Low Bond Number Research Articles

Future Green Technology: A Freezing Water Micro-Droplet as an Optical Switch Based on a Time-Domain Photonic Hook.

This paper pays attention to the broader interest of freezing water droplets in mesotronics, particularly to their use as a new all-optical device platform. Here, we show that a freezing mesoscale water droplet with a low Bond number can behave as fully biocompatible natural microlense to form a photonic hook for application in a tunable temperature-controlled optical switch. We first introduced and demonstrated the basic concepts of an optical switch without changes in the wavelength of illumination of a particle or any moving parts being involved. The principle of the operation of the switch is based on the temperature-induced phase change inside the water droplet's refractive index. The simulation results show that the optical isolation of switched channels for an optical switch with linear dimensions of about 15 λ3 based on a freezing water droplet can reach 10 dB in the process of temperature variation at a fixed wavelength. The use of freezing mesoscale droplets acting as a time-domain photonic hook generator open an intriguing route for optical switching in multifunctional green electronics tools for sensing, integrated optics and optical computers.

A transient hydrodynamic model of screen channel liquid acquisition devices for in-space cryogenic propellant management

Screen channel liquid acquisition devices (LADs) will play a crucial role in future deep space travel. It is essential that vapor-free delivery of propellants during tank-to-tank transfer is ensured to maximize yield from storage tanks and prevent potential combustion instabilities. The screen channel LAD utilizes a fine screen wire mesh that can separate phases in a low Bond number (i.e., microgravity) environment using surface tension forces. This study presents the development and verification of a new model for transient screen compliance, one of the influential factors for screen channel LAD design. Screen compliance is crucial during LAD channel outflow transients because the slight deflection of the screen can provide needed mass to satisfy rapid outflow demands and reduce the pressure difference across the screen. The model is successfully verified against computational fluid dynamics simulations. In addition, the characteristic speed for the governing screen compliance equations is derived, which allows for numerical stability criteria to be established. As shown in this study, the transient maximum pressure difference across the screen can greatly exceed the steady state maximum pressure difference across the screen in many cases.

Cold pressurant gas bubble point experiments for coarse mesh screen channel liquid acquisition devices in liquid nitrogen

Screen channel liquid acquisition devices (LADs) will play a crucial role in future deep space travel. It is essential that vapor-free delivery of propellants during in-space transfer is ensured to maximize yield from storage tanks and prevent potential combustion instabilities. The screen channel LAD utilizes a fine screen wire mesh that can separate phases in a low Bond number (i.e. microgravity) environment using surface tension forces. This paper presents cryogenic liquid nitrogen (LN2) screen channel LAD bubble point experiments focused on coarser mesh screens. A total of 14 different screen types are tested across a range of cryogenic liquid temperatures near the screen (70.8 − 80.0 K) and dewar pressures (∼101.3 – 156.5 kPa or ∼0 – 8 psig) for two pressurization schemes (noncondensable gaseous helium and autogenous pressurization with gaseous nitrogen) to examine the effect of pressurant gas type on LAD performance. Cold pressurant gas bubble point test results for coarser meshes shown here agree almost perfectly with historical trends using fine mesh screens. Pressurization with a noncondensable generally leads to higher bubble point due to evaporative cooling at the LAD screen while pressurization with the vapor generally leads to degradation in performance due to condensation of the vapor and subsequent interfacial heating.

Oscillatory Signatures in the Raindrop Motion Relative to the Air Medium with Terminal Velocity

This paper aims to study the path oscillations of single, spherical water droplets levitated in a wind tunnel to better comprehend the mechanical motion of small raindrops. The observations were carried out in the Mainz vertical wind tunnel. The discrete, fast Fourier transform was used to determine the oscillatory frequencies of the droplet paths, and the Hilbert transform was applied to analyze their instantaneous frequency stability. Both transversal and streamwise components of the path oscillations are described with typical frequencies of several tens of Hertz. The studied oscillations elongate the paths, reduce the terminal velocity of the smallest droplets, and make the rain droplet fall in a non-uniform motion even after reaching terminal velocity. The terminal velocity can be considered as physically having been reached if our proposed practical criterion is satisfied. From a fluid mechanics perspective, the paper fills an experimental gap in the studies of the paths oscillations of single, liquid spheres of microscopic sizes at low Bond numbers <1 and relatively low Reynolds numbers ~102.

Lateral drift of liquid droplets sliding down substrates with nonuniform dissipation

Understanding and controlling the motion of liquid droplets is important both from a fundamental point of view and in many applications. In this work, we theoretically analyze the dynamics of a liquid droplet sliding down a ramp under the action of gravity. We show that when the dissipation (which we account for by means of an effective contact line viscosity) has a gradient in the direction orthogonal to gravity, the droplet, while sliding down, drifts laterally in the direction opposite to the dissipation gradient. We validate our analytical results, obtained in the limit of low Bond numbers, by a numerical solution of our model.

Numerical investigation of high-frequency pulsating electrohydrodynamic jet at low electric Bond numbers

Electrohydrodynamic jet printing is a highly promising technology for the fabrication of three-dimensional micro/nanoscopic structures, but the advancement of this technology is hindered by the insufficient understanding of many aspects of its mechanisms. Here we conduct a numerical investigation on high-frequency (∼1 kHz) pulsating electrohydrodynamic jet at low electric Bond numbers (Boe = 0.15–0.7). By analyzing the entire jetting process using the voltage distribution, electric charge density, and flow field obtained from the numerical results, we overcome the limitations of experimental approach and demonstrate the influences of electric voltage (Φ), nozzle-to-substrate distance (H), and liquid surface tension coefficient (γ) on the dynamic behaviors and durations of the three jetting stages: (1) cone formation, (2) jetting, and (3) meniscus oscillation. Furthermore, as a measure of the relative significance of the electric force to the surface tension force, the impacts of Boe on the jetting process are also examined. Results show that some critical aspects of the pulsating jetting process are closely related to Boe: (1) the transitional values of Boe between the four observed jetting regimes on the variations of Φ, H, and γ apply to all three parameters; (2) the nondimensionalized Taylor cone length scales with Boe according to a power law; (3) the jetting processes that have similar Boe collapse onto a universal profile. These new findings of pulsating electrohydrodynamic jet provide a useful supplement to the currently inadequate comprehension of the complicated electrohydrodynamic jet printing process.

Archimedes’ principle with surface tension effects in undergraduate fluid mechanics

The general physics of how objects float is only partly covered in most undergraduate fluid mechanics courses. Although Archimedes’ principle is a standard topic in fluid statics, the role of surface tension in floating is rarely discussed in detail. For example, very few undergraduate textbooks, if any, mention that the total buoyancy force on a floating object includes the weight of the fluid displaced by the meniscus. This leaves engineering students without an understanding of a wide range of phenomena that occur at a low Bond number (the ratio of buoyancy to interfacial tension forces), such as how heavier-than-water objects can float at a gas-liquid interface. This article makes a case for teaching a more unified version of Archimedes’ principle, which combines the effects of surface tension and hydrostatic pressure in calculating the total buoyancy on floating objects. Sample problems at the undergraduate level and two classroom demonstrations are described that reinforce the basic science concepts.

Dynamics of inner gas during the bursting of a bubble at the free surface

In the present study, simulations are directed to capture the dynamics of evacuating inner gas of a bubble bursting at the free surface, using Eulerian based volume of fluid (VOF) method. The rate by which surrounding air rushing inside the bubble cavity through the inner gas evacuation is estimated and compared by the collapsing bubble cavity during the sequential stages of the bubble bursting at the free surface. Further, the reachability of inner gas at different horizontal planes over the unperturbed free surface is estimated. The evacuating inner gas accompanies vortex rings, which entrains the surrounding gas-phase. During the successive stages of air entrainment, spatiotemporal characteristics of the vortex ring are obtained. At low Bond numbers (Bo < 1), the axial growth pattern of gas jet and the radial expansion of jet tip are studied with the phase contour of evacuating inner gas. Furthermore, the axial growth of rising inner gas over the free surface and the radial expansion of vortex rings of a bubble bursting at the free surface is compared with the quiescent surrounding air under the respiration process. At last, the effects of various possible asymmetric perforation of the bubble cap keeping the same Bo are studied. The cause of the bent gas jet, as a consequence of the perforation of the bubble cap, asymmetrically, is explained by plotting the velocity vectors. The effect of miscibility on the spreading dynamics of inner gas has been found to be minimal at the early stage of the bursting process.

A Morawetz Inequality for Gravity-Capillary Water Waves at Low Bond Number

This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the solutions. Furthermore, it is uniform both in the infinite depth limit and the zero surface tension limit.

Inertial and buoyancy effects on the flow of elongated bubbles in horizontal channels

When a long gas bubble travels in a horizontal liquid-filled channel of circular cross-section, a liquid film is formed between the bubble and the channel wall. At low Reynolds and Bond numbers, inertial and buoyancy effects are negligible, and the liquid film thickness is a function of the capillary number only. However, as the tube diameter is increased to the millimetre scale, both buoyancy and inertial forces may become significant. We present the results of a systematic analysis of the bubble shape, inclination, and liquid film thickness for a wide range of capillary, Bond, and Reynolds numbers, namely 0.024≤Cal≤0.051, 0.11≤Bo≤3.5, and 1≤Rel≤750. Three-dimensional numerical simulations of the flow are performed by employing the Volume-Of-Fluid method implemented in OpenFOAM. In agreement with previous studies, we observe that buoyancy lifts the bubble above the channel axis, making the top liquid film thinner, and thickening the bottom film. As the Bond number approaches unity, the cross-sectional shape of the bubble deviates significantly from a circular shape, due to flattening of the bottom meniscus. The simulations demonstrate the existence of a cross-stream film flow that drains liquid out of the top film and drives it towards the bottom film region. This drainage flow causes inclination of the bubble, with a larger inclination angle along the bottom plane of the bubble than the top. As buoyancy becomes even more significant, draining flows become less effective and the bubble inclination reduces. A theoretical model for the liquid film thickness and bubble speed is proposed embedding dependencies on both capillary and Bond numbers, which shows good agreement with the reported numerical results. Inertial forces tend to shrink the bubble cross-section and further lift the bubble above the channel centreline, so that the bottom film thickness increases significantly with the Reynolds number, whereas the top film thickness is less sensitive to it.

Experimental study of the evolution of water-entry cavity bubbles behind a hydrophobic sphere

This paper describes an experimental investigation of the cavity evolution and shedding wake behind a hydrophobic sphere during the water-entry process. Two distinct shedding phenomena are confirmed by varying the impact velocity and sphere size: regular air-bubble shedding and unstable air-cloud shedding. Both of these modes are highly dependent on the Weber and Bond numbers. Under the air-bubble shedding mode, approximately periodic big bubble shedding and low-frequency oscillation signals are observed. The relationship between big bubble shedding events and the corresponding acoustic signals is derived, and an empirical method for predicting the shedding period is proposed. The in-phase relationship between small bubble shedding and cavity rippling is confirmed, and we refer to the cavity shedding phenomenon as “acoustic” shedding. Unlike the observations of air-bubble shedding, the air-cloud shedding mode produces a group of disordered small bubbles from the rear of the cavity. Moreover, the cavity seal type has a significant effect on the cavity shedding mode. A deep seal always promotes the onset of air-cloud shedding, whereas surface seals with relatively low Bond numbers result in the air-bubble shedding mode. A surface seal suppresses resonance in the cavity volume. By observing the cavity motion, we find that air-cloud shedding is always accompanied by severe cavity resonance and a rapid decrease in cavity length. Under the air-bubble shedding mode, the cavity motion exhibits relatively weak oscillations.

Submersion of impacting spheres at low Bond and Weber numbers owing to a confined pool

We numerically investigate the mechanism resulting in fate change of a hydrophobic sphere impacting onto a confined pool, that is, at the same impact speed, it does not submerge in a wide pool but does in a narrow pool. We find that the reflection of the impact-induced gravity-capillary waves from the pool boundary is responsible for this phenomenon. In particular, the return of the wave to the symmetry axis may coincide with the rising of the impacting sphere to the water surface, which corresponds to the critical conditions of the fate change. Moreover, for the spheres at the onset of submersion in a wide pool, our analysis suggests that this scenario also accounts for an interesting observation in the numerical simulations. That is, the effective pool size $S_{c}$ , beyond which the submersion of impacting spheres is no longer affected by the pool size $S$ , is mainly dependent on the sphere diameter, no matter whether the surface waves are the capillary or gravity waves. For $S<S_{c}$ , two important pool sizes ( $S_{w,1}$ and $S_{w,2}$ , and $S_{w,2}\geqslant S_{w,1}$ ) are identified at the same impact speed, and the sphere submersion takes place at $S_{w,1}\leqslant S\leqslant S_{w,2}$ . Based on the flow features identified in simulations, a scaling law is proposed to correlate the Weber number and Bond number with $S_{w}$ . The theoretical prediction is shown to agree well with the numerical results.

Investigation of bubble formation and its detachment in shear-thinning liquids at low capillary and Bond numbers

In the present paper, the formation of an air bubble in a shear-thinning non-Newtonian fluid was investigated numerically. For modeling, an algebraic volume of fluid (VOF) solver of $$\hbox {OpenFOAM}^\circledR $$ was improved by applying a Laplacian filter and was evaluated using the experimental results from the literature. The enhanced solver could compute the surface tension force more accurately, and it was important especially at low capillary and Bond numbers due to the dominance of surface tension force relative to the other forces. The adiabatic bubble growth was simulated in an axisymmetric domain for $$\hbox {Bo}=0.05,0.1,0.5$$ and $$\hbox {Ca}=10^{-1},10^{-2},10^{-3},10^{-4}$$ , and the bubble detachment time and volume were examined. According to the results, for Newtonian fluids, there is a critical capillary number for a given Bond number, and for lower values of this critical number, no difference is observed between the bubble detachment volumes and also bubble detachment times. Similarly, the results indicated that for non-Newtonian fluids, if apparent capillary number (obtained by apparent viscosity) is less than the critical capillary number, the detachment volume is the same as the corresponding Newtonian case. The velocity of the bubble during its formation in Newtonian and shear-thinning fluids was also studied. Moreover, the bubble formation and detachment characteristics such as instantaneous contact angle and necking radius were investigated for Newtonian and non-Newtonian liquids, and the shear-thinning effect was examined as well. The results indicated that changing the ambient fluid to a non-Newtonian liquid has no effect on the trend of the contact angle; however, the minimum contact angle has a higher value when the shear-thinning effect increases. Also, the variation of neck radius with the time until detachment was presented by the power relation $${R}_{\mathrm{n}} =\left( {{t}_{\mathrm {det}} -{t}} \right) ^{{\eta }}$$ , and the effect of shear thinning on its exponent $${\eta }$$ was investigated.

Investigation of the evaporation heat transfer mechanism of a non-axisymmetric droplet confined on a heated micropillar structure

In scaling computer performance, closely integrating heterogeneous components in a single system-in-package assembly provides faster signal communication and a tighter footprint – but also generates more heat: state-of-the-art microelectronic devices can produce more than 1 kW cm−2. The next generation high-powered electronics will need two-phase liquid cooling, such as droplet evaporation, which utilizes the latent heat of vaporization to remove excessive heat. Compared to traditional single-phase cooling techniques, two-phase cooling offers both high efficiency and an exceptionally high heat dissipation rate. Although droplet evaporation has been explored for over a hundred years, many fundamental transport mechanisms are still not well understood. For example, most droplet evaporation studies focused only on spherical droplets, which possess a uniform curvature, κ (for a low Bond Number), along the meniscus interface. Evaporation from a non-spherical droplet, due to the change in perimeter-to-area ratio and the meniscus curvature, exhibits very different interfacial mass transport features from a spherical droplet. In particular, a higher perimeter-to-solid-liquid-area ratio will yield a larger fraction of thin film region and therefore a smaller thermal resistance in an evaporating droplet, while a high local curvature will facilitate a stronger local vapor diffusion rate. In this study, we develop a numerical model to investigate the evaporation behavior of asymmetrical microdroplets suspended on heated porous micropillar structures. We explore the equilibrium profiles and mass transport characteristics of droplets with circular, triangular, and square contact shapes, using the Volume of Fluid (VOF) method, and we use a simplified Schrage model [1] to study the evaporative mass transport at the liquid-vapor interface. The results show that microdroplets evaporating on a triangular substrate possess 12.8% smaller effective film thickness compared to that on a circular substrate, due to a longer length of the contact line. During the evaporative heat transfer process, the triangular-based droplets also exhibit the smallest temperature difference between the droplets’ solid-liquid interface and ambient temperature, which leads to a higher heat transfer coefficient (21% larger than a spherical droplet at a supplied heat flux of 500 W/cm2). When the supplied heat increases to a higher value (e.g., 1000 W/cm2), the shape effect becomes less significant where the diffusion resistance is dominated by the liquid-vapor interfacial temperature instead of the meniscus curvature.

Investigation of Two-Phase Flow in a Hydrophobic Fuel-Cell Micro-Channel

This paper presents a quantitative visualization study and a theoretical analysis of two-phase flow relevant to polymer electrolyte membrane fuel cells (PEMFCs) in which liquid water management is critical to performance. Experiments were conducted in an air-flow microchannel with a hydrophobic surface and a side pore through which water was injected to mimic the cathode of a PEMFC. Four distinct flow patterns were identified: liquid bridge (plug), slug/plug, film flow, and water droplet flow under small Weber number conditions. Liquid bridges first evolve with quasi-static properties while remaining pinned; after reaching a critical volume, bridges depart from axisymmetry, block the flow channel, and exhibit lateral oscillations. A model that accounts for capillarity at low Bond number is proposed and shown to successfully predict the morphology, critical liquid volume and evolution of the liquid bridge, including deformation and complete blockage under specific conditions. The generality of the model is also illustrated for flow conditions encountered in the manipulation of polymeric materials and formation of liquid bridges between patterned surfaces. The experiments provide a database for validation of theoretical and computational methods.

The water entry of multi-droplet streams and jets

Water entry has been studied for over a century, but few studies have focused on multiple droplets impacting on a liquid bath sequentially. We connect multi-droplet streams, jets and solid objects with physical-based scaling arguments that emphasize the intrinsically similar cavities. In particular, the cavities created by the initial impact of both droplet streams and jets on an initially quiescent liquid pool exhibit the same types of cavity seal as hydrophobic spheres at low Bond number, some of which were previously unseen for jets and droplet streams. Low-frequency droplet streams exhibit an additional three new cavity seal types unseen for jets or solid spheres that can be predicted with a new non-dimensional frequency. The cavity depth and cavity velocity for both droplet and jet impact are rationalized by an energy scaling analysis and the Bernoulli equation.

Simulation of deformation and fragmentation of a falling drop under electric field

Physical properties and especially the size of drops are important parameters in many industrial and medical applications. High voltage electric field is one of the effective means to control the final size of drops during the fabrication process which could greatly influence the final size of the product. Therefore a detailed study of electric field effect on a liquid drop is very important. In this work deformation and fragmentation of a falling droplet under gravity and electric force have been studied numerically. Electric force is used as an effective external controlling mechanism to influence the deformation of a drop. The three-dimensional deformation of a falling droplet is studied numerically using the open-source volume-of-fluid solver, Gerris. The numerical results are compared with previous analytical, experimental and numerical data and great agreements between the results are obtained. The results are presented for broad range of Bond numbers (Bo) from low Bond number (drop with small deformation) to large Bond number (drop breakup and fragmentation). The results revealed that the electric field can be used as a powerful controlling tool in delaying and expediting the falling drop breakup process. The results also showed that falling drop deforms severely by increasing Bo number which leads to the breakup and fragmentation compared to the cases of low Bo number in which the drop deforms mildly without breakup.

Detailed finer features in spectra of interfacial waves for characterization of a bubble-laden drop

This article describes unexplored details of the intriguing spectral manifestation of the small-amplitude waves at the surfaces of a bubble-laden drop. Its natural frequencies of interfacial pulsation reveal a non-trivial variation with the position of the cavity inside the liquid. This configurational dependence of spectra is calculated for arbitrary location of the void by using a novel approach under low capillary number and low Bond number limits. The analysis is based on expansion in two sets of basis functions where their mutual transformations are utilized to enforce interfacial boundary conditions. The obtained results quantify a few important features which have both scientific and technological significance. For a concentric geometry, the inherent azimuthal degeneracy makes the frequencies for a number of vibrational modes exactly the same. For an eccentric position of the bubble, however, this degeneracy disappears, creating small deviations in the spectral values corresponding to different azimuthal modes. Such behaviour is akin to fine-structure split in an atomic system, where different quantum numbers ensure small deviation in energy levels of the states. The formulated mathematical procedure can determine the individual frequency values for the interfacial oscillation even if these are grouped closely together in bands. The paper shows how the number of fine structures inside a band and their specific values can be exploited to predict the size and position of the cavity in an opaque drop without any direct visualization of its interior.

Influence of operating conditions and liquid phase viscosity with volume of fluid method on bubble formation process

In this paper, a numerical study of air bubble formation and detachment in water (S-1) and in glycerin solutions (S-2 and S-3) was investigated using the volume of fluid (VOF) method. The full formation process was studied under the effect of operating conditions such as wettability of the orifice plate (static contact angle θS), orifice diameter and orifice air velocity. In addition, the influence of the viscosity of the liquid phase was examined. The numerical simulations were carried out for different orifice velocities satisfying the quasi-static bubble growth condition at low Capillary and Bond numbers. Under such conditions, the surface tension effect is dominant over viscous drag and buoyancy effects. The bubble growth at different instants predicted by the VOF simulation was experimentally validated in water. During the expansion/elongation stage significant bubble shape oscillations have been observed by the simulation. In water, bubble shape is mainly dominated by inertial and surface tension forces, and the influence caused by the viscous drag force could be neglected. However, when the viscosity of the solution increases, the bubble shape oscillations can be reduced due to the effect of viscous drag force.

Experimental study of bubbling regimes on submerged micro-orifices

The dynamic bubble behaviors on submerged micro-orifices are investigated experimentally. The orifice diameters are within the range of 0.11–0.24mm, and the range of gas flow rate is between 0.167 and 25ml/min. The experimental data were recorded using a high-speed camera, and images were analyzed with the help of MATLAB. The bubble formation processes on the submerged micro-orifices show some new and different dynamic behaviors, such as multi-bubble coalescence, and the alternate appearance of paring and single bubbles. At low Bond numbers (variation in the orifice radii), with an increase in Weber number (variation in the gas flow rate), a bubble first emerges during a single period stage followed by a multi-bubble coalescence stage. The bubble coalescence processes can be divided into five cases according to the number of coalescing bubbles. With an increase in the gas flow rate, it was found that the complete bubble detachment time and the detachment volume increase. Meanwhile, based on an analysis of the bubble aspect ratio and rising velocity, departing bubbles were shown to suffer stronger oscillations. At a higher Bond number, a bubble first appears in a single form, followed by a period of double bubble behaviors, and back to a period of single form. Finally, a bubbling regime map was constructed to describe the dynamic behaviors of bubbles under the conditions of different Weber and Bond numbers.