Oblate Drops Research Articles

Exploring Electrohydrodynamic Flows Inside Leaky Dielectric Drops: A Laser Velocimetry Approach

A novel experimental methodology based on laser velocimetry for the investigation of electrohydrodynamic (EHD) flows inside a neutrally buoyant leaky dielectric drop of 2.25 mm of radius is proposed. Utilizing fluorescent particles, 2D and 3D Lagrangian Particle Tracking (LPT) measurements of the EHD flows inside an initially spherical drop under a constant electric field are reported. Two leaky dielectric fluids, namely Silicone and Castor oils, were used as either the drop or the medium phases, depending on the case investigated, thus covering oblate and prolate drops both with small and larger deformations. 2D measurements of the trajectories and velocities of the tracer particles in a Lagrangian referential enabled a direct comparison with the leaky dielectric model (LDM), a well-stablished theory covering drops with smaller deformations, showing a good agreement with our measurements. The symmetry inside a drop with increased deformation is investigated from an analysis of the mean 3D velocity field, whose measurements converge to the two-dimensional analysis at the plane of symmetry of the drop, suggesting symmetrical flow structures inside the drop.

Numerical implementation of three-dimensional vectorial complex ray model and application to rainbow scattering of spheroidal drops.

The rainbow patterns of oblate spheroidal drops have been observed in experiments nearly forty years ago [Nature312, 529 (1984)10.1038/312529a0]. However, the prediction for those complex patterns has been a challenge for conventional light scattering models. The vectorial complex ray model (VCRM) allows to account for the direction, the polarization, the phase, the amplitude and the wavefront curvature of waves and provides a powerful tool for the study of the light/electromagnetic wave interaction with a homogeneous object of any shape with smooth surface. In [Opt. Lett.46, 4585 (2021)10.1364/OL.434149], the authors have reported an important breakthrough of VCRM for the three-dimensional scattering (VCRM3D) and the simulated rainbow patterns of oblate drops. The present paper is devoted to the detailed description of the numerical implementation allowing the simulation of the 3D scattering field by a nonspherical particle. Its ability to predict both the fine and coarse intensity structures of the rainbows and the near-backward scattering patterns of spheroids is demonstrated. This work opens perspectives for exploring the 3D scattering characteristics of large objects with any smooth shape and developing relevant optical techniques for particle characterization.

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.

Scattering by relatively small oblate spheroidal drops of water in the rainbow region: T-matrix results and geometric interpretation

Late in 1999 Michael Mishchenko carried out T-matrix calculations of light scattered by oblate spheroidal drops of water at Marston's request. The purpose of the investigation was to determine if the caustic features, known by that time to be visible in the far-field scattering near the primary rainbow region for horizontally illuminated relatively large drops, would also be present in T-matrix calculations of the scattering by small drops. The axis of rotational symmetry of the drop was taken to be vertical, and the illumination was taken to be horizontal. Scattering into the relevant angular region was evaluated for drop aspect ratios D/H = 1, 1.23, 1.311, and 1.37 where D denotes the equatorial diameter and H the vertical diameter. The results are limited to polarized monochromatic illumination with ka = 140 where a = D/2 and k = 2π/λ with λ being the wavelength of the illumination. As expected from geometrical analysis, a horizontal “V” shaped caustic is visible for D/H = 1.311. The T-matrix results, gray-tone intensity displays provided by Mishchenko, remain instructive.

Generalized rainbow patterns of oblate drops simulated by a ray model in three dimensions.

The scattering patterns near the primary rainbow of oblate drops are simulated by extending the vectorial complex ray model (VCRM) [Opt. Lett.36, 370 (2011)OPLEDP0146-959210.1364/OL.36.000370] to three-dimensional (3D) calculations. With the curvature of a wavefront as an intrinsic property of a ray, this advanced ray model permits, in principle, to predict the amplitudes and phases of all emergent rays with a rigorous algebraic formalism. This Letter reports a breakthrough of VCRM for 3D scattering with a line-by-line triangulation interpolation algorithm allowing to calculate the total complex amplitude of a scattered field. This makes possible to simulate not only the skeleton (geometrical rainbow angles, hyperbolic-umbilic caustics), but also the coarse (Airy bows, lattice) and fine (ripple fringes) structures of the generalized rainbow patterns (GRPs) of oblate drops. The simulated results are found qualitatively and quantitatively in good agreement with experimental scattering patterns for drops of different aspect ratios. The physical interpretation of the GRPs is also given. This work opens up prominent perspectives for simulating and understanding the 3D scattering of large particles of any shape with a smooth surface by VCRM.

The role of oscillation in ellipsoidal drop impact on a solid surface

Ellipsoidal shapes of drops can significantly modify the impact dynamics and suppress the rebound by inducing symmetry breaking in the mass and momentum distributions compared to the axisymmetric dynamics of typical drops. However, the previous works have assumed that the drop oscillation at the moment of impact only slightly affects the post-dynamics although the oscillation must be involved in the spreading. Here, we study the impact dynamics of the oscillating ellipsoidal drops on non-wetting surfaces as a function of the ellipticity, oscillation phase, and Weber number (We) experimentally and numerically. The spreading dynamics show notable hysteretic features in the maximal spreading diameters at the four regions of the oscillation phase. The hysteresis appears more prominently in prolate drops than in oblate drops and becomes remarkably suppressed at the four phases as We increases. Momentum analysis shows that the phases for shaping the drops spherically can drive higher asymmetry in the horizontal momenta than the other phases for shaping the drops ellipsoidally. The momentum asymmetry in the horizontal axes indicates that the oscillation phase as well as the ellipticity can play an important role in altering the hydrodynamics and reducing the bounce magnitude.

Bouncing dynamics of spheroidal drops on macro-ridge structure

Bouncing drops on solid surfaces have gained increasing attention for various practical applications, such as self-cleaning and anti-icing strategies. Breaking the circular symmetry in bouncing dynamics on a ridge enables drop dynamics to be modified significantly and the residence time of drops on surfaces to be reduced. Here, we numerically investigate the asymmetric bouncing dynamics of oblate and prolate spheroidal drops on a superhydrophobic surface decorated with a rectangular ridge to demonstrate the feasibility of further reducing the residence time by shaping raindrop-like drops. The residence time is investigated for various aspect ratios and Weber numbers, which are discussed based on impact stages of spreading, splitting, and retraction. The underlying principle behind the residence time reduction is analyzed by quantifying the temporal variations in the widths and the axial momenta of the drops. The bouncing directions of the spheroidal drops are closely related to the momentum distributions during the retraction. We investigate how to change the residence time for ridges of different heights and widths. The symmetry-breaking bouncing of the spheroidal drops on ridge surfaces will provide fundamental and practical inspiration for the efficient control of drop mobility.

The Effect of Drop Shape, Sensing Volume and Raindrop Size Statistics to the Scattered Field on 300 GHz

Scattering and attenuation properties of rain using various drop models are calculated using Mie theory, the T-matrix method and numerical technique respectively. Ellipsoidal and oblate drops use the axis ratio of raindrops reported in literature. Scattering are presented both for small volumes containing a single drop and for large volumes with multiple drops in which the statistics of drop size distribution need to be taken into account. The angular dependence of scattering, absorption and scattering cross sections and polarization ratios are investigated. For single scattering the angular dependence and cross sections of nonspherical drops differed from those of a spherical drop. Differences between ellipsoid and oblate drops were minimal. Drop shape affected the polarization mostly at 40-140 degree detection angles. The averaged linear attenuation and rain induced cross channel signal were studied. Attenuation was close to the ITU-R.838 model and in the same order of magnitude as the attenuation due to atmospheric moisture. The cross channel signal was calculated as a function of rain rate and transmitter/receiver angle. Vertical polarization was observed to produce a higher cross channel signal than horizontal polarization. It was concluded that drop shape is not an important factor at 300 GHz due to small drops dominating the scattering signal. Overall, the results showed that in single scattering, the ellipsoid is a good approximation for raindrops and that for large volumes, a spherical drop approximation and a Mie solution may be sufficient at frequencies of 300 GHz and higher.

Simulation of the optical caustics associated with the primary rainbow for oblate spheroidal drops illuminated by a Gaussian beam.

A vector ray-tracing model (VRT) has been developed to compute the optical caustics associated with the primary rainbow for an oblate spheroidal water drop illuminated by a Gaussian beam. By comparing the optical caustic structures (in terms of limiting rainbow and hyperbolic umbilic fringes) for a water drop with a Gaussian beam (GB) illumination with that for the same drop, but with parallel beam (PB) illumination, the influence of the Gaussian beam on the optical caustics is investigated. For a water drop with GB illumination and different drop/beam ratios (i.e., the ratio between the drop equatorial radius and the Gaussian beam waist), the location of cusp points and the curvature of the limiting rainbow fringe are also studied. We anticipate that these results not only confirm the approach to compute optical caustics for oblate spheroidal drops illuminated by a shaped beam, but may also lead to a new method for measuring the aspect ratio of spheroidal drops.

Annular stratification of acoustically levitated aqueous two-phase-system drops

The phase separation of an aqueous two-phase system (ATPS) consisting of 20 wt. % polyethylene glycol (PEG) and 12 wt. % ammonium sulfate (AMS) was investigated under the condition of acoustic levitation. An annular stratification occurs in the acoustically levitated and severely flattened ATPS drops, where the AMS-rich phase is horizontally encircled by the PEG-rich phase. This is different from the conventional top-bottom layered stratification usually taking place in test tubes or levitated oblate drops. Calculation of the total surface and interface energy of a biphasic drop based on two simplified geometric models of stratification demonstrates that the annular structure is more stable than the top-bottom one if the drop is flattened enough. Two stages of phase separation are revealed through videographic and statistical analyses on the PEG-rich globules that are spontaneously formed inside the ATPS drops. The first stage is characterized by rapid reduction in the population of small PEG-rich globules, which is governed by frequent collision and coalescence and can be speeded up by drop oscillation and acoustic streaming during levitation. The second stage shows a slow reduction in the population of PEG-rich globules, a process controlled by integration of those globules into the already-formed bulk PEG-rich layer.

Transient electrohydrodynamics of a liquid drop at finite Reynolds numbers

Transient electrohydrodynamics of a liquid drop at finite Reynolds numbers is studied by computer simulations. The governing equations of the problem are solved using a parallelized front tracking/finite difference method in the framework of the Taylor–Melcher leaky dielectric theory. The effect of the dielectric properties of the fluids on the dynamic response of the drop is studied by considering three representative fluid systems, which encompass all the regions of the deformation–circulation map. A parametric study is performed over a range of capillary number , the Ohnesorge number squared and the viscosity ratio to explore the effect of these parameters on the dynamic response. At low to intermediate capillary numbers, the dynamic response resembles that of a first- or second-order mechanical system, where the deformation settles to a steady state monotonically or through oscillations. However, beyond a threshold capillary number, the dynamic response becomes nonlinear. It is shown that the Ohnesorge number squared and the viscosity ratio are the key parameters that determine the transition from a monotonic response to an oscillatory one (and vice versa) in density-matched fluid systems. To predict the dynamic response of nearly spherical drops at arbitrary , an analytical equation is developed, which results in a critical Ohnesorge number squared that demarks a monotonic response from an oscillatory one. Investigations on the effect of the viscosity ratio on the dynamic response shows that the relaxation time (for monotonic response) increases with an increase in and that the steady state deformation correlates positively with for oblate drops; however, depending on the corresponding regions of prolate drops on the deformation–circulation map, correlates positively or negatively with . A long-standing misconception regarding the oblate region is also rectified by development of a normal stress map, which determines the relative importance of the electric pressure and the normal hydrodynamic stress in setting the sense of the deformation of the drop.

Electrohydrodynamic settling of drop in uniform electric field: beyond Stokes flow regime

The electrohydrodynamics of a weakly conducting buoyant drop under the combined influence of gravity and a uniform electric field is studied computationally, focusing on the inertia-dominated regime. Numerical simulations are performed for both perfectly dielectric and leaky dielectric drops over a wide range of dimensionless parameters to explore the interplay of fluid inertia and electrical stress to govern the drop shape and charge convection. For perfectly dielectric drops, the fluid inertia alters the drop shape and the deformation behaviour of the drop follows a non-monotonic path. The drop shape at steady state exhibits the transition from oblate to prolate shape on increasing the electric field strength, in sharp contrast to the cases concerning the Stokes flow regime. Similar behaviour is also obtained for leaky dielectric drops for certain fluid properties. For leaky dielectric drops, the fluid inertia also affects the convective transport of charges at the drop surface and thereby alters the drop dynamics. Unlike the Stokes flow regime, where surface charge convection has little effect on the settling speed, the same modifies the drop settling speed quite significantly in the finite inertial regime depending on the combination of electrical conductivity ratio and permittivity ratio. For oblate drops at low capillary number, charge convection alters drop shape, while keeping the nature of deformation unaltered. However, for relatively large capillary number, the oblate drop transforms into a dimpled shape due to charge convection. For all cases, an interesting fact is noticed that under the combined action of electric and inertial forces, the resultant deformation is less than the summation of the deformations caused by individual effects, when inertial effects are strong. These results are likely to provide deep insights into the interplay of various nonlinearities towards altering electrohydrodynamic settling of drops and bubbles.

Spreading Dynamics and the Residence Time of Ellipsoidal Drops on a Solid Surface.

Controlling bouncing drops on solid surfaces has gained significant attention because of the benefit of low residence time in anti-icing and self-cleaning strategies. Given that the drop shape at the moment of impact is classically assumed to be spherical, the residence time on a flat surface is bounded by a theoretical Rayleigh limit. In this study, we investigated the impact dynamics of oblate and prolate ellipsoidal drops to demonstrate the concept of modifying the residence time by shaping like raindrops. Experimental and numerical studies show that the initial shape plays a vital role in an increase or reduction in bounce speed of the drop, which is explained by scaling the maximum spreading time. The hydrodynamic features of ellipsoidal drops are analyzed by quantifying the temporal variations in diameters, heights, velocity fields, momenta, and energy dissipation. We believe that the ellipsoidal drop impact can provide an efficient pathway for controlling the residence time in practical applications.

Three dimensional numerical simulation of drops suspended in a channel under uniform electric field

Abstract In this paper, numerical study of emulsion of drops in a channel under uniform electric field is investigated. The electric field is created by imposing an electric potential difference across the channel walls. The leaky dielectric model developed by Taylor is used to compute the electric force. The electric conductivity ratio σ r = σ i σ o and electric permittivity ratio e r = e i e o are used to study oblate and prolate drops The interaction of drop-to-wall and drop-to-drop are investigated by changing the dimensionless numbers. Oblate drops and prolate drops with σ r e r , are always attracted to the walls, but prolate drops with σ r > e r , are attracted by or repelled from the walls depending on the initial distance and electric properties. Drop-to-drop interaction depends on the initial configuration of drops. Drop pairs that are placed at an angle relative to the direction of the electric field, are always attracted towards each other. If they have an initial separation distance in the direction of the electric field, they are attracted to the walls or towards each other depending on their separation distance. Prolate drops with σ r > e r , obtain a different configuration in their final state. They may stick together or have a small separation distance, depending on their electric properties. Increasing the electric capillary number, changes the vertical and horizontal separation distance of drop pairs at steady state configuration. It also decreases the time required to reach steady state configuration. The behavior of emulsion of drops is studied by considering oblate drops, prolate drops with σ r e r and prolate drops with σ r > e r . All drops form fiber chains in the direction of the electric field that will break in stronger Poiseuille flow. Oblate drops either settle on the walls or migrate to the channel center line, but prolate drops form chains of fibers that sometimes extend between the walls. Presence of drops will change the well-known parabolic velocity profile especially for prolate drops that are suspended in the channel. The channel flow rate is also studied for different emulsion of drops and electric field. At a constant electric field and pressure gradient, the channel flow rate for oblate emulsion is higher than prolate ones. As the electric field strength increases, the channel flow rate will also increase for oblate emulsions.

Model for computing optical caustic partitions for the primary rainbow from tilted spheriodal drops.

A model is proposed to compute the salient optical caustic partitions occurring in the primary rainbow for oblate spheroidal drops. By computing the boundary limits of outgoing rays, the optical caustic structures (termed rainbow and hyperbolic umbilic fringes) for tilted drops are calculated and compared with those for aligned (untilted) drops. The curvature of the rainbow fringe and the shifts of cusp caustics are discussed as well. The observed properties of the caustics can potentially be used for drop measurements. The model could also be applied to compute the optical caustics for drops with arbitrary shape, arbitrary orientation, and shaped beam illumination.

Modelling of heat and mass transfer from spheroidal drops with non-uniform surface temperature

The heating and evaporation of single component spherical and spheroidal (oblate and prolate) drops in gaseous quiescent environment are modelled, accounting for the effect of a non-uniform distribution of the temperature at the drop surface. The analytical solution to the species conservation equations in the proper coordinate system (spherical/spheroidal) is used to numerically solve the energy equation. The effect of temperature disuniformity on the local non-dimensional heat transfer coefficient and global heat and evaporation rates is calculated for different drop temperature disuniformity, mean surface temperature, drop deformation, gas temperature and pressure and liquid species.

A drop in uniaxial and biaxial nonlinear extensional flows

In this theoretical report, we explore small deformations of an initially spherical drop subjected to uniaxial or biaxial nonlinear extensional creeping flows. The problem is governed by the capillary number (Ca), the viscosity ratio (λ), and the nonlinear intensity of the flow (E). When the extensional flow is linear (E = 0), the familiar internal circulations are obtained and the same is true with E > 0, except that the external and internal flow rates increase with increasing E. If E < 0, the external flow consists of some unconnected regions leading to the same number of internal circulations (−3/7 < E < 0) or twice the number of internal circulations (E < −3/7), when compared to the linear case. The shape of the deformed drop is represented in terms of a modified Taylor deformation parameter, and the conditions for the breakup of the drop by a center pinching mechanism are also established. When the flow is linear (E = 0), the literature predicts prolate spheroidal drops for uniaxial flows (Ca > 0) and oblate spheroidal drops for biaxial flows (Ca < 0). For the same |Ca|, if E > 0, the drop is more elongated than the linear case, while E < 0 results in less elongated drops than the linear case. Compared to the linear case, for both uniaxial and biaxial extensional flows, E > 0 tends to facilitate drop breakup, while E < 0 makes drop breakup more difficult.

Nonlinear electrohydrodynamics of slightly deformed oblate drops

The transient deformation of a weakly conducting (‘leaky dielectric’) drop under a uniform DC electric field is computed via an axisymmetric boundary integral method, which accounts for surface charge convection and a finite relaxation time scale over which the drop interface charges. We focus on drops that attain an ultimate oblate (major axis normal to the applied field) steady-state configuration. The computations predict that as the time scale for interfacial charging increases, a shape transition from prolate deformation (major axis parallel to the applied field) to oblate deformation occurs at intermediate times due to the slow buildup of charge at the surface of the drop. Convection of surface charge towards the equator of the drop is shown to weaken the steady-state oblate deformation. Additionally, convection results in sharp shock-like variations in surface charge density near the equator of the drop. Our numerical results are then compared with an experimental system consisting of a millimetre-sized silicone oil drop suspended in castor oil. Agreement in the transient deformation is observed between our numerical results and experimental measurements for moderate electric field strengths. This suggests that both charge relaxation and charge convection are required, in general, to quantify the time-dependent deformation of leaky dielectric drops. Importantly, accurate prediction of the observed modest deformation requires a nonlinear model. Discrepancies between our numerical calculations and experimental results arise as the field strength is increased. We believe that this is due to the observed onset of rotation and three-dimensional flow at such high electric fields in the experiments, which an axisymmetric boundary integral formulation naturally cannot capture.

Surprises and anomalies in acoustical and optical scattering and radiation forces

Experiments on radiation torques and negative radiation forces by various researchers display how the underlying wave-field geometry influences radiation forces. Other situations strongly influenced by wave-field geometry include high-order caustics present in light-scattering patterns of objects as simple as oblate drops of water or oblate bubbles of air in water. Related theoretical and experimental investigations are considered. Acoustic scattering enhancements associated with various guided waves are also examined. These include guided waves having negative group velocities and guided wave radiating wavefronts having a vanishing Gaussian curvature.

Mass transfer around oblate spheroidal drops in a biaxial stretching motion

AbstractMass transfer around a small eccentricity oblate spheroidal drop in a biaxial stretching and creeping flow, was investigated theoretically. The results at small capillary (Ca ≪ 1) and large Peclet numbers (Pe ≫ 1), obtained via regular perturbations in Ca, suggest that: at very short times, the mass transfer rate to or from the drop represents, at O(Ca), mass transfer by diffusion only around a sphere. At long times or at steady‐state, the mass transfer rate is, at O(Ca), slightly larger than that of a spherical drop, it increases with increasing the capillary number and decreases with increasing the viscosity ratio. Steady‐state is established faster, as the viscosity ratio decreases and as the capillary number increases. Exact and approximate analytical solutions, valid at all times, which converge to the asymptotic solutions for short and long times, are also presented.