Spheroidal Drop Research Articles

Laminar-turbulent transition in an electromagnetically levitated droplet

During experiments on the MSL-1 (first microgravity science laboratory) mission of the space shuttle (STS-83 and STS-94, April and July 1997), a droplet of palladium-silicon alloy was electromagnetically levitated for viscosity measurements. For the nondeforming droplet, the resultant magnetohydrodynamic (MHD) flow inside the drop can be inferred from motion of impurity particulates on the surface. In the experiments, subsequent to melting, Joule heating produces a continuous reduction of viscosity of the fluid resulting in an acceleration of the flow with time. These observations indicate formation of a pair of co-rotating toroidal flow structures inside the spheroidal drop that undergo flow instabilities. As the fluid temperature rises, the amplitude of the secondary flow increases, and beyond a point, the tracers exhibit noncoherent chaotic motion signifying emergence of turbulence inside the drop. Assuming that the observed laminar-turbulent transition is shear-layer type, the internal structure of the toroidal loops is used to develop a semiempirical correlation for the onset of turbulence. Our calculations indicate that the suggested correlation is in modest agreement with the experimental data, with the transition occurring at a Reynolds number of 600.

Relaxation of nonspherical sessile drops towards equilibrium

We present a theoretical study related to a recent experiment on the coalescence of sessile drops. The study deals with the kinetics of relaxation towards equilibrium, under the action of surface tension, of a spheroidal drop on a flat surface. For such a nonspherical drop under partial wetting conditions, the dynamic contact angle varies along the contact line. We propose a new nonlocal approach to the wetting dynamics, where the contact line velocity depends on the geometry of the whole drop. We compare our results to those of the conventional approach in which the contact line velocity depends only on the local value of the dynamic contact angle. The influence on drop dynamics of the pinning of the contact line by surface defects is also discussed.

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.

Influence of charge relaxation on the capillary oscillations of a charged viscous spheroidal drop

A dispersion relation is derived for the spectrum of capillary modes of a charged spheroidal drop of a viscous liquid with allowance for charge relaxation. It is shown that the finite charge transport rate leads to lowering of the instability growth rates for various capillary modes of a spheroidal drop of a low-viscosity liquid. As the degree of deformation of the drop increases, the magnitude of the absolute change in the growth rate caused by the finite rate of charge redistribution decreases.

Local enhancement of a uniform electrostatic field near the tip of a spheroidal drop

The enhancement in a uniform electrostatic field at the tip of a spheroidal drop is shown to depend on the dielectric constant of the drop material, its initial radius, and the external electric field and to become greater as these increase. The loss of stability of a drop in an external electrostatic field that is accompanied by a very rapid growth in the magnitude of the spheroidal deformation causes a rapid, transient enhancement of the field at its tip.

Critical conditions for instability of a highly charged oblate spheroidal drop

The spectrum of capillary oscillations of a charged oblate spheroidal drop is calculated in neglect of the interaction between modes by means of a perturbation expansion in the small deviation of the equilibrium shape of the drop from spherical. The critical conditions for instability of its nth mode with respect to the self-charge are calculated in the form of an analytical function describing how the dimensionless Rayleigh parameter characterizing the stability of the drop depends on the value of the spheroidal deformation.

Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number

The electrohydrodynamic flow associated with a fluid drop in an electric field is a consequence of the tangential electric stress at the fluid interface. The tangential viscous stress due to the electrohydrodynamic flow arises to just balance the tangential electric stress at the fluid interface so that the traction boundary condition is satisfied. Influenced by both the local electric stress and viscous stress, the drop interface may exhibit various shapes. The presence of fluid flow also leads to charge convection phenomena. The relative significance of the charge convection effect is usually measured in terms of the electric Reynolds number, Re E , defined as the ratio of the timescales of charge convection by flow and that for charge relaxation by ohmic conduction. This work presents a quantitative analysis of the charge convection effects in a framework of the leaky dielectric model at finite Re E , which has not been considered in previous investigations. Axisymmetric steady flows driven by an applied uniform electric field about a deformable fluid drop suspended in an immiscible fluid are studied by computational means of the Galerkin finite–element method with supplementary asymptotic analysis. The results of finite–element computations are in general agreement with the prediction by the asymptotic analysis for spherical drops at vanishingly small Re E . A common effect of charge convection is found to reduce the intensity of electrohydrodynamic flow. As a consequence, oblate drops are predicted to be less deformed in an electric field when charge convection is taken into account. The prolate drops are often associated with an equator–to–pole flow, which convects charges toward the poles to form a charge distribution resembling that in a highly conducting drop immersed in an insulating medium. Therefore, charge convection tends to enhance the prolate drop deformation. In many cases, charge convection effects are found to be significant even at apparently small Re E , corresponding to the charge relaxation time–scale about 10 −3 s, suggesting that many experimental results reported in the previous publications could have been influenced by charge convection effects.

Oscillations of a viscous spheroidal drop with allowance for capillary forces

Oscillations of a viscous spheroidal drop with allowance for capillary forces

Transient heat transfer to a spheroidal liquid drop suspended in an electric field

Heat transfer to a spheroidal drop of a dielectric fluid suspended in another dielectric fluid in the presence of an electric field is investigated in this paper. The effect of drop deformation on the heat transport is analyzed. A range of prescribed drop deformations from ba=0.99 to ba=0.4 is considered. The electric potential distribution is numerically obtained for each deformed drop configuration. The resulting flow field is determined by the solution of the Navier-Stokes equations in the continuous and the dispersed phase. The transient heat transfer is studied in the limit of bulk of the resistance to the heat transport being in the droplet. An alternating-direction-implicit (ADI) method is used to obtain the transient temperature field for drop Peclet number from 5 to 1500. In the limiting case of negligible drop deformation, we recover the flow field and the electrically induced interfacial stresses for a liquid sphere calculated by analytical methods. Heat transfer results for a spherical drop (ba=0.99) show excellent agreement with results available in the published literature. Study of drop deformation reveals interesting features in the flow and heat transport. The location of the maximum surface velocity moves toward the equatorial plane for a deformed drop compared to that for a liquid sphere. It is found that the increase in drop deformation results in higher steady state Nusselt numbers for very large as well as very small equivalent Peclet numbers. However, for intermediate equivalent drop Peclet numbers, the equivalent steady state Nusselt numbers for a deformed drop may be lower than for a sphere.

E 6 diffraction catastrophe in light scattered near the rainbow region of an acoustically levitated spheroidal water drop

Light scattering by oblate acoustically levitated drops of water gives a method of exploring a variety of weak foci known as diffraction catastrophes. The present research concerns laser light observations of a formerly unidentified caustic exhibited by white light scattering of a highly oblate drop [H. J. Simpson and P. L. Marston, Appl. Opt. 30, 3468–3473 (1991), plate 16]. The scattering pattern exhibits a rich structure that changes rapidly over a narrow range of drop aspect ratio D/H. The caustic patterns are those predicted of an E6 catastrophe in the range of D/H explored [J. F. Nye, Proc. R. Soc. London (accepted for publication)]. The focal section of the E6 is observed though the E6 has a control space of five dimensions while the experiment explores only three dimensions. The observations illustrate how special symmetrics and geometries can constrain the caustics produced in far-field scattering from a penetrable spheroid. [Work support by ONR.]

Lips caustics in light backscattered from an acoustically levitated spheroidal water drop

Cusp diffraction catastrophes that open up roughly transverse to the propagation direction are known to exist for both light and sound [P. L. Marston, J. Acoust. Soc. Am. 81, 226–232 (1987)]. They may be produced by reflecting high-frequency sound from curved surfaces. In the present research, a closely related caustic in which the cusp curves join to form a pair of lips is studied. This caustic was predicted to exist [J. F. Nye, Nature 312, 531–532 (1984)] in light backscattered from horizontally illuminated oblate drops of water, provided the axis ratio q = D/H was within certain ranges. The associated rays are diffracted at the drop's surface and have only one internal reflection. As q is increased above the critical value q4≈1.31 associated with a hyperbolic umbilic focal section, the cusp points at the ends of the lips caustic were predicted to merge in the backward direction at a “lips event” when q = qL1 ≈ 1.42. For qL1 < q < qL2 ≈ 1.58, no caustics for this class of ray are expected while a second lips event occurs at qL2. Observations of farfield scattering from levitated drops support Nye's analysis and illustrate a mechanism for producing lips caustics. The observed backscattering is weak for q between qL1 and qL2. [Work supported by ONR.]

Heat-transfer process in direct contact evaporator. (3rd report. Behavior and heat transfer of R113 liquid floating on flowing water surface).

To explain the negative temperature dependence of heat-transfer coefficient in a column-like boiling regime, where a vapor column is formed just behind a nozzle outlet, simplified model experiments were carried out. Optical observation of R113 liquid behavior floating on a flowing water surface gave the following results. R113 liquid on the water surface alternatively could take three states, such as spheroidal drop, lens and continuous thin film. When the temperature difference between water and saturation temperature of R113 was raised, the appearance time of the spheroidal drop state became longer and the average thickness of the lens or continuous thin film states with shorter appearance time increased. Heat transfer to evaporating R113 liquid was enhanced in this order of three states. These results lead to a lower heat-transfer coefficient at a higher temperature difference.

Numerical analysis of drag coefficients and the heat and mass transfer of spheroidal drops.

Numerical analyses of the drag coefficients and the heat and mass transfer from a spheroidal liquid drop under low mass flux conditions were made by use of a finite difference method for Rep=1-200, Pr or Sc=0.5-2.0 and aspect ratio=0.3-2.0. The present numerical data were compared with existing experimental and numerical data and showed good agreement. The effect of deformation from spherical to spheroidal shape on drag coefficients and heat and diffusion fluxes of a spheroidal particle or a drop were studied. A new correlation for the effect of the aspect ratio (b/a) on heat and mass transfer of a spheroidal drop was proposed.

Capillary oscillations of a drop in an electric field

The capillary oscillations of a conducting spheroidal drop with small eccentricity in an electric field are studied. The effect of the electric field is to lower the natural frequencies of oscillations and the damping coefficient.

Characteristics of rain induced attenuation and phase shift at cm and mm waves using a tropical raindrop size distribution model

The tropical raindrop size distribution model developed by Ajayi and Olsen has been employed to study some characteristics of rain induced attenuation and phase shift for a tropical location for spherical, oblate spheroidal and Pruppacher-Pitter drop shapes. Parameters such as the a and b values for the power law relation between the specific attenuation and rainfall rate as well as differential attenuation and phase shift and their normalized values, were computed. A single power law between the specific phase shift and the rain rate was found to be adequate for vertical polarization, whilst a two-segment power law fitting is required for horizontal polarization between 1GHz and about 100GHz. The results were compared in many cases with those obtained with the Laws and Parsons drop size distribution, currently adopted by the CCIR for scattering applications.

Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci

An experiment by Marston and Trinh1, showed how an oblate spheroidal water drop illuminated by a parallel beam of light forms a caustic in the far field characterized by the hyperbolic umbilic catastrophe—a generalization of the primary rainbow formed by a spherical drop. As the axis-ratio of the drop was changed, they observed a singular section of the catastrophe appearing for D/H = 1.305 ± 0.016—D being the diameter in the horizontal equatorial plane and H the diameter along the vertical axis of rotational symmetry. I show here how this ratio may be calculated to confirm quantitatively the arguments1 that the hyperbolic–umbilic catastrophe is associated with skew rays, and predict the occurrence of novel ‘lips’ phenomena, yet to be observed.

Stability of spheroidal drops in homogeneous electric fields

The deformation of dielectric drops by uniform electric field is studied theoretically. A first-order approximation shows that if the effects of viscosity are neglected the deformed drop takes the shape of a spheroid. The resulting changes in electrostatic and surface energies are calculated and the stability of the spheroidal drop is discussed. The results are compared with those obtained from earlier studies.

The stability of a water drop oscillating with finite amplitude in an electric field

By assuming that an uncharged drop situated in a uniform electric fieldEretains a spheroidal shape while oscillating about its equilibrium configuration, two approximate equations of motion are derived for the deformation ratio γ expressed as the ratioa/bof the major and minor axis of the drop. Solutions of these equations of motion indicate that the stability of a drop of undistorted radiusRand surface tensionTdepends uponE(R/T)½and the initial displacement of γ from its equilibrium value. The predictions of the two equations are compared to assess the accuracy of the spheroidal assumption as applied to such a dynamical situation. The analysis is used to determine the stability criterion of a drop subject to a step function field. Finally, the limit of validity of the spheroidal assumption is discussed in terms of Rayleigh's criterion for the stability of charged spherical drops. By applying Rayleigh's criterion to the poles of a spheroidal drop, the stage at which the drop departs from spheroidal form to form conical jets was approximately determined.

Numerical computations of the dynamics of the disintegration of a drop situated in an electric field

The marker-and-cell technique, which has recently been developed for modelling the dynamics of incompressible fluid flow by means of a high-speed computer, has been applied to a study of the instability of an uncharged liquid drop of radius R and surface tension T situated in an electric field of strength E . This problem, which is of considerable importance in certain cloud physical situations, was previously treated analytically by Taylor who assumed that the drop retained a spheroidal shape throughout the period of deformation until the instability point was attained. His calculated instability criteria, namely that E(R/T) ½ = 1.625 when the ratio of the semi-major to semi-minor axes a/b = 1.9, agree well with experimental measurements. The present numerical calculations permit a quantitative assessment to be made of the validity of the spheroidal assumption and, of greater importance, provide a description of the dynamics of the disintegration of a drop subjected to intense electrical forces. In order to conserve computer time the initial condition was assumed to be that a spheroidal drop of undistorted radius 0.2 cm and surface tension 70 dyn cm –1 , possessing a degree of deformation represented by a/b = 1.9, was introduced into a field of strength E = 9500 V cm –1 , which is 4% greater than the critical value deduced on the basis of the spheroidal assumption. Computed cross-sections through the axis of the drop at appropriate intervals of time illustrate the onset of instability at the poles of the drop and demonstrate that the spheroidal assumption provides an extremely accurate representation of the shape of a drop situated in an electric field. This latter conclusion is reinforced by the calculated distribution of pressure over the surface of a spheroidal drop introduced into a critical field. Calculated values of the outward velocity of the fluid at the poles of the disintegrating drop show that capillary waves generated on the surface increase with amplitude until the final stage of instability is initiated, whereupon the velocity increases extremely rapidly, culminating in the ejection of fluid from each pole of the drop, probably in the form of a jet which subsequently breaks up to produce a number of droplets. The corresponding inward velocities at the equatorial points undergo much less variation than the polar velocity and do not exhibit a particularly pronounced increase at the time of instability. The computations indicate that the velocity of ejection of fluid from the poles of the drops is of the order of 100 cm s –1 . This value is in excellent agreement with experimental measurements made by several workers.

Steady motion of a prolate spheroidal drop

An expression for the mass “induced” by a prolate spheroid moving in an ideal fluid is obtained. The drag force on a prolate spheroidal drop is determined by using the effective mass approach. The drag force on a prolate drop is compared with the drag force on an oblate drop of the same volume and the same semi-axis ratio. The limits of applicability of the theory are discussed.