ALBERT

Notes: As a preliminary to understanding the complicated interactions between two electrified drops, this paper analyzes the simpler but instructive problem of the electrohydrostatic interactions between two parallel, translationally symmetric supported liquid columns. The columns are taken to be electrically conducting, surrounded by an insulating fluid, and pinned at their contact lines on the surface of an insulating solid support. The issue of the shapes and stability of the columns is mathematically a two-dimensional, nonlinear free boundary problem, which is solved here by means of the Galerkin/finite element method. Despite the generality of the formulation of the problem and that of the numerical scheme used to solve it, attention is focused here on situations in which the two columns have the same volume per unit length and their undeformed cross sections correspond to semicircles bounded by a straight line that represents the solid plane on which the columns are pinned. Computational results are reported that show different behaviors of electrified columns obtained by varying the direction in which the external field is applied or connecting/disconnecting the columns from power supplies that maintain them at fixed electric potentials. When the potentials of the columns are held fixed, an externally applied field always tends to pull them apart. For isolated columns, however, an externally applied field within a large range of oblique angles tends to pull the columns together. In the absence of an externally applied field, electrostatic interactions can also arise when the columns bear net electrical charges. Whether the interaction forces are attractive or repulsive then depends on the relative amounts of charge on the columns. An attractive force can result even when the two columns bear net charges of the same sign, provided that the relative difference in the amounts of charge is large enough. © 1995 American Institute of Physics.

Notes: During drop formation from a tube, a thin liquid thread—the precursor to satellites—connects an about-to-form primary drop to the remainder of the liquid hanging from the tube at the incipience of breakup. Whether the thread, once it detaches from the primary and pendant drops, evolves into a sphere or breaks into several subsatellites has heretofore been inadequately explored due to experimental and theoretical difficulties. These challenges are resolved here with an ultrafast digital imaging system and a novel computational algorithm. New findings range from the discovery of unexpected dynamics to the first demonstration of the transition from one scaling law governing interface rupture to another. © 2001 American Institute of Physics.

Notes: A liquid being ejected from a nozzle emanates from it as discrete, uniformly sized drops when the flow rate is sufficiently low. In this paper, an experimental study is presented of the dynamics of a viscous liquid drop that is being formed directly at the tip of a vertical tube into ambient air. The evolution in time of the drop shape and volume is monitored with a time resolution of 1/12 to 1 ms. Following the detachment of the previous drop, the profile of the new growing drop at first changes from spherical to pear-shaped. As time advances, the throat of the pear-shaped drop takes on the appearance of a liquid thread that connects the bottom portion of the drop that is about to detach to the rest of the liquid that is pendant from the tube. The focus here is on probing the effects of physical and geometric parameters on the universal features of drop formation, paying special attention to the development, extension, and breakup of the liquid thread and the satellite drops that are formed subsequent to its breakup. The role of surfactants in modifying the dynamics of drop formation is also studied. The effects of finite inertial, capillary, viscous, and gravitational forces are all accounted for to classify drastically different formation dynamics and to elucidate the fate of satellite drops following thread rupture. © 1995 American Institute of Physics.

Notes: The lack of a simple method for generating drops whose radii (Rd) are much smaller than those (R) of nozzles which produce them has heretofore been a major limitation of the drop-on-demand technique. Therefore, the only reliable way to reduce Rd to date has been to reduce R. A new method is reported which allows an order of magnitude reduction in drop volume while using the same nozzle. It is shown that the key to forming drops with Rd〈R is to judiciously control the capillary, viscous, and inertial time scales that govern the flow within the nozzle and the forming drop. The time scales are controlled in experiments by appropriately driving a piezoelectric sleeve surrounding a microcapillary tube, and the interplay between them is elucidated through computation. © 2002 American Institute of Physics.

Notes: Dynamics of formation of a drop of a Newtonian liquid from a capillary tube into an ambient gas phase is studied computationally and experimentally. While this problem has previously been studied computationally either (a) using a set of one-dimensional equations or (b) treating the dynamics as that of irrotational flow of an inviscid fluid or creeping flow, here the full nonlinear, transient Navier–Stokes system subject to appropriate initial and boundary conditions is solved in two dimensions to analyze the dynamics at finite Reynolds numbers. The success of the computations rests on a finite element algorithm incorporating a multiregion mesh which conforms to and evolves with the changing shape of the drop. The new algorithm is able to capture both the gross features of the phenomenon, such as the limiting length of a drop at breakup and the volume of the primary drop, and its fine features, such as a microthread that develops from a main thread or a neck in a viscous drop approaching breakup. The accuracy of the new calculations is verified by comparison of computed predictions to old and new experiments. With the new algorithm, it is shown for the first time that the interface of a viscous drop can overturn before the drop breaks. Calculations have also been carried out to determine the range of parameters over which algorithms that treat the drop liquid as inviscid and the flow inside it as irrotational can accurately predict the dynamics of formation of drops of low viscosity liquids. Limiting lengths of drops and primary drop volumes are computed over a wide range of the parameter space spanned by the relevant dimensionless groups. © 1999 American Institute of Physics.

Notes: Oscillations of supported liquid drops are the subject of wide scientific interest, with applications in areas as diverse as liquid–liquid extraction, synthesis of ceramic powders, growing of pure crystals in low gravity, and measurement of dynamic surface tension. In this study, axisymmetric forced oscillations of arbitrary amplitude of a viscous liquid drop of fixed volume which is pendant from or sessile on a rod with a fixed contact line and surrounded by an inviscid ambient gas are induced by moving the rod in the vertical direction sinusoidally in time. This nonlinear free boundary problem is solved by a method of lines using Galerkin/finite element analysis for discretization in space and an implicit, adaptive finite difference technique for discretization in time. The variation of the drop response over a wide range of the governing parameters (Reynolds number Re, gravitational Bond number G, volume, and forcing frequency and amplitude) is analyzed. The results show that as the forcing frequency is increased, a sequence of oscillation modes is observed, each with its own resonance frequency ωrn, n=1,2,..., at which drop response amplitude reaches a local maximum. While resonance frequencies depend strongly on drop size and on forcing amplitude, the effect of Reynolds number on ωrn is large when Re is small and diminishes when Re is large, in accord with observations for free oscillations. At high Re, a sharp increase in drop deformation can occur for drops forced to oscillate in the vicinity of their resonance frequencies, indicating the incipience of hysteresis. The maximum observed drop deformations increase with Re, G, and forcing amplitude, while the value of the drop deformation as a function of drop size is determined by a balance between the magnitude of the viscous shear stress imposed on the drop liquid by the solid rod relative to the capillary pressure due to surface tension acting on the fluid interface. The effects of viscous dissipation are also seen in the damping of various oscillation modes and in the creation, evolution in time, and disappearance of zones of fluid recirculation within the drop. © 1997 American Institute of Physics.

Notes: Whereas oscillations of free drops have been scrutinized for over a century, oscillations of supported (pendant or sessile) drops have only received limited attention to date. Here, the focus is on the axisymmetric, free oscillations of arbitrary amplitude of a viscous liquid drop of fixed volume V that is pendant from a solid rod of radius R and is surrounded by a dynamically inactive ambient gas. This nonlinear free boundary problem is solved by a method of lines using Galerkin/finite element analysis for discretization in space and an implicit, adaptive finite difference technique for discretization in time. The dynamics of such nonlinear oscillations are governed by four dimensionless groups: (1) a Reynolds number Re, (2) a gravitational Bond number G, (3) dimensionless drop volume V/R3 or some other measure of drop size, and (4) a measure of initial drop deformation a/b. In contrast to free drops whose frequencies of oscillation ω decrease as the amplitudes of their initial deformations increase, the change in frequency Δω of pendant drops with increasing initial deformation is drop size dependent. As the average linear size of pendant drops characterized by V1/3 becomes large compared to the rod radius, V1/3/R(very-much-greater-than)1, Δω falls as a/b rises, in accordance with results for free drops. The dynamics of very small drops, i.e., ones for which V1/3/R(very-much-less-than)1, however, are profoundly affected by the presence of the solid rod. For such small drops, Δω rises as a/b rises, a remarkable fact. The results show that for drops of a given size, the frequency is insignificantly affected by viscosity over a wide of range of Reynolds numbers. However, when Re falls below a critical value, the nature of drop motion changes from underdamped oscillations to an aperiodic return to the rest state. Detailed examination of flow fields inside oscillating drops and decomposition of drop shapes into their linear modes supply further insights into the underlying physics. The effect of finite G in modifying the frequencies of oscillations and the rate at which they are damped is also investigated.