Small Weber Numbers Research Articles

Oriented bouncing of droplets with a small Weber number on inclined one-dimensional nanoforests.

Asymmetric superhydrophobic structures with anisotropic wettability can achieve directional bouncing of droplets and thus can have applications in directional self-cleaning, liquid transportation, and heat transfer. To achieve convenient large-scale preparation of asymmetric superhydrophobic surfaces, inclined nanoforests are prepared in this work using a technique of competitive ablation polymerization, which allows the control of the inclined angles, diameters, and heights of the nanostructures. In this study, such asymmetric structures with the smallest dimension (230 nm diameter) known are achieved by a simple etching method to guide droplet unidirectional bouncing. With such nanoforests, the mechanism of droplet bouncing on their surface is investigated, and controllable droplet bouncing over a long distance is achieved using droplets with a low Weber number. The proposed structure has a promising future in directional self-cleaning, liquid transportation and heat transfer.

Experimental and Numerical Investigation on the Dynamics of Impacting Droplet Spreading at Small Weber Numbers

The dynamic of droplet spreading on a free-slip surface was studied experimentally and numerically, with particularly interest in the impacts under relatively small droplet inertias (We≤30). Our experimental results and numerical predictions of dimensionless droplet maximum spreading diameter βmax agree well with those of Wildeman et al.’s widely-used model at We>30. The “1/2 rule” (i.e., approximately one half of the initial kinetic energy Ek0 finally transferred into surface energy) was found to break down at small Weber numbers (We≤30) and droplet height is non-negligible when the energy conservation approach is employed to estimate βmax. As We increases, surface energy and kinetic energy alternately dominates the energy budget. When the initial kinetic energy is comparable to the initial surface energy, competition between surface energy and kinetic energy finally results in the non-monotonic energy budget. In this case, gas viscous dissipation contributes the majority of the dissipated energy under relatively large Reynolds numbers. A practical model for estimating βmax under small Weber numbers (We≤30) was proposed by accounting for the influence of impact parameters on the energy budget and the droplet height. Good agreement was found between our model predictions and previous experiments.

A lattice Boltzmann study on the bouncing behavior of equal-sized droplet collision

The bouncing behavior of equal-sized droplet collision is simulated by the recent multiphase lattice Boltzmann model with self-tuning equation of state. The nonmonotonic coalescence-bouncing-coalescence transition is successfully reproduced. The effects of Weber number, Ohnesorge number, liquid-to-gas density ratio, and impact factor are investigated. It is found that when the Reynolds number or Ohnesorge number is fixed, the nonmonotonic coalescence-bouncing-coalescence transition can be observed as gradually increasing the Weber number. The increase in the Ohnesorge number is beneficial to the occurrence of the bouncing behavior and leads to the increase in the largest Weber number for the bouncing behavior. The lowest Ohnesorge number for the bouncing behavior is approximately 0.2. Considering that the bouncing behavior is caused by the resistance effect of the gas film between droplets, the decrease in the liquid-to-gas density ratio can promote the bouncing behavior and thus expand the range of the corresponding Weber number. For the off-center collision, the increase in the impact factor can trigger the coalescence-bouncing transition under both relatively small and large Weber numbers. For the coalescence-bouncing transition with a relatively large Weber number, the phase diagram of the collision outcome is in qualitative agreement with the prediction by the previous theoretical model.

Experimental analysis on the evaporator startup behaviors in a trapezoidally grooved heat pipe

The startup behaviors of evaporation heat transfer in a trapezoidal groove are experimentally studied by using an axial heat pipe with trapezoidal grooves. The effects of heat load and inclined angles on the evaporation startup behaviors are analyzed and discussed. The startup regime diagram is provided to quantitatively describe the behaviors of evaporation heat transfer based on the Weber number and Bond number. The results indicate that, only the gradual startup is observed for the evaporation heat transfer in trapezoidal grooves, however the gradual startup and overshoot startup is experienced in sequence in the tilt condition as the heat load increases. The appearance of evaporation startup regime transition from the gradual startup to overshoot startup is induced by the evaporation regime transition from the fin-film evaporation to the corner-film evaporation inside the evaporator section. The gradual startup occurs for a small Weber number and the overshoot startup occurs for a large Weber number. Increase in Bond number introduces a larger transition Weber number for the startup regime variation of evaporation heat transfer in a trapezoidal groove. Interestingly, the occurrence of the temperature overshoot of few Celsius degrees can be observed to verify whether the grooved heat pipe is in a horizontal position.

Experimental investigation on impact and spreading dynamics of a single ethanol–water droplet on a heated surface

In the present study, the impact and spreading dynamics of a single droplet on a heated surface are experimentally investigated. Ethanol-water droplets with ethanol concentrations of 0–8 vol% are adopted as the working fluid with a wide range of surface temperature from 75 °C to 450 °C and the Weber number ranges from 7 to 63. The instantaneous droplet evolution after impact is studied via visualization. All the observed phenomena implied that ethanol additive has a significant influence on both impact dynamics and spreading characteristics. A new phenomenon is observed that the addition of ethanol can compensate for the negative influence of small Weber numbers, i.e., some ethanol–water droplets with smaller Weber numbers show better spreading dynamics, higher breakup and atomization possibility but weaker rebound compared with water droplets with higher Weber numbers. It is speculated that droplet impact is no longer solely dependent on the Weber number for ethanol–water droplet, and lower surface tension caused by ethanol additive and Marangoni effect due to concentration gradient along the interface are mainly responsible for this difference. The result also shows that impact behavior exhibits complex dependence on the surface temperature and Weber number, including deposition, rebound, rebound with atomization, breakup with atomization and breakup, while the regime map is significantly affected by ethanol additive.

Effects of Nanoparticle Additives on Spray Characteristics of Liquid Jets in Gaseous Crossflow

Nanofluids are attracting attention as future energy carriers owing to their high performance for improving combustion and heat transfer. In this study, the macroscopic characteristics of nanofluid jets in a subsonic gaseous crossflow were investigated by focusing on the influence of nanoparticle additives on the breakup process. Based on a distribution map of the image grayscale standard deviation, we propose an improved method to process transverse injection shadowgraphs. A simplified model of the transition mechanism from column breakup to surface breakup at a small Weber number was established. The effects of nanoparticles on the jet trajectory and column fracture position were analyzed according to the deviations from the pure liquid. To interpret the effects of the nanoparticles, a new nondimensional parameter was introduced into the empirical correlation of the column fracture position. The results indicated that at low concentrations of nanoparticles, the surface tension of the nanofluids increased slightly, while the viscosity increased significantly (by up to 23%). These changes in the physical properties had little effect on the breakup regimes or jet trajectory. Moreover, the nanoparticles promoted cavitation inside the liquid column, resulting in an additional primary breakup mode for the nanofluids. Consequently, the length of the column fracture was reduced by up to 20% compared with that of the basic fluid.

Faceted and Circular Droplet Spreading on Hierarchical Superhydrophobic Surfaces.

Bouncing of water droplets on the post-built superhydrophobic surfaces was studied. The topography of the surfaces was constituted by PDMS conical posts decorated with ZnO nanoparticles. Droplet impact on surface topographies built of posts with varied configuration and separation was studied under different Weber numbers. Faceted spreading and retraction of droplets were observed. Square-, pentagon-, and hexagon-shaped droplets were registered. It was shown that the nature of droplet spreading depended on both the Weber number and the topography of the post arrays. Even bouncing under small Weber numbers We ≅ 6.5 resulted in the Cassie-Wenzel transitions, starting from the area adjacent to the axis of droplets, and the area exposed to the wetting transitions scaled as [Formula: see text]. During spreading, two main stages were recorded as the kinematic (inertial) stage and the viscous stage. The viscous stage, in turn, appeared as a consequence of two substages governed by various time scaling laws. The faceted triple line was observed for the Cassie-like retraction of droplets.

Numerical Interpretation to the Roles of Liquid Viscosity in Droplet Spreading at Small Weber Numbers.

Droplet impacting a free-slip plane at small Weber numbers (We < 30) was numerically investigated by a front tracking method, with particular emphasis on clarifying the roles of the liquid viscosity and the "left-over" internal kinetic energy in droplet spreading. The most interesting discovery is that there exists a certain range of We in which the maximum diameter rate, D̃m, shows a nonmonotonic variation with the Reynolds number, Re. This non-monotonic variation is owing to the dual role of liquid viscosity in influencing droplet spreading. Specifically, when the initial surface energy is comparable to the initial kinetic energy (the corresponding We is around 10-30), the high strain rates of the droplet internal flow dominate its viscous dissipation at a relatively large Re, while the liquid viscosity dominates the viscous dissipation at a relatively small Re. Furthermore, to unravel the influence of droplet attachment and detachment on droplet spreading, we considered two limiting situations such as full attachment (with no gas film throughout droplet spreading) and full detachment (with a gas film throughout droplet spreading). The results show that the droplet with a gas film tends to generate a stronger vortical motion in its rim, results in a larger left-over kinetic energy, and hence causes a smaller spreading.

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.

Linear stability of confined swirling annular liquid layers in the presence of gas velocity oscillations with heat and mass transfer

The linear stability of confined swirling annular liquid layers is studied in the presence of gas velocity oscillations with heat and mass transfer. The dispersion relation is obtained and is solved using Floquet theory. Results show that parametric instability regions appear when oscillations are considered and the oscillations play a destabilizing role. The forcing frequency controls the location of parametric instability regions and plays a slight destabilizing role. Heat and mass transfer has a stabilizing effect at small Weber numbers but a destabilizing effect at large Weber numbers. In addition, heat and mass transfer promotes the dispersion effect, vanishing the gap between the parametric and Kelvin-Helmholtz (K-H) instability regions. The effect of mode order is complex. The order n=0 is the most unstable. Rotating restrains the instability. The effect of confinement is complex. Confinement plays a stabilizing role at all the range of Weber numbers when ignoring heat and mass transfer. However, confinement promotes the instability at small Weber numbers but suppresses it at large Weber numbers when considering heat and mass transfer.

Direct numerical simulation of droplet breakup in homogeneous isotropic turbulence: The effect of the Weber number

We report the direct numerical simulation of droplet breakup in forced homogeneous isotropic turbulence with a mass-conserving level set method. In particular, the effect of the Weber number on the interface morphology of droplet breakup and the interaction between vortical structures and the interface are addressed. The densities and viscosities are the same for both phases in the present simulations. It is found that with increasing the Weber number, the initially large-scale spherical droplet tends to break down into dispersed small droplets. Compared with the flow local topology in the carrier-phase turbulence, the local topology of the bi-axial strain is suppressed at the statistical stationary state inside the droplet. During the droplet breakup process, the vorticity tends to be tangent to the large-scale interface at the early stage, and subsequently the alignment is mitigated for large Weber numbers, because the small droplets with relatively strong surface tension and small local Weber numbers can resist the deformation induced by local turbulent straining motion in the statistically stationary state.

Experimental analysis of the evaporation regimes of an axially grooved heat pipe at small tilt angles

Using a high-speed digital microscope, visualization experiments of the vapor-liquid two-phase flow and heat transfer in an axially grooved heat pipe are conducted to investigate the evaporation regimes that occur at small tilt angles. The effects of the heat load and tilt angle on these evaporation regimes are analyzed. The evaporation regimes are further quantitatively characterized with a regime diagram developed based on the Bond number and Weber number. The results indicate that, unlike in a horizontal position, the evaporation regime of a grooved heat pipe with a small tilt angle can include corner-film evaporation induced by the combination of gravity and capillary force in addition to pool-surface evaporation and fin-film evaporation. Corner-film evaporation is the characteristic evaporation regime of an axially grooved heat pipe with a small tilt angle, and it exhibits random oscillatory motion of the vapor-liquid interface. Pool-surface evaporation is observed at small Weber numbers, corner-film evaporation occurs at large Weber numbers when the Bond number is approximately greater than 0.003, and fin-film evaporation occurs at medium Weber numbers.

Experimental study on thermo-hydrodynamic behaviors in miniaturized two-phase thermosyphons

In this paper, the thermo-hydrodynamic behaviors in miniaturized two-phase thermosyphons are experimentally investigated by a visualization system. The effects of heat load and channel dimension on the thermo-hydrodynamic behaviors are examined and further quantitatively recognized by a state diagram depending on the Weber number and Bond number. In addition, the vapor–liquid two-phase flow, heat transfer regimes and thermal performances are analyzed and discussed. The results indicate that, differing from the conventional two-phase thermosyphon, the state of working fluid in a miniaturized two-phase thermosyphon is no longer only in a fashion of pool and annular-flow states, but also in a fashion of oscillation state. The oscillation state is the characteristic two-phase flow regime of a miniaturized two-phase thermosyphon and possesses the feature of random formation and oscillatory motion of liquid plug in minichannels. The pool state occurs at small Weber number, the annular-flow state is observed at large Weber number, and the oscillation state takes place at medium Weber number when Bond number is approximately smaller than 0.7. The thermal performance of oscillation state is better than that of pool state but is less powerful than that of annular-flow state. The heat transfer regimes are the film evaporation and condensation combined with thermal conduction in the pool state, the forced evaporation and condensation in the oscillation state, and the film evaporation and condensation in the annular-flow state, respectively.

Isothermal dissolution of small rising bubbles in a low viscosity liquid

The isothermal dissolution of small single rising bubbles in a low viscosity liquid is numerically and experimentally studied. We propose a tracking-interface numerical method to analyze the dissolution of single bubbles rising in an isothermal liquid bath in the limit of small Weber numbers. The key elements of the method are the use of a frame of reference moving with the bubble and the application of different meshes to solve the mechanical and mass-diffusion problems. In addition, the gas concentration in the atmosphere over the bath, determined by mass balance of species in the gas column, is shown to be an essential component of the global dissolution problem in steady regime. Comparison with small oxygen bubbles rising in water has been carried out with remarkable agreement.

Gas–liquid–liquid three-phase flow pattern and pressure drop in a microfluidic chip: similarities with gas–liquid/liquid–liquid flows

We report a three-phase slug flow and a parallel-slug flow as two major flow patterns found under the nitrogen-decane-water flow through a glass microfluidic chip which features a long microchannel with a hydraulic diameter of 98 μm connected to a cross-flow mixer. The three-phase slug flow pattern is characterized by a flow of decane droplets containing single elongated nitrogen bubbles, which are separated by water slugs. This flow pattern was observed at a superficial velocity of decane (in the range of about 0.6 to 10 mm s(-1)) typically lower than that of water for a given superficial gas velocity in the range of 30 to 91 mm s(-1). The parallel-slug flow pattern is characterized by a continuous water flow in one part of the channel cross section and a parallel flow of decane with dispersed nitrogen bubbles in the adjacent part of the channel cross section, which was observed at a superficial velocity of decane (in the range of about 2.5 to 40 mm s(-1)) typically higher than that of water for each given superficial gas velocity. The three-phase slug flow can be seen as a superimposition of both decane-water and nitrogen-decane slug flows observed in the chip when the flow of the third phase (viz. nitrogen or water, respectively) was set at zero. The parallel-slug flow can be seen as a superimposition of the decane-water parallel flow and the nitrogen-decane slug flow observed in the chip under the corresponding two-phase flow conditions. In case of small capillary numbers (Ca ≪ 0.1) and Weber numbers (We ≪ 1), the developed two-phase pressure drop model under a slug flow has been extended to obtain a three-phase slug flow model in which the 'nitrogen-in-decane' droplet is assumed as a pseudo-homogeneous droplet with an effective viscosity. The parallel flow and slug flow pressure drop models have been combined to obtain a parallel-slug flow model. The obtained models describe the experimental pressure drop with standard deviations of 8% and 12% for the three-phase slug flow and parallel-slug flow, respectively. An example is given to illustrate the model uses in designing bifurcated microchannels that split the three-phase slug flow for high-throughput processing.

A quasi-static model of drop impact

We develop a conceptually simple theoretical model of non-wetting drop impact on a rigid surface at small Weber numbers. Flat and curved impactor surfaces are considered, and the influence of surface curvature is elucidated. Particular attention is given to characterizing the contact time of the impact and the coefficient of restitution, the goal being to provide a reasonable estimate for these two parameters with the simplest model possible. Approximating the shape of the drop during impact as quasi-static allows us to derive the governing differential equation for the droplet motion from a Lagrangian. Predictions of the resulting model are shown to compare favorably with previously reported experimental results.

Axisymmetric annular curtain stability

A temporal stability analysis was carried out to investigate the stability of an axially moving viscous annular liquid jet subject to axisymmetric disturbances in surrounding co-flowing viscous gas media. We investigated in this study the effects of inertia, surface tension, the gas-to-liquid density ratio, the inner-to-outer radius ratio and the gas-to-liquid viscosity ratio on the stability of the jet. With an increase in inertia, the growth rate of the unstable disturbances is found to increase. The dominant (or most unstable) wavenumber decreases with increasing Reynolds number for larger values of the gas-to-liquid viscosity ratio. However, an opposite tendency for the most unstable wavenumber is predicted for small viscosity ratio in the same inertia range. The surrounding gas density, in the presence of viscosity, always reduces the growth rate, hence stabilizing the flow. There exists a critical value of the density ratio above which the flow becomes stable for very small viscosity ratio, whereas for large viscosity ratio, no stable flow appears in the same range of the density ratio. The curvature has a significant destabilizing effect on the thin annular jet, whereas for a relatively thick jet, the maximum growth rate decreases as the inner radius increases, irrespective of the surrounding gas viscosity. The degree of instability increases with Weber number for a relatively large viscosity ratio. In contrast, for small viscosity ratio, the growth rate exhibits a dramatic dependence on the surface tension. There is a small Weber number range, which depends on the viscosity ratio, where the flow is stable. The viscosity ratio always stabilizes the flow. However, the dominant wavenumber increases with increasing viscosity ratio. The range of unstable wavenumbers is affected only by the curvature effect.

Fully implicit time discretization for a free surface flow problem

AbstractOne of the challenges in the numerics of free surface flows is the coupling of the flow field to the geometry of the domain. The most simple approach is an explicit decoupling, i.e. computing the flow field with geometrical information of a prior time step and then updating the geometry. This widely used approach leads to a severe CFL condition of the type $\Delta t \le C\sqrt{{\rm We}}\;h^{3 \over 2}$, which may prescribe infinitesimally small time step sizes in the interesting case of a small Weber number (i.e. high surface tension). A semi‐implicit approach utilizing the fact that $x^{n+1} = x^n + \Delta t\, u^{n+1}_{|\Gamma}$, where xk is a parametrization of the capillary boundary Γ, is also available [1]. This approach can be proven to be unconditionally stable but is of first order only. It also suffers from relatively strong numerical dissipation.We present a fully implicit approach using a backward differentiation formula to achieve a time discretization method that is of second order and only minimally dissipative. A numerical example of an oscillating drop showing very low numerical dissipation and second order convergence as well as numerical evidence for the stability of the method is presented. Since the method requires the solution of a highly nonlinear coupled system, possible preconditioners for this system are discussed, including a lower order decoupling. (© 2011 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A microscale model for thin-film evaporation in capillary wick structures

A numerical model is developed for the evaporating liquid meniscus in wick microstructures under saturated vapor conditions. Four different wick geometries representing common wicks used in heat pipes, viz., wire mesh, rectangular grooves, sintered wicks and vertical microwires, are modeled and compared for evaporative performance. The solid–liquid combination considered is copper–water. Steady evaporation is modeled and the liquid–vapor interface shape is assumed to be static during evaporation. Liquid–vapor interface shapes in different geometries are obtained by solving the Young–Laplace equation using Surface Evolver. Mass, momentum and energy equations are solved numerically in the liquid domain, with the vapor assumed to be saturated. Evaporation at the interface is modeled by using heat and mass transfer rates obtained from kinetic theory. Thermocapillary convection due to non-isothermal conditions at the interface is modeled for all geometries and its role in heat transfer enhancement from the interface is quantified for both low and high superheats. More than 80% of the evaporation heat transfer is noted to occur from the thin-film region of the liquid meniscus. The very small Capillary and Weber numbers resulting from the small fluid velocities near the interface for low superheats validate the assumption of a static liquid meniscus shape during evaporation. Solid–liquid contact angle, wick porosity, solid–vapor superheat and liquid level in the wick pore are varied to study their effects on evaporation from the liquid meniscus.

Analyses on a charged electrolyte droplet in a dielectric liquid under non-uniform electric fields

Most of the analyses on the behaviors of a charged electrolyte droplet in a dielectric fluid have been performed based on the model of perfect conductor droplet. However, if the thickness of the electrical double layer (EDL) is not negligibly small with a finite thickness, the perfect conductor model must be corrected to some degrees. In this work, two analyses are performed for an electrolyte droplet in non-uniform electric fields based on the electrokinetic model. Firstly, the exact solution of electric potential is obtained for a spherical electrolyte droplet in a non-uniform electric field (uniform field + general linear field) under the Debye–Huckel approximation. The solution is used to derive the formula of the electrical force exerted on the droplet. The solution is also used for prediction of the first order deformation of droplet in the limit of small Weber number. Secondly, a new method is proposed for obtaining the first order correction to the solution of conductor model. This method is applicable to any deformed droplet shape. The key idea is that the volume charge density inside the thin EDL is integrated in the normal direction to the surface to be treated as the effective surface charge density. From the analysis, important features of the thin EDL are also revealed. When the EDL thickness decreases with O( κ −1), the difference between the droplet surface potential and the bulk potential also decreases with O( κ −1). Therefore, the slope of the electric potential change in the EDL remains as O(1). As a first concrete application, an ellipsoidal electrolyte droplet under a non-uniform electric field is studied.