Droplet Oscillation Research Articles

Nonwetting droplet oscillation and displacement by viscoelastic fluids

Simulations and theory reveal oscillations of droplets when driven by viscoelastic fluids because of their elastic memory effect. The viscoelastic oscillation helps release trapped droplets from hard-to-displace positions.

On a non-linear droplet oscillation theory via the unified method

Presently, the oscillation of a liquid droplet in a dynamically negligible outer medium subject to surface tension and small viscosity is investigated. By using the potential flow assumption, the unified transform method by Fokas is employed to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations (IDEs). This new system depends on one less spatial variable and is now defined on a time-independent domain. Most importantly, the resulting set of equations governs the general droplet oscillation with arbitrarily large deviations from the spherical shape. As the nonlinearity of the above IDE system up to now prevented an analytical solution, the Poincaré expansion technique is employed, retaining terms up to the second order. By decomposing the unknowns into normal modes, these equations are uncoupled and the resulting ordinary differential equations for the mode amplitudes are solved, and the results are compared to those of previous works. It should be stressed that the present analysis is limited to small viscosity, or, in other words, for small Ohnesorge numbers. The reason for this is that, inside of the droplet, a potential flow is assumed and the viscous effect is taken into account only at the droplet surface by the jump condition of momentum. This is only reasonable for a small viscosity and a short time. Otherwise, vorticity is generated at the interface and diffuses toward the inside of the droplet.

Characterization of oscillation amplitude of contact angle during AC electrowetting of water droplets

Most studies on electrowetting (EW) involve the use of AC electric fields, which cause droplets to oscillate in response to the sinusoidal waveform. Oscillation-driven mixing in droplets is the basis for multiple microfluidic applications. Presently, we study the voltage and AC frequency-dependent oscillations of electrowetted water droplets on a smooth, hydrophobic surface. We introduce a new approach towards analyzing droplet oscillations, which involves characterization of the oscillation amplitude of the contact angle (CA). An experimentally validated, fundamentals-based model to predict voltage and frequency-dependent CA oscillations is developed, which is analogous to the Lippmann’s equation for predicting voltage-dependent CAs. It is seen that this approach can help estimate the threshold voltage more accurately, than from experimental measurements of CA change. Additionally, we use a coplanar electrode configuration with high voltage and ground electrodes arranged on the substrate. This configuration eliminates measurement artefacts in the classical EW configuration associated with a wire electrode protruding into the droplet. An interesting consequence of this configuration is that the system capacitance is reduced substantially, compared to the classical configuration. The coplanar electrode configuration shows a reduced rate of CA change with voltage, thereby increasing the voltage range over which the CA can be modulated.

Effect of Superimposed Vibrations on Droplet Oscillation Modes in Prilling Process

This article was aimed to solve an urgent problem of ensuring quality for prilling processes in vibrational prilling equipment. During the research, the need for the application of vibrational prilling to create a controlled impact on the process of jet decay on droplets with the proper characteristics was substantiated. Based on the experimental and theoretical studies of the process of decay of a liquid jet into drops, axisymmetric droplet oscillation modes for the different frequencies were observed. Frequency ranges of transition between modes of decay of a jet into drops were obtained. As a result, the mathematical model of the droplet deformation was refined. The experimental research data substantiated this model, and its implementation allowed determining the analytical dependencies for the components of the droplet deformation velocity. The proposed model explains the existence of different droplet oscillation modes depending on the frequency characteristics of the superimposed vibrational impact. Based on an analytical study of the droplet deformation velocity components, the limit values of the characteristics defining the transition between the different droplet oscillation modes were discovered. Analytical dependencies were also obtained to determine the diameter of the satellites and their total number.

A consistent method for direct numerical simulation of droplet evaporation

A consistent method for direct numerical simulations of evaporation processes is presented. It is implemented into the CFD code FS3D which solves the incompressible Navier-Stokes equations and is based on a Volume-of-Fluid method. In the new framework convection of the fluid and the gaseous phase is conducted first, whereas the physics of the phase change is captured subsequently in a fully consistent loop.We present several validation cases and show, that the method is able to simulate evaporation processes for a large range of ambient conditions, ranging from supercooled droplets to extreme environments with high evaporation rates. Comparisons to analytical models from literature and other experimental investigations are in very good agreement.Then, the method is applied for several three-dimensional direct numerical simulations of evaporating droplets. We present simulations of supercooled droplets in an airflow and compare them to experimental data. Furthermore, an oscillating droplet with evaporation is simulated and evaluated regarding the influence of the phase change on the droplet oscillation behavior. The potential of the method is also demonstrated for a free falling droplet in a high pressure and temperature atmosphere. Vapor fields and temperature regimes are evaluated in this case and show very good agreement with experiments.

Shape oscillations of a viscoelastic droplet suspended in a viscoelastic host liquid

The small-amplitude oscillation of a liquid droplet suspended in an immiscible host liquid is studied, where both liquids are assumed viscoelastic. An analytical characteristic equation is derived, and the damping rate and angular frequency that describe the droplet oscillation are numerically solved. The effect of the properties of the host liquid, including its density, viscosity and elasticity, on the viscoelastic droplet oscillation is examined for the quadrupole mode. The host liquid is found to make the droplet oscillation more complex and the mechanisms behind that are discussed.

Moving surface mesh-incorporated particle method for numerical simulation of a liquid droplet

In this study, a new particle method for simulating the dynamics of a liquid droplet in a two-dimensional space has been developed. The proposed method incorporates a moving surface mesh to represent a deformable free-surface boundary. The domain enclosed by the surface mesh is defined as a liquid volume (droplet), and the outer region is a gas phase with constant pressure. Fluid particles are seeded inside the liquid domain, and also, discrete nodes along the surface mesh are defined as additional computational points. The incompressible flow is solved based on the LSMPS (least squares moving particle semi-implicit) method, where all differential operators are discretized by means of consistent schemes. The stress balance equations are solved along free surfaces, where the surface tension force is directly evaluated by the geometry of the surface mesh at each surface node. As numerical tests, various problems including a hydrostatic pressure problem, circular and square patch tests, droplet oscillations and static droplets suspended by solid walls have been simulated. As a result, the proposed method shows excellent agreement with the reference solutions, even under large free-surface deformations, which verifies the validity of the current developments.

A numerical study of a hollow water droplet falling in air

We numerically study the dynamics of a hollow water droplet falling in the air under the action of gravity. The focus of our study is to investigate the effects of the difference in radii (thickness) of the hollow droplet, gravity and surface tension at the air–water interface on shape oscillations and the breakup dynamics of the hollow droplet. We found that the oscillations of the inner interface (inner air bubble) are mostly periodic, while the outer interface undergoes irregular oscillations due to the interaction with the surrounding air. Increasing the ‘thickness’ of the hollow droplet decreases the amplitude of oscillations which further decays with time for high surface tension. It is observed that for a fixed value of the ‘thickness’ and low surface tension, the hollow droplet undergoes transition from the oscillatory regime to the dripping regime as it falls. The velocity contours are used to explain the behaviour observed in the present study. The deformation and shape oscillations of the hollow droplet are also compared with those observed in the case of a normal droplet of equal liquid volume falling in air.

Oscillation behaviors of oil droplets adhered on the solid surfaces with different wettability in a laminar flow field

The oscillation of oil droplets on surfaces with different wettability in laminar flow field affected the initial movement behaviors of oil droplets, which was studied experimentally and theoretically in the paper. In the initial oscillation stage, the contact line of the oil drop on the substrate did not move, while the center mass and contact angle hysteresis of the droplet have changed. It was found that there were two oscillation patterns for adhered oil drop in the flow water, one was the up-down oscillation, the other was left-right oscillation, which can be transformed into each other. The contact angle evolution of oscillating oil drop was analyzed. It was found that with the increase of water flow velocity, the difference between the maximum and minimum contact angles of adhered oil drops on different surfaces was increasing, and the contact angle of oil drops on oleophobic surfaces underwater was more likely to exceed the advancing contact angle during the oscillation process. The force analysis of the three-phase contact point of the oil drop on the surfaces was carried out, and the oscillation behaviors of the oil drop were explained theoretically. The sensitivity analysis of the factors affecting the oil drop oscillation behaviors was carried out, and the influences of the oil drop height and the surface wettability on the oil drop oscillation frequency and amplitude was obtained.

Effects of hysteresis window on contact angle hysteresis behaviour at large Bond number

Contact angle hysteresis, defined as the difference between advancing and receding contact angles, is an important phenomenon in multiphase flow on a wetting surface. In this study, a modified pseudo-potential lattice Boltzmann (LB) multiphase model with tunable surface tension is proposed, which is further coupled with the geometrical formulation contact angle scheme to investigate the motion of droplets invoking the contact angle hysteresis. We focus on the dynamic behaviour of droplets driven by a body force at the Bond number ranging from 1 to 6, which is defined as the ratio of the body force to the capillary force. The droplet morphology change is examined by varying (i) the Bond number and (ii) the hysteresis window. Results show the droplet morphology evolution can be classified into different stages, including stretch, relaxation, and equilibrium. The droplet oscillation phenomenon at large Bond numbers at the equilibrium stage is observed for the first time. In addition, it is found that such oscillation can lead to the breakup and/or coalescence of droplets when the surface waves spread on the top of the droplet. Furthermore, there is slight oscillation of the normalized length, width and height at the equilibrium stage for the neutral hysteresis window while more dramatic oscillation will appear for the hydrophobic hysteresis window.

Small droplet bouncing on a deep pool

Droplet bouncing on liquid surfaces frequently occurs for low-Weber-number impacts. Previous studies typically used large droplets with oscillation initiated by their creation process but without determining the effects of these oscillations. Here, we use small droplets, providing the means to reduce oscillations to show that the probability of the droplet bounce does not depend on the droplet oscillations. The time from the moment of contact to the maximum penetration depth was found to be independent of the Weber number for droplets of fixed diameter but increased with an increase in diameter. Both the maximum penetration depth and the maximum rebound height increased monotonically with the Weber number. A simple model predicting the maximum penetration depth was proposed and validated through comparison with experimental data.

Oscillation and self-propulsion of Leidenfrost droplets enclosed in cylindrical cavities.

Leidenfrost droplets can be considered as soft engines capable of directly transforming heat into mechanical energy. Despite remarkable advancements in understanding the propulsion of Leidenfrost droplets on asymmetric structures, the complex dynamics of droplets in enclosed structures is not fully understood. To address this fundamental gap, we investigated the dynamics of Leidenfrost droplets restricted by metal disks. The disk alters the accumulation and release of the vapour generated by the droplet, and substantially changes its dynamic characteristics. Our experiments reveal the formation of oscillating multi-lobed structures when restricting the droplet within a disk. In comparison, patterning offset radial grooves on the surface of the disk rectifies the vapour flow and facilitates the self-propulsion of the droplet along the edge of the disk. Our work offers opportunities for developing soft and short-living actuators, which can operate at high temperatures.

Interface oscillation of droplets upon impact on a heated surface in the Leidenfrost state

When a droplet impacts on a heated surface above the dynamic Leidenfrost temperature, it bounces off the surface by levitating on a vapour layer. The vapour layer is of great importance for impact morphology and heat transfer. In this study, by focusing on the vapour layer, we identify that the bottom interface of the droplet, rather than remaining stable, may oscillate rapidly during the rebounding stage of the droplet in the Leidenfrost state. The interface oscillation refers to repeated formation and discharge of dimple depression at the bottom of the droplet. The results of heat transfer analysis confirm that the interface oscillation can remarkably affect the flow in the vapour layer and the heat transfer between the droplet and the heated surface. The effects of important dimensionless numbers controlling the process are analysed, including the Jakob number, the Ohnesorge number, and the Weber number.

Ionic Tuning of Droplet Motion on Water Surface.

Herein, the oscillation of an oil droplet on the surface of water is studied. The droplet contains an anionic surfactant that can react with the cations present in water. The oscillation starts after a random motion, and the oscillation pattern apparently depends on the cation species in the water phase. However, a common pattern is included. The cation species only affects the amplitude and frequency and sometimes perturbs the regular pattern owing to the instability at the oil/water interface. This common pattern is explained by a simple model that incorporates the surfactant transport from the droplet to the surrounding water surface. The dependency of the amplitude and frequency on cation species is expressed quantitatively by a single parameter, the product of the amplitude and square of frequency. This parameter depends on the cationic species and can be understood in terms of the spreading coefficient. The simple model successfully explains this dependency.

Oscillatory motion and merging responses of primary and satellite droplets from Newtonian liquid jets

Numerical simulations of laminar Newtonian liquid jets are presented that divulge several breakup and merging responses for both primary and satellite droplets. Using an axisymmetric VOF model, we predict the presence of surface oscillations of droplets disintegrated from jets, which at times lead to merging of droplets when travelling downstream. Our simulations indicate that when primary droplets undergo merging, they exhibit characteristics such as (i) permanent and (ii) partial coalescence. We find that the partial coalescence of droplets can lead to a reflexive-like separation and then show re-merge responses. The formation of an air-bubble during the merging of binary droplets is predicted. Characteristics such as forward, rear, and simultaneous merging patterns of satellites with primary droplets are captured in our numerical simulations. Furthermore, a new merging pattern is reported wherein, satellites formed from the swellings of the jet can be absorbed back into the moving core of the liquid jet.

Large-eddy simulation of an air-assisted liquid jet under a high-frequency transverse acoustic forcing

The present contribution falls in the scope of high-frequency combustion instabilities occurring in liquid rocket engines. Under subcritical operating conditions, numerical simulations have to render the effects of acoustic waves on the atomisation of liquid jets since these may impact the stability of the engine. Therefore, the present contribution aims at evaluating the ability of a particular numerical strategy adapted to the simulation of two-phase flows to render these interaction mechanisms. The selected strategy is based on the coupling between a diffuse interface method for the simulation of large liquid structures, and a kinetic-based Eulerian model for the description of droplets. The numerical simulation of an air-assisted liquid jet submitted to a transverse acoustic modulation is performed. The flattening of the liquid core under acoustic constraints is retrieved and induces an intensification of its stripping. In addition, thanks to appropriate coupling source terms, the modification of the spray shape as well as periodic oscillations of droplets are retrieved. The numerical strategy is thus proved to be adapted to deal with atomised liquid jets under transverse acoustic modulation and can be used for future numerical studies of high-frequency combustion instabilities.

Internal flow during mixing induced in acoustically levitated droplets by mode oscillations

In this paper, we describe a mixing method with mode oscillation on the internal flow field of a levitated droplet. The effect of internal flow on the mixing performance of droplets acoustically levitated via ultrasonic phased arrays remains unclear. To better understand the mixing mechanism of a levitated droplet, clarifying the effect of the internal flow field on droplet mixing from mode oscillation during acoustic levitation is necessary. We used a 50 wt. % glycerol aqueous solution with 6th mode oscillation. We applied particle image velocimetry (PIV) to study the internal flow fields under interfacial oscillation. The PIV results indicated that the visualized flow field enhanced mixing performance with increasing Reynolds number. We demonstrated the nonlinear characteristics of droplet mixing compared to potential flow. The nonlinearity of the droplet oscillation was driven by the nonlinear acoustic field exerted on the levitated droplet. Mode oscillation on the droplet surface induced a pressure gradient and caused internal flow in the droplet. The pressure gradient in the droplet from the interfacial oscillation was quantitatively analyzed. Pressure induced by the interfacial oscillation, which can be roughly ten times larger than the hydrostatic pressure in the droplet, drastically enhanced the mixing performance in the droplet. Our experimental findings provide deeper physical insights into noncontact fluid manipulation for potential lab-in-a-drop applications.

Droplet oscillation mechanism and its free surface behavior on impacting a heated hydrophobic surface at low Weber numbers

Oscillation mechanism and free surface behavior of micro-droplets (300, 400, and 500 µm in diameter), impinging on a dry, hydrophobic heated surface was investigated using the volume of fluid (VOF) method. It was observed that the droplets exhibit unique oscillation characteristics at low Weber numbers for different contact angles and surface temperatures. At higher contact angles, droplet deformation was retarded, and it tends to retain its shape. This increased its chances of bouncing away from the surface. With increasing surface temperature, the tendency of the droplet to bounce away from the surface also increased. The oscillation behavior was significantly influenced by droplet size, contact angle with the surface and by surface temperature. Larger sized droplets had lower oscillation frequency. An increase in contact angle or surface temperature led to the lowering of the oscillation frequency of droplets. However, droplet Weber number was found to have no significant effect on oscillation frequency. At these low value of Weber numbers, all droplet sizes investigated showed slightly under-damped harmonic motion with amplitude exponentially decaying toward zero.

Experimental study on fusion and break-up motion after droplet collision

Abstract The interactions between droplets have an important influence on the atomization of liquid fuel, the combustion efficiency, and the reduction of particulate matter emissions for an engine. For this reason, this paper presents results from an experimental study on the coalescence and break-up of droplets after collision. According to the shape and parameters of the droplets at different times after the collision of the droplets was captured by a high speed camera, analysis was done for the following effects of droplet collisions: the collision-coalescence motion for the collision between the droplets, the change history of the dimensionless length-to-width ratio of the oscillation motion, the critical size ratio of the breakup motion, and the liquid physical properties of the particles. The results show that the droplets collide and exhibit two forms of coalescence oscillation and break-up: for oscillating motion, at higher droplet collision velocities and dimensionless size ratios, there will be a larger dimensionless length-to-width ratio for the droplet oscillation; for the break-up motion, at higher collision velocities, there will be lower dimensionless size ratios, and lower liquid surface tension, shorter times over which the droplet breaks, and facilitated droplet break-up. The research results presented here can be used for atomization in engine cylinder, increasing the gas/liquid contact area and enhancing the combustion efficiency of gas/liquid heat transfer to improve the combustion efficiency of the engine.

Computational and experimental analysis of droplet transportation/jetting behaviours driven by thin film surface acoustic waves

A Coupled Level Set Volume of Fluid (CLSVOF) approach has been applied to investigate severe deformation/transportation/jetting behaviours of sessile droplet driven by thin-film surface acoustic waves (SAW) devices. For validation of this computational method, a series of experimental studies of droplet transportation/jetting were performed using ZnO/Si thin film based SAW devices with resonant frequencies ranging from 64.49 MHz to 271.36 MHz. Good agreements between the computational and experimental results showed the capability of the developed CLSVOF method in modelling complex acoustofluidics phenomena such as significant internal streaming, pumping and jetting of the droplet driven by the propagating SAW.Results obtained from the computational model are used to clarify the fluidic mechanisms of droplet oscillation and wobbling behaviours during transportation. Numerical results reveal the liquid streaming patterns and airflow velocity field around the droplet at different stages of transportation/jetting process. Effects of droplet volume, the resonant frequency of SAW devices and applied SAW power on droplet transportation/jetting were investigated both theoretically and experimentally. In particular, comparisons between experimental and computational results showed that the model predicted well the minimum RF power to start droplet pumping and jetting at various resonant frequencies.