Oblate Droplets Research Articles

Role of surface charge convection on oblate droplets in different conductivity regimes

This paper presents a robust numerical model to simulate the electro-hydrodynamic flows of a neutrally buoyant liquid droplet suspending in another liquid for varying electrical conductivities ranging from near-dielectric to highly conductive fluids. The effect of such conductivity on the interfacial charge transport in the droplets has been investigated. The model is first validated with the theory of electro-rotation corresponding to the strong electric field. The results are compared with the theoretical predictions from the Quincke theory. The angular velocities at different electric field ratios agree well with theoretical predictions. Furthermore, droplet investigations are performed in distinct conductivity regimes with the electric Reynolds number ranging from 10−2 to 104. The findings reveal that conductivity influences the evolution of diverse droplet shapes in the corresponding regimes through surface charge convection. We observe prolate shapes at very low electric conductivities, while larger conductivities of droplets and suspending media lead to oblate drop shapes. With increasing electrical conductivity of the droplet and the medium, we observe the onset of distinct droplet shapes similar to the existing literature. The mechanism for the onset of different regimes is adequately explained by quantifying surface properties like tangential stress and velocity.

Extremely large water droplet impact onto a deep liquid pool.

Most studies of droplet impact on liquid pools focus on droplet diameters up to the capillary length (0.27cm). We break from convention and study extremely large water droplets (1 to 6cm diameter) falling into a pool of water. We demonstrate that the depth and width of the cavity formed by large droplet impact is greatly influenced by the deformed shape of the droplet at impact (i.e., prolate, spherical, and oblate), and larger droplets amplify this behavior by flattening before impact. In particular, the maximum cavity depth is a function of the Froude number and axis ratio of the droplet just before impact. Further, the cavity depth is more dependent on the droplet height than width, and the maximum cavity diameter is independent of the droplet height. In general, we observe that more oblate droplets result in decreasing cavity depths for a fixed liquid volume. This is because an increase in horizontal droplet diameter results in a reduced impact energy flux and therefore reduced cavity depth.

Heating and evaporation of a mono-component spheroidal droplet with non-uniform surface temperature

A new mathematical model for spheroidal droplet heating and evaporation is proposed. This model takes into account the effect of liquid finite thermal conductivity and is based on the previously obtained analytical solution for the vapour mass fraction at the droplet surface and a new correlation for the convective heat transfer coefficient incorporated into the numerical code. The heat transfer equation in the liquid phase is solved numerically using the finite-element heat transfer module of COMSOL Multiphysics. It is shown that the lifetime of spheroidal (prolate and oblate) droplets is shorter than that of spherical droplets of the same volume. The difference in the lifetimes of spheroidal and spherical droplets, predicted by the new model, is shown to increase with increasing aspect ratios for prolate droplets and decreasing aspect ratios for oblate droplets. As in the case of stationary spherical droplets, the d2-law is shown to be valid for spheroidal droplets after the completion of the heat-up period. The predictions of this model agree with experimental observations. The duration of the heat-up period is shown to decrease with increasing aspect ratios for prolate droplets and decreasing aspect ratios for oblate droplets. The maximal surface temperatures are predicted near the regions where the surface curvature is maximal. The aspect ratios are shown to be weak functions of time, in agreement with experimental observations.

Marangoni instability in oblate droplets suspended on a circular frame

We study theoretically internal flows in a small oblate droplet suspended on the circular frame. Marangoni convection arises due to a vertical temperature gradient across the drop and is driven by the surface tension variations at the free drop interface. Using the analytical basis for the solutions of Stokes equation in coordinates of oblate spheroid, we have derived the linearly independent stationary solutions for Marangoni convection in terms of Stokes stream functions. The numerical simulations of the thermocapillary motion in the drops are used to study the onset of the stationary regime. Both analytical and numerical calculations predict the axially symmetric circulatory convection motion in the drop, the dynamics of which is determined by the magnitude of the temperature gradient across the drop. The analytical solutions for the critical temperature distribution and velocity fields are obtained for the large temperature gradients across the oblate drop. These solutions reveal the lateral separation of the critical and stationary motions within the drops. The critical vortices are localized near the central part of a drop, while the intensive stationary flow is located closer to its butt end. A crossover to the limit of the plane film is studied within the formalism of the stream functions by reducing the droplet ellipticity ratio to zero value. The initial stationary regime for the strongly oblate drops becomes unstable relative to the many-vortex perturbations in analogy with the plane fluid films with free boundaries.

Phase behavior and structure of a biocompatible microemulsion based on Tween 20, 2-ethylhexylglycerine and isopropyl palmitate in water

Microemulsions are thermodynamically stable mixtures of water and oil. For many applications, such as in cosmetics, it is important that the components are biocompatible. For the formulation of a new biocompatible microemulsion based on Tween 20 (Tw20) surfactant, the glycerol ether 2-ethylhexylglycerin (EHG) is employed as an alternative to commonly used short- to medium chain alcohols as cosurfactant. Tw20 itself is not capable of any significant solubilization but the addition of EHG allows to enhance the solubilization capacity for the polar oil isopropyl palmitate (IPP) largely. At a Tw20 to EHG molar ratio of 1:12, i.e., very large excess of cosurfactant, the solubilization capacity is increased by nearly three orders of magnitude compared to a pure Tw20 solution. The resulting oil-in-water (O/W) microemulsion is studied using static and dynamic light scattering (SLS/DLS) and small-angle neutron scattering (SANS). The SANS spectra are analyzed with a core–shell ellipsoid model, which indicates formation of oblate droplets at higher EHG content. Dynamic light scattering experiments confirm the presence of microemulsion droplets with hydrodynamic radii increasing from 3.8 to 19.8 nm upon addition of cosurfactant. In summary, this study shows that EHG is a cosurfactant that enhances the solubilization properties of a biocompatible surfactant largely and thereby renders it very useful for formulations.

The deformation and breakup of a droplet under the combined influence of electric field and shear flow

A lattice Boltzmann and finite-difference hybrid method is used to simulate the droplet deformation and breakup under the combined action of shear flow and electric field. The hybrid method is first used to validate for the droplet deformation in the combined action of shear flow and electric field. It is then used to simulate the droplet deformation and breakup in two different electric systems. Results of prolate droplets show that the droplet height and deformation both increase with increasing electric capillary number (). In addition, for the breakup mode of prolate droplets, increasing makes the long axis of the droplet incline more towards the wall electrodes and droplet breaks up into more daughter droplets. Results of oblate droplets show that the droplet height decreases with increasing . However, the droplet deformation first decreases and then increases with increasing , and its minima occurs at . For the breakup mode of oblate droplets, the droplet deforms into a more oblate shape with a longer neck and finally breakup into more daughter droplets with increasing .

Heat transfer for a single deformed evaporating droplet in the internal combustion engine

A droplet injected from the engine nozzle at high speed generally exhibits a non-spherical shape due to aerodynamic pressure on the droplet surface. The non-spherical shape of a droplet affects its convective heat transfer, heating, and evaporation. Droplet evaporation is critical in heat transfer. The effect of a droplet's non-sphericity on evaporation is investigated in this paper, under the assumption that droplet shapes are prolate or oblate spheroids. Analytical evaporation rates and heat transfer enhancement factors under forced convection for both prolate and oblate spheroidal droplets are derived. The effects of droplet aspect ratio, volume, and Reynolds number on the droplet evaporation under different conditions are numerically investigated. The results indicate that the evaporation conversion rates for both prolate and oblate spheroids are enhanced compared to a spherical droplet of the same volume. In addition, larger droplets show an increased enhancement effect on the evaporation process. Assuming that the droplet shrinkage reduces the dimension perpendicular to the droplet surface, the spheroidal droplet maintains its original aspect ratio during the evaporation process in a quiescent environment, while the nonspherical droplet tends to increase its sphericity under a forced convection situation.

Numerical analysis of two-phase electrohydrodynamic flows in the presence of surface charge convection

The mechanism of surface charge convection (SCC) reveals inherently nonlinear coupling between the electrostatic problem and the hydrodynamic flow in the Melcher–Taylor electrohydrodynamic (EHD) model. Considering that most previous numerical models are based on decoupled leaky dielectric equations, the quantitative effect of SCC on two-phase EHD under different parameters remains unclear. In the present study, we propose a new numerical scheme to solve the two-phase EHD problems in the framework of the lattice Boltzmann method. The fully coupled equations including the Navier–Stokes equations, the Nernst–Planck equations, and the Poisson equation are solved using three well-designed lattice Boltzmann equations. The problem of droplet deformation under a uniform electric field is studied. By neglecting SCC at a small electric Reynolds number ReE ≪ 1, our model successfully reproduces previous theoretical and numerical results. When considering the SCC mechanism at finite values of ReE, the intensity of the EHD flow is reduced. Consequently, oblate droplets are predicted to be less deformed, while prolate droplets are enhanced. In addition, the SCC effect increases as the values of both ReE and the electric capillary number, Ca, increase. In addition, a sharp variation in surface charge density is observed near the equator of the droplet due to SCC.

Numerical investigation of evaporation and radiation absorption of a non-spherical water droplet under asymmetrically radiative heating

A numerical model for evaporation along with internal radiation absorption of a non-spherical droplet is presented. In this model, the VOF method is utilized to trace the interface and the radiative transfer is taken into account by using the Monte Carlo method combined with a contour based interface reconstruction algorithm. Meanwhile, the dynamic mesh technique is applied to increase the computational efficiency. The validated model is used to investigate the radiation absorption inside prolate and oblate water droplets under infrared laser irradiation, and a parametric study for the effects of droplet’s shape and size is presented. The results reveal that the shape of droplet dominates the spectral absorptance when the droplet’s optical thickness is small, but for optically thick droplets, the radiation absorption turns to be dominated by the spectral reflectivity of the irradiated surface. For a deformable droplet under asymmetrically radiative heating, the incident radiation can significantly enhance the heat transfer and evaporation processes. In addition, the assumption of uniform radiation absorption is found to be valid to optical thin droplets, whereas this assumption will underestimate the average interface temperature and evaporation rate of droplets with large optical thickness.

Theoretical study on combustion of non-spherical particles in nano-aluminized propellant in air**Foundation Item: Science & Department of Heibei Province (No. 16211013D), Handan Science & Technology Development Bureau (No. 1511103015).

To investigate the deflagration phenomenon of an aluminized solid propellant during the combustion process, an improved theoretical combustion model has been developed. The model uses non-spherical particles for predicting the instantaneous mass evaporation rates of droplets as well as the burning time. The burning behavior of individual prolate and oblate spheroidal droplets, including the surface-area ratio, Nusselt numbers, temperature, and size change rate, was theoretically analyzed to elucidate the effect of non-spherical grains on the burning stability. Models for the Al evaporation rates and burnout time of both prolate and oblate spheroids in the diffusion-kinetics controlled regimes were established. According to the results, the particle morphology, size, and Reynolds numbers significantly affect the burning characteristics of nano-aluminized propellants. The proposed model is useful for steady-state combustion-performance assessment of aluminized solid propellants.

Effects of droplet shape on impact force of low-speed droplets colliding with solid surface

Experimental studies of low-speed droplets colliding with a flat solid surface are performed to record the impact force and deformation by using a highly sensitive piezoelectric transducer and a high-speed camera. The experimental data are used to verify the accuracy of a 3D numerical model via the smoothed particle hydrodynamics method, and the numerical method is used to explore the effects of the droplet morphology on the collision force. As the horizontal–vertical ratio of the droplets increases, the peak impact force increases by a power function trend and the time to reach the collision force peak decreases. The relationship between the equivalent volume of a spherical droplet and the volume of an ellipsoid droplet with different horizontal–vertical ratios under the same peak impact force is obtained. Self-similar theory is also suitable for droplets with ellipsoid shape. Finally, the stresses inside the material after large-sized spherical and small-sized oblate droplets hit the wall surface are compared. Results indicate that the curvature radius of droplets is a key factor that affects initial impact force and material erosion.

Numerical study of droplet dynamics in a steady electric field using a hybrid lattice Boltzmann and finite volume method

A hybrid method is developed for simulation of electrohydrodynamics interfacial flows. This method uses a lattice Boltzmann color model to describe the immiscible two-phase flow and a finite volume method to solve the Poisson equation for electric potential. The lattice Boltzmann and finite volume simulations are coupled by the leaky dielectric model. The method is applied to simulate a single droplet subject to a steady electric field, in which the influence of electric capillary number (CaE), dielectric constant ratio (Q) inside and outside of the droplet, and electric conductivity ratio (R) is studied for both oblate and prolate droplets. For a droplet undergoing small deformation, our numerical results are found to agree well with theoretical predictions, justifying the numerical method. Results of oblate droplets show that at low R, the droplet undergoes the transition from steady deformation to breakup with CaE, and the critical electric capillary number for droplet breakup, CaEB, decreases with increasing Q, whereas at high R, the droplet does not break up but finally reaches a steady shape regardless of the value of CaE. For prolate droplets, the droplet state may undergo the transition from steady shape to periodic oscillation and finally to breakup as CaE increases. Increasing Q increases both CaEB and the critical electric capillary number CaEO, which characterizes the transition from steady shape to periodic oscillation, but the increase in CaEO is less significant. In the CaE-R diagram, the periodic oscillation is limited to a small range, and increasing R decreases CaEB.

Evaporation of oblate spheroidal droplets: A theoretical analysis

ABSTRACTThe time-dependent evaporation of an oblate spheroidal droplet that undergoes forced convection was numerically investigated by solving the governing equations of energy, mass, and momentum. The research results elucidate the effects of oblate spheroid shape on droplet evaporation. A quantitative evaluation of spheroidal geometrical parameters (i.e., equatorial diameter, surface area, and volume) reveals that the evaporation rate increases at a much higher rate for an oblate droplet that approaches a disk-like shape of the same equivalent volume as a sphere. The variation of local Sherwood number on the surface of an oblate spheroid was obtained for the Reynolds number (2 ≤ Re ≤ 150) and the droplet axis ratio (0.1 ≤ Axr ≤ 0.9).

Electrically induced structure transition in nematic liquid crystal droplets with conical boundary conditions.

Polymer-dispersed liquid crystal composites have been a focus of study for a long time for their unique electro-optical properties and manufacturing by "bottom-up" techniques at large scales. In this paper, nematic liquid crystal oblate droplets with conical boundary conditions (CBCs) under the action of electric field were studied by computer simulations and polarized optical microscopy. Droplets with CBCs were shown to prefer an axial-bipolar structure, which combines a pair of boojums and circular disclinations on a surface. In contrast to droplets with degenerate planar boundary conditions (PBCs), hybridization of the two structure types in droplets with CBCs leads to a two-minima energy profile, resulting in an abrupt structure transition and bistable behavior of the system. The nature of the low-energy barrier in droplets with CBCs makes it highly sensitive to external stimuli, such as electric or magnetic fields, temperature, and light. In particular, the value of the electric field of the structure reorientation in droplets with CBCs was found to be a few times smaller than the one for droplets with PBCs, and the droplet state remained stable after switching off the voltage.

Simulation of optical caustics associated with the tertiary rainbow of oblate droplets.

In this paper, a vector ray tracing (VRT) model is used to simulate optical caustic structures, including rainbow and hyperbolic umbilic (HU) fringes, in the tertiary rainbow region of light scattering from oblate spheroidal droplets. In order to apply the optical caustic structures to particle diagnostics, the evolution of rainbow and HU fringes with an increase in the aspect ratio of oblate spheroidal droplets is investigated in detail, and the curvature of rainbow fringes are calculated. Next, on the basis of the VRT model, the location of cusp caustics is calculated and compared with theoretical prediction.

Experimental validation of the vectorial complex ray model on the inter-caustics scattering of oblate droplets.

We report the first experimental validation of the Vectorial Complex Ray Model (VCRM) using the scattering patterns of large oblate droplets trapped in an acoustic field. The two principal radii and refractive index of the droplets are retrieved with a minimization method that involves VCRM predictions and experimental light scattering patterns. The latter are recorded in the droplet equatorial plane between the primary rainbow region and the associated hyperbolic-umbilic diffraction catastrophe. The results demonstrate that the VCRM can predict the fine and coarse stuctures of scattering patterns with good precision, opening up perspectives for the characterization of large non-spherical particles.

Shape evolution of free falling deformed droplets

We have investigated the shape evolution of pre-compressed oblate droplets during free fall by using a high speed camera. In the experiments, the properties of the droplets were tuned by sodium dodecylsulfate (SDS). It has been found, the shape evolution strongly depends on the concentration of SDS. Interestingly, the droplets with concentration of 0.8 cmc can maintain its compressed state for long time (>50 ms) without recovering to spherical shape. While the droplets of other concentrations exhibit obvious recovery and oscillation. The Gibbs elasticity of the droplet surface is not sufficient to balance the Laplace pressure which drives the droplets recover. It is deduced that the Marangoni effect caused by the SDS concentration gradient which is induced by the acoustic field plays an important role. This effect results in larger additional surface elasticity, accompanying the Gibbs one, to the droplet which balances its recover force.

Structure transitions in oblate nematic droplets

We consider the structure transitions in oblate supramicrometer nematic droplets related to reorientation of the line defect in the electric field. These transitions can be used in optical devices based on polymer dispersed liquid crystal materials with high contrast ratio. We suggest a simple method for determination of director distribution in nematic droplets of an arbitrary shape with surface interaction and in the presence of constant electric field. Point and linear defects are taken into account. This method does not require any presuppositions about symmetry of the director distribution. The elasticity continuum theory is treated with Monte Carlo annealing on a simple lattice. A special triangulation-based technique is applied for accurate representation of the droplet boundaries. The method is tested on 5CB material.

Simulation of optical caustics associated with the secondary rainbow of oblate droplets

A vector-ray tracing (VRT) model is employed to simulate optical caustic structures [in terms of rainbow and hyperbolic umbilic (HU) fringes] in the secondary rainbow region of light scattering from oblate droplets. The changes of the rainbow fringe and HU fringe in response to shape deformation of oblate droplets are investigated and the curvature of the rainbow fringe is calculated. Then the location of cusp caustics is calculated from VRT simulations and compared with analytic solutions.

Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement

A vector ray tracing (VRT) model is developed to simulate optical caustic structures in the primary rainbow region of light scattering from oblate droplets. The changes of the optical caustic structures in response to shape deformation of oblate droplets are investigated. Then the curvature calculated from the simulated rainbow fringes is compared with that from the measured rainbow fringes and good agreement is found. Furthermore, according to the generalized rainbow patterns and the relation between aspect ratio and curvature of the rainbow fringe, non-sphericities in terms of aspect ratio of an oblate water droplet is measured with high measurement accuracy.