface is found not to affect the scaling of the temporal evolution of the liquid bridge, which follows the same 2/3 power law as that of miscible droplets. In stage II, by developing a new capillary time scale considering both surface and interfacial tensions, we show that the interfacial tension between the two immiscible liquids functions as a non-negligible resistance to the oscillation which decreases the oscillation periods. In stage III, a modified Ohnesorge number is developed to characterize the visco-capillary and inertia-capillary time scales involved during the displacement of water by oil; a new model based on energy balance is proposed to analyse the maximum retraction velocity, highlighting that the viscous resistance is concentrated in a region close to the contact line.

Fragmentation and solidification of fusible alloy melt by water spray

To elucidate the fragmentation and solidification processes that occur during the production of metal powders from a thermo-fluid perspective, we conducted experiments to visualize the water atomization of molten 42 Sn– 58 Bi fusible alloy and made measurements of the circularity of the resulting metal powder. The fragmentation process inside a fully developed fast water spray with a pressure up to 5 MPa was clearly captured. The time-resolved images revealed fragmentation patterns involving extending molten ligaments resulting from splashing, vapor explosions, and splitting. The time variation of the thinning of the neck diameter of the ligaments revealed that the fragmentation is rate-controlled by the capillarity of the molten drop itself, and solidification and viscosity have no influence on the pinching-point dynamics. The produced metal powder displayed size-dependent circularity, and larger particles tended to be less spherical than small particles. The mechanism is attributed to non-uniform solidification, which is enhanced at the pinching part of a droplet, consistent well with the images obtained. • Successful visualization of disintegrating alloy ligaments inside a water spray. • Fragmentation follows capillary timescale with partial stretching by impinging water. • Non-uniform solidification of elongated drops creates particles with less circularity.

High-speed visualization measurement for breakup process of liquid metal with solidification

In recent years, Additive Manufacturing (AM) technology such as 3D printing has attracted much attention. In order to produce high-quality metal powders as printing materials, it is necessary to elucidate the phenomena of liquid metal splitting accompanied by solidification, which is manifested in the powder manufacturing process. So far, various experiments have been conducted on the atomization method, but the mechanism for determining powder properties has not been clarified. In particular, there is insufficient knowledge about the phenomenon of molten metal splitting along with solidification. Therefore, in this study, we investigate the breakup process of a single ligament of 42Sn-58Bi, which is a kind of fusible alloy with a melting point of 139 ℃. A high-speed camera is used to visualize the pinch-off process of the ligament. The variation of the neck diameter of the ligament is traced during the fragmentation process. When the initial temperature is high, we found that the neck diameter thins down following a power law of 2/3, corresponding to the trend derived from capillary time scale. On the other hand, as the initial temperature gets lower, the neck diameter becomes thinner than that predicted by the capillary time scale just before breaking up, which is attributed to the influence of solidification.

Miscibility and wettability: how interfacial tension influences droplet impact onto thin wall films

Abstract

Droplet dynamics on viscoelastic soft substrate: Toward coalescence control

We study the dynamical behavior of droplets on a viscoelastic soft substrate. Using thin film approximation for the hydrodynamics and time-dependent Winkler’s substrate model, we show numerically how droplet growth depends strongly on the viscous damping characteristic of the substrate, leading to asymmetric stick-slip dynamics corroborated by experimental observations. Scaling arguments are presented to rationalize radial growth and the underlying substrate response to viscoelastic limits. Using an adjacent pair of inflating droplets, we report strongly diverse coalescence outcomes with non-linear coalescence times, including attraction, repulsion, and remarkably, a separation regime, within which the two droplets grow away from each other and remain separated due to intervening wetting ridges. Together, our results indicate strong interactions between the substrate and the droplet across viscoelastic and capillary timescales, with practical implications for smart surface engineering, condensation, and coalescence control.

Hole expansion from a bubble at a liquid surface

For millimetre to micron sized bubbles, floating at the free surface of different low viscosity fluids with different surface tensions, and then collapsing, we study the ensuing expansion of the outer radius of the hole (ro) at the free surface, as well as its velocity of expansion (uo). Since the thin film cap of the bubble disintegrates before the hole in it reaches the static rim, the hole expansion at intermediate times occurs as if it initiates at the bubble’s static rim of radius Rr; the evolution of ro then results to be a strong function of gravity, since Rr depends strongly on the bubble radius R. A scaling analysis, which includes the increase in the tip radius due to mass accumulation and the resulting change in the retraction force, along with the gravity effects by considering the hole radius in excess of its initial static radius, re = ro − Rr, results in a novel scaling law re/R∼(t/tc)4/7, where tc=ρR3/σ is the capillary time scale; this scaling law is shown to capture the evolution of the hole radii in the present study. The dimensionless velocities of hole expansion, namely, the Weber numbers of hole expansion, Weo=ρuo2R/σ, scale as Weo∼(t/tc)−6/7, independent of gravity effects, matching the observations. We also show that these Weber numbers, which reduce with time, begin with a constant initial Weber number of 64, while the viscous limit of the present phenomena occurs when the bubble Ohnesorge number Oh=μ/σρR≃0.24.

Micro-scale extensional rheometry using hyperbolic converging/diverging channels and jet breakup.

Understanding the elongational rheology of dilute polymer solutions plays an important role in many biological and industrial applications ranging from microfluidic lab-on-a-chip diagnostics to phenomena such as fuel atomization and combustion. Making quantitative measurements of the extensional viscosity for dilute viscoelastic fluids is a long-standing challenge and it motivates developments in microfluidic fabrication techniques and high speed/strobe imaging of millifluidic capillary phenomena in order to develop new classes of instruments. In this paper, we study the elongational rheology of a family of dilute polymeric solutions in two devices: first, steady pressure-driven flow through a hyperbolic microfluidic contraction/expansion and, second, the capillary driven breakup of a thin filament formed from a small diameter jet ([Formula: see text]). The small length scale of the device allows very large deformation rates to be achieved. Our results show that in certain limits of low viscosity and elasticity, jet breakup studies offer significant advantages over the hyperbolic channel measurements despite the more complex implementation. Using our results, together with scaling estimates of the competing viscous, elastic, inertial and capillary timescales that control the dynamics, we construct a dimensionless map or nomogram summarizing the operating space for each instrument.

Numerical simulations of bubble formation from a submerged orifice and a needle: The effects of an alternating electric field

In many applications, such as bubble column reactors, electric field is employed to provide a greater control on the sizes of bubbles forming at orifices and needles. In this study, we investigate the effects of an alternating electric field on the bubble dynamics. We perform numerical simulations of an alternating electric field coupled with two-phase flow using a Coupled Level-Set and Volume-of-Fluid method. We show that bubbles forming at orifices and needles decrease in size (up to 30%) only for a range of applied frequency and for other frequencies, the size of bubbles can be much bigger compared to the bubbles forming in the corresponding DC electric field case. The oscillating electric forces excite capillary waves on the bubble interface resulting in applied frequency dependent bubble oscillations. The numerically observed resonance for the needle case corresponds to 2ωτc=0.75, where 2ω is the frequency of the oscillation of the electric field force at the interface and τc is the capillary time scale, indicating that the resonance behavior is indeed governed by the interactions between the capillary and electric field force. A decomposition of bubble profile shapes into Legendre modes shows that for orifice as well as the needle case, second mode is most dominant followed by the fourth mode.

Droplet impact on deep liquid pools: Rayleigh jet to formation of secondary droplets.

The impact of droplets on a deep pool has applications in cleaning up oil spills, spray cooling, painting, inkjet printing, and forensic analysis, relying on the changes in properties such as viscosity, interfacial tension, and density. Despite the exhaustive research on different aspects of droplet impact, it is not clear how liquid properties can affect the instabilities leading to Rayleigh jet breakup and number of daughter drops formed after its pinch-off. In this article, through systematic experiments we investigate the droplet impact phenomena by varying viscosity and surface tension of liquids as well as impact speeds. Further, using numerical simulations, we show that Rayleigh-Plateau instability is influenced by these parameters, and capillary time scale is the appropriate scale to normalize the breakup time. Based on Ohnesorge number (Oh) and impact Weber number (We), a regime map for no breakup, Rayleigh jet breakup, and crown splash is suggested. Interestingly, crown splash is observed to occur at all Ohnesorge numbers; however, at high Oh, a large portion of kinetic energy is dissipated, and thus the Rayleigh jet is suppressed regardless of high impact velocity. The normalized required time for the Rayleigh jet to reach its peak varies linearly with the critical height of the jet.

Dynamics of Collapse of Free Surface Bubbles

We experimentally study the collapse of cavities formedby bubbles of different radius R at the free surface of five different fluids. The tangential and normal velocities of the disturbance front on the bubble surface were measured andshown that the tangential velocities scale as the capillary velocity (σ/ρR)1/2. The total time for the completion of cavity collapse was measured and shown to scale as capillary time scale R3/2/(σ/ρ)1/2. We hence infer thatthe cavity collapse is dominated by surface tension and inertia with viscosity not playing a role in the regime of our investigation. Such a behavior of cavity dynamics is different from the behavior of jet velocity discussed by Sangeeth et.al.[9] wherein the jet velocity showed a clear viscous cut off. The rate of change of cavity volume was found to be constant with time, the implications of which needs to be investigated further.

Early regimes of capillary filling

In this paper we analyze the inviscid regime (for which viscosity is unimportant and the flow occurs due to the balance between the capillary and the inertial effects) that invariably precedes the classical century-old Washburn regime during capillary filling. We demonstrate that a new nondimensional number, namely, the product of the Ohnesorge number and the ratio between the filling length (ℓ) and the radius of the capillary (R), dictates the occurrence of this regime and the other well-known regimes in a capillary filling problem. We also identify that this inviscid regime occurs for the time that is of the order of the capillary time scale and, as has been quantified before [Quere, Eur. Phys. Lett. 39, 533 (1997); Joly, J. Chem. Phys. 135, 214705 (2011)], is characterized by the filling length being linearly proportional to the filling time. We establish the universality of this regime by pinpointing the existence of this regime (showing appropriate dependencies of the capillary radii and density) from existing experimental and Molecular Dynamics Simulation results that signify disparate ranges of length and time scales.

Effects of dynamic surface tension and fluid flow on the oscillations of a supported bubble

Oscillating bubble techniques are commonly used to infer dynamic surface tension (DST) from the measurement of the dynamic pressure difference across an interface. In inferring DSTs from such measurements, the hydrodynamic effects are assumed to be negligible, so that the surface tension is uniform and the static Young-Laplace (Y-L) equation is valid. To examine these assumptions, the Navier-Stokes and continuity equations governing the flow of a liquid outside an axisymmetric bubble supported by a narrow tube are solved simultaneously by a finite element method with the equations governing surfactant transport and adsorption. Surface densities of adsorbed surfactant monolayers are examined to find the limits for accurate DST measurements. At low pulsation or oscillation rates (compared to the reciprocal of capillary time scale) and moderate area amplitudes, surface densities on the interface are found to be uniform, and DST measurements from pressure differences are quite accurate unless low surface tensions are attained. At high frequencies, however, because of strong convection around the surface of the bubble, the surface density along the liquid–gas interface becomes non-uniform while the bubble shape becomes highly deformed. The latter findings have analogs in the closely related problem of oscillating supported drops and can have profound ramifications for popular techniques for measuring DSTs.

Linear stability analysis of axisymmetric perturbations in imperfectly conducting liquid jets

A discussion is presented on the role of limited conductivity and permittivity on the behavior of electrified jets. Under certain conditions, significant departures with respect to the perfect-conductor limit are to be expected. In addition, an exploration is undertaken concerning the validity of one-dimensional average models in the description of charged jets. To that end, a temporal linear modal stability analysis is carried out of poor-conductor viscous liquid jets flowing relatively to a steady radial electric field. Only axisymmetric perturbations, leading to highest quality aerosols, are considered. A grounded coaxial electrode is located at variable distance. Most available studies in the literature are restricted to the perfect-conductor limit, while the present contribution is an extension to moderate and low electrical conductivity and permittivity jets, in an effort to describe a situation increasingly prevalent in the sector of small-scale free-surface flows. The influence of the electrode distance b, a parameter α defined as the ratio of the electric relaxation time scale to the capillary time scale, and the relative permittivity β on the growth rate has been explored yielding results on the stability spectrum. In addition, arbitrary viscosity and electrification parameters are contemplated. In a wide variety of situations, the perfect-conductor limit provides a good approximation; however, the influence of α and β on the growth rate and most unstable wavelength cannot be neglected in the general case. An interfacial boundary layer in the axial velocity profile occurs in the low-viscosity limit, but this boundary layer tends to disappear when α or β are large enough. The use of a one-dimensional (1D) averaged model as an alternative to the 3D approach provides a helpful shortcut and a complementary insight on the nature of the jet’s perturbative behavior. Lowest-order 1D approximations (average model), of widespread application in the literature of electrified jets, are shown to be inaccurate in low-viscosity imperfect-conductor jets.

Macroscopic and microscopic capillary length and time scales from field infiltration

Estimates of characteristic times to approach steady state flow in multidimensional infiltration in the landscape depend on the magnitude and character of the capillary length scale λc and the associated capillary time scale tc. Here we derive relationships between λc and tc and readily measured field properties sorptivity S and hydraulic conductivity K or S at two supply heads. We explore the relations between λc and tc and other macroscopic and microscopic length, potential, and time scales. In addition, we show that the microscopic characteristic length λm associated with λc gives physically plausible estimates of flow‐weighted mean pore dimensions. We contrast values of λc, tc, and λm for undisturbed field soils with those of repacked materials for water supply potentials close to zero. Large λm for the undisturbed surface soils are attributed to preferential flow. Data from here and elsewhere reveal no apparent trend of λc with soil texture, with most λc of the order of 100 mm. We suggest that the characteristic size of devices used to determine hydraulic properties of field soils should be greater than or equal to λc for representative measurements. The geometric mean time of approach to steady state flow when water is supplied at potentials near or greater than zero is found to be 1.7 hours. This value together with published results suggest that the time of approach to steady state flow from multidimensional cavities is of the order of 1 hour for many field situations.