Spheroidal Drops Research Articles

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.

Modeling the evaporation of multicomponent drops of general shape on the basis of an analytical solution to the Stefan–Maxwell equations

This paper proposes an analytical approach to model the evaporation of multicomponent drops of general shape, which is based on the solution to Stefan–Maxwell equations. The model predicts the quasi-steady molar fractions and temperature distributions in the gas phase as well as the heat rate and the species evaporation rates. The model unifies previous approaches to this problem, namely, for spherical and spheroidal drops, under a unique model and proposes solutions for other shapes and geometries, such as sessile drops and drop pairs. To assess the model, a comparison with a numerical solution to the conservation equations is also reported for both different drop configurations and different compositions.

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.

An analytical model for the evaporation of multi-component spheroidal drops based on Stefan–Maxwell equations

An analytical evaporation model for multi-component spherical and spheroidal drops, based on the analytical solution of the Stefan–Maxwell equations is proposed to study the multi-component steady diffusion in the gas phase. A comparison with the ‘classical’ approach based on the Fick’s law is reported for three families of liquid mixtures, namely alkanes, alcohols and water solutions of alcohols under a wide range of operating conditions. The local vapour fluxes of each vaporising species on the drop surface are correlated with the local Gaussian curvature. The effects of drop shape and of the modelling of constitutive equations on the evaporation rates are discussed. Comparison of the present model with experimental measurements of different compositions of tetradecane–hexadecane drops vaporising in stagnant air shows a rather satisfactory agreement.

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.

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.

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.

An analytical approach to model heating and evaporation of multicomponent ellipsoidal drops

An analytical model for heating and evaporation of non-spherical multicomponent drops is reported, based on the solution of energy and species conservation equations in orthogonal curvilinear coordinates, accounting also for the effect of convective conditions according to the film theory approach. The instantaneous heat and evaporation rates of multicomponent drops is calculated as function of increased surface area due to drop deformation. The model is applied to predict evaporation characteristics of spheroidal drops under quasi-steady and transient conditions, analysing the effect of drop size, composition and velocity on the temperature evolution, for a range of drop and gas temperature conditions

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.

Analytical Solution for Spheroidal Drop under Axisymmetric Linearized Boundary Conditions

A liquid spheroidal drop freely suspended in another fluid is considered under arbitrary axisymmetric boundary conditions, which are linearized with respect to the velocity field and can result, in particular, from an axisymmetric external flow and an electric field being applied either separately or in combination. All nonlinear effects including inertia and surface charge convection are assumed to be negligible, whereas the drop and the ambient fluid are assumed to be leaky dielectrics and to have different viscosities. Central to the analysis are the reformulated stress boundary conditions and representation of the velocity field inside and outside the drop in terms of non-Stokes stream functions. In the prolate spheroidal coordinates, the stream functions are expanded into infinite series of spheroidal harmonics, and the reformulated velocity and stress boundary conditions yield a first-order difference equation for the series coefficients, which admits an exact nonrecursive solution. “Steady” spheroidal drops then correspond to minimums of a kinematic condition error that admits a simple efficient approximation. Under the simultaneous presence of aligned linear flow and a uniform electric field with corresponding capillary numbers $Ca$ and $Ca_E$, a spherical drop is stationary when ${Ca}=k\,{Ca}_E$ and becomes prolate/oblate when ${Ca}\gtrless k\,{Ca}_E$, where $k$ is proportional to the Taylor discriminating function and depends on ratios of viscosities, dielectric constants, and electric conductivities of the two phases. A spheroidal drop is “steady” when ${Ca}=k_1\,{\rm Ca}_E+k_2$, where $k_1$ and $k_2$ depend on the spheroid's axes ratio $d$ and approach $k$ and 0, respectively, as $d\to1$. The results show that when the Taylor deformation parameter $D$ is in the range $[-0.5,0.4]$, this relationship can be used for finding any of the three $Ca$, $Ca_E$, and $D$ when the other two are given.

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.

A Liquid Spheroidal Drop in a Viscous Incompressible Fluid under a Steady Electric Field

An analytical approach to the problem of a liquid spheroidal drop in a viscous incompressible fluid under a steady electric field is presented. It is assumed that the drop and ambient fluid are slightly conducting (leaky dielectrics) with no net charge. The velocity field inside and outside the drop is governed by the Stokes equations and is represented in terms of four generalized analytic functions which are found from the velocity and stress boundary conditions. For the case of equal viscosities, the velocity field admits an integral-form solution, whereas for an arbitrary viscosity ratio $\lambda$, it is given by a series of spheroidal harmonics in the prolate spheroidal coordinates. Both solution forms hold for prolate and oblate spheroids. The axes ratio of a spheroid closest to a steady shape (or “steady” spheroid) is a minimizer of the kinematic condition error. However, the entire dependence of the “steady” spheroid's axes ratio on an electric capillary number ${\rm Ca}_E$ can be estimated analytically with the inverse problem: for each spheroid's axes ratio, find ${\rm Ca}_E$ that minimizes the kinematic condition error. The inverse problem, being quadratic with respect to $1/{\rm Ca}_E$, has a simple analytical solution, which identifies critical ${\rm Ca}_E$ for spheroidal drops and includes an “unstable” solution. Remarkably, as functions of ${\rm Ca}_E$, the deformation parameters $D$ for the “steady” spheroids and for “true” steady shapes (obtained based on boundary-integral equations) have the same qualitative behavior for various viscosity, conductivity, and permittivity ratios ($\lambda$, $R$, and $Q$, respectively), and are virtually identical within the range $-0.4\le D\le0.4$ for all investigated $\lambda$, $R$, and $Q$. This shows that the presented analytical approach for spheroidal drops is by far superior to Taylor's small-perturbation theory and Ajayi's second-order correction ($O({\rm Ca}_E)$ and $O({\rm Ca}_E^2)$ theories, respectively) and can replace numerical solutions from boundary-integral equations up to large deformations.

Mass transfer around prolate spheroidal drops in an extensional flow

Mass transfer in the continuous phase around a small eccentricity prolate spheroidal drop in an axisymmetric extensional creeping flow and at large Peclet numbers was investigated theoretically. The results show that, at very short times, the total quantity of solute transferred to or from the drop represents, at O(Ca1), mass transfer by diffusion only around a sphere. For long times, or at steady-state, the total quantity of solute transferred is, at O(Ca1), slightly smaller than that of a spherical drop, and it decreases with an increase of the capillary number or the viscosity ratio. © 2012 Canadian Society for Chemical Engineering

Mass transfer around oblate spheroidal drops at low Reynolds numbers

Steady and unsteady mass transfer in the continuous phase around slightly deformed oblate spheroidal drops at low (but not zero) Reynolds numbers was investigated theoretically. Asymptotic analytical solutions for short and long times, at large Peclet numbers, were obtained by the useful equations derived by Lochiel and Calderbank and by Favelukis and Mudunuri for axisymmetric drops of revolution, with the only requirements being the shape of the drop and the tangential velocity at the surface of the drop. As expected, the result, although complicated, represents a small correction to the classical problem of mass transfer around a spherical drop under creeping flow conditions, since the physical problem presented here requires both the Reynolds and the Weber number to be much smaller than one.

Unsteady mass transfer around spheroidal drops in potential flow

Unsteady mass transfer in the continuous phase around spheroidal drops in potential flow and at high Peclet numbers has been theoretically studied. Analytical solutions for the concentration profile, the molar flux, the concentration boundary layer thickness, and the time to reach steady state are presented. The solution to the problem was obtained by the useful equations derived by Favelukis and Mudunuri for axisymmetric drops of revolution, with the only requirements being the shape of the drop and the tangential velocity at the surface of the drop. The solution suggests that, as the eccentricity increases, the total quantity of material transferred to or from the drop decreases (for prolate spheroids) and increases (for oblate spheroids). It was also determined that when the dimensionless time is greater than 2, then steady state is in practice obtained, with prolate drops attain steady-state conditions faster than oblate drops.

Energy analysis of possible channels for decomposition of a charged drop into two parts

Based on numerical analysis of the mathematical expression for potential energy of an isolated charged spheroidal drop and two approximate spheroidal daughter drops, mechanisms for decomposition of a multiply charged drop in two nearly equal parts were studied taking into account the electrostatic interaction of the daughter drops. It was shown that, as the distance between the daughter drops increased, both the probability of spontaneous decomposition of a multiply charged drop in two daughter drops and the decomposition symmetry degree increase at the moment of breaking the connection between the daughter drops.

Gas Absorption into a Moving Spheroidal Water Drop

Abstract Theoretical and experimental studies have been carried out to describe the absorption of sulfur dioxide by moving spheroidal water drops under transient flow conditions. These investigations allow the determination of the rate at which SO2 is scavenged from air by deformed spheroidal freely falling water drops. First, the theoretical models in the case of spheroidal water drops will be presented. For drops smaller than 1 mm in diameter, modeling is based on numerical solutions of Navier–Stokes and convective-diffusion equations, which describe transient flow and concentration fields, respectively, both in liquid and gas phases. For drops larger than 1 mm in diameter, modeling is based on scaling of interfacial agitation taking into account the effect of turbulent mixing and oscillation inside the drops. Second, an experimental device involving a 5-m rain shaft is presented in order to study SO2 absorption and to verify the theoretical models for spheroidal water drops. The predicted sulfur concen...

Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops: errata

Light scattered by an oblate drop of water has been observed to produce cusp caustics in the general vicinity of the rainbow region [ P. L. Marston E. H. Trinh , Nature London312, 529– 531 ( 1984)]. The principal curvatures of the generic local wave front that produces the far-field transverse cusp are examined. This wave front is shown to generate a transverse cusp curve (U − Uc)3 = − d∞V2, where U and V are horizontal and vertical scattering angles and Uc is the cusp point direction. The far-field opening rate d∞ is calculated for the transverse cusp. It is shown that d∞ has a simple dependence on the parameters of the generic wave front. We define the aspect ratio of the drop q = D/H, where H is the height and D is the equatorial width for the scattering drop. The method of generalized ray tracing is used to relate q to principal curvatures and shape parameters of the outgoing wavefront and hence to d∞. Measurements of d∞. for scattering laser light from acoustically levitated drops appear to support the calculation. As q goes to q4 ≈ 1.31, the critical value for generation of a hyperbolic–umbilic focal section, the predicted d∞ goes to infinity. The nature of the divergence was numerically investigated as was the rate at which d∞ vanishes as q approaches critical values for lips and transition events.

Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops

Light scattered by an oblate drop of water has been observed to produce cusp caustics in the general vicinity of the rainbow region [P. L. Marston and E. H. Trinh, Nature London 312, 529-531 (1984)]. The principal curvatures of the generic local wave front that produces the far-field transverse cusp are examined. This wave front is shown to generate a transverse cusp curve (U - U(c))(3) = - d(infinity)V(2), where U and V are horizontal and vertical scattering angles and U(c) is the cusp point direction. The far-field opening rate d(infinity) is calculated for the transverse cusp. It is shown that d(infinity) has a simple dependence on the parameters of the generic wave front. We define the aspect ratio of the drop q = D/H, where H is the height and D is the equatorial width for the scattering drop. The method of generalized ray tracing is used to relate q to principal curvatures and shape parameters of the outgoing wave front and hence to d(infinity). Measurements of d(infinity) for scattering laser light from acoustically levitated drops appear to support the calculation. As q goes to q(4) approximately 1.31, the critical value for generation of a hyperbolic-umbilic focal section, the predicted d(infinity) goes to infinity. The nature of the divergence was numerically investigated as was the rate at which d(infinity) vanishes as q approaches critical values for lips and transition events.