Osman A. Basaran

An experimental study of dynamics of drop formation

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...

Nonlinear oscillations of viscous liquid drops

A fundamental understanding of nonlinear oscillations of a viscous liquid drop is needed in diverse areas of science and technology. In this paper, the moderate- to large-amplitude axisymmetric oscillations of a viscous liquid drop, which is immersed in dynamically inactive surroundings, are analysed by solving the free boundary problem comprised of the Navier–Stokes system and appropriate interfacial conditions at the drop–ambient fluid interface. The means are the Galerkin/finite-element technique, an implicit predictor-corrector method, and Newtons method for solving the resulting system of nonlinear algebraic equations. Attention is focused here on oscillations of drops that are released from an initial static deformation. Two dimensionless groups govern such nonlinear oscillations: a Reynolds number, Re , and some measure of the initial drop deformation. Accuracy is attested by demonstrating that (i) the drop volume remains virtually constant, (ii) dynamic response to small-and moderate-amplitude disturbances agrees with linear and perturbation theories, and (iii) large-amplitude oscillations compare well with the few published predictions made with the marker-and-cell method and experiments. The new results show that viscous drops that are released from an initially two-lobed configuration spend less time in prolate form than inviscid drops, in agreement with experiments. Moreover, the frequency of oscillation of viscous drops released from such initially two-lobed configurations decreases with the square of the initial amplitude of deformation as Re gets large for moderate-amplitude oscillations, but the change becomes less dramatic as Re falls and/or the initial amplitude of deformation rises. The rate at which these oscillations are damped during the first period rises as initial drop deformation increases; thereafter the damping rate is lower but remains virtually time-independent regardless of Re or the initial amplitude of deformation. The new results also show that finite viscosity has a much bigger effect on mode coupling phenomena and, in particular, on resonant mode interactions than might be anticipated based on results of computations incorporating only an infinitesimal amount of viscosity.

Shapes and stability of pendant and sessile dielectric drops in an electric field

Axisymmetric equilibrium shapes and stability of linearly polarizable dielectric (ferrofluid) drops of fixed volume which are pendant/sessile on one plate of a parallelplate capacitor and are subjected to an applied electric (magnetic) field are determined by solving simultaneously the free boundary problem comprised of the Young-Laplace equation for drop shape and the Laplace equation for electric (magnetic) field distribution. When the contact angle that the drop makes with the plate is fixed to be 90 and the distance between the plates is infinite, the results are identical to those of a free drop immersed in a uniform field and resolve discrepancies between previously reported theoretical predictions and experimental measurements. Remarkably, regardless of the value of the ratio of the permittivity (permeability) of the drop to that of the surrounding fluid, K, drop shapes develop conical tips as drop deformation increases. However, three types of behaviour are found, depending on the value of K. When K K~ > K~, families of equilibrium drop shapes become unstable at turning points with respect to field strength. Beyond the turning points, the unstable families terminate : the mean curvature at the virtually conical drop tip grows without bound. However, in the range K~ < K < K~, the new results predict that drop deformation exhibits hysteresis, in accord with experiments of Bacri, Salin & Massart (1982) and Bacri & Salin (1982, 1983). Such hysteresis phenomena have been surmized previously on the basis of approximate theories, though they have not been calculated systematically until now. Moreover, detailed computations reveal the importance of varying the drop size and plate spacing, and whether, along the three-phase contact line, the contact line is fixed or the contact angle is prescribed.

Nonlinear oscillations of pendant drops

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...

Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field

Axisymmetric shapes and stability of nonlinearly polarizable dielectric (ferrofluid) drops of fixed volume which are pendant/sessile on one plate of a parallel-plate capacitor and are subjected to an applied electric (magnetic) field are determined by solving simultaneously the free boundary problem comprised of the Young-Laplace equation for drop shape and the Maxwell equations for electric (magnetic) field distribution

Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall

Steady states of a translationally-symmetric cylindrical bubble protruding from a slot in a solid wall into a liquid undergoing a simple shear flow are investigated. Deformations of and the flow past the bubble are determined by solving the nonlinear free-boundary problem comprised of the two-dimensional Navier–Stokes system by the Galerkin/finite element method. Under conditions of creeping flow, the results of finite element computations are shown to agree well with asymptotic results. When the Reynolds number Re is finite, flow separates from the free surface and a recirculating eddy forms behind the bubble. The length of the separated eddy measured in the flow direction increases with Re , whereas its width is confined to within the region that lies between the supporting solid surface and the separation point at the free surface. By tracking solution branches in parameter space with an arc-length continuation method, curves of bubble deformation versus Reynolds number are found to exhibit turning points when Re reaches a critical value Re c. Therefore, along a family of bubble shapes, solutions do not exist when Re > Re c. The locations of turning points and the structure of flow fields are found to be governed virtually by a single parameter, We = Ca Re , where We and Ca are Weber and capillary numbers. Two markedly different modes of bubble deformation are identified at finite Re. One is dominant when Re is small and is tantamount to a plain skewing or tilting of the bubble in the downstream direction; the other becomes more pronounced when Re is large and corresponds to a pure upward stretching of the bubble tip.

Hysteresis in forced oscillations of pendant drops

A hysteresis phenomenon has been revealed through experiments conducted with large‐amplitude forced oscillations of pendant drops in air. Under strong excitation, the frequency response of a drop forced at constant amplitude exhibits jump behavior; a larger peak response amplitude e↓ appears at a lower frequency ω↓ during a downward (↓) variation of forcing frequency than during an upward (↑) variation, viz. e↓≳e↑ and ω↓<ω↑. Similar results are obtained when forcing amplitude is varied at constant frequency. This behavior is characteristic of a system with a soft nonlinearity. These findings indicate that oscillating pendant drops constitute a convenient system for studying nonlinear dynamics.

Oscillation Frequencies of Droplets Held Pendant on a Nozzle

Abstract Small-amplitude oscillations of liquid droplets held pendant on a nozzle and surrounded by either air or another liquid were investigated experimentally. The oscillations were induced by mechanical means, and the natural frequencies of oscillations were visually determined. Results are compared to existing related theories of Miller and Scriven and of Strani and Sabetta. As may be expected, the presence of the solid support causes measured oscillation frequencies for the lowest oscillation mode to deviate greatly from the Miller and Scriven low viscosity approximation (n = 2) for free spherical drops. Experiments are in qualitative agreement with the first (n = 1) mode predicted by Strani and Sabetta, with the best correspondence occurring under circumstances where the ratio of nozzle to drop radii is small. The difference between the experimental results and the theory of Strani and Sabetta can be attributed to the restrictive boundary conditions of their analytic model. Thus, better theoretical...

Wall effects on flow past fluid spheres at finite Reynolds number : wake structure and drag correlations

Abstract Following the pioneering works of Leal and coworkers, a detailed report is made on the dynamics of recirculating wakes that form at finite Reynolds number Re due to vorticity accumulation at the rear of a fluid sphere that is either suspended in a tube by an upflowing fluid ( the fluidized drop problem ) or falling in a tube ( the falling drop problem ). The axisymmetric, steady flow of a Newtonian fluid past the fluid sphere is determined by finite element analysis using a consistent penalty formulation. By way of example, the flow past a fluid sphere that is falling in a tube for which the ratio of the tube radius to the drop radius 1 λ = 5 undergoes remarkable transitions when the ratio of the viscosity of the drop to that of the ambient fluid, κ, varies over a narrow range. When κ ≤ 2.75, no wake forms behind the sphere as Re increases. When 3 ≤ κ Re = Re c (1) , grows, and eventually disappears as Re rises above a certain amount Re c (2) . Remarkably, when κ = 3, these critical Reynolds numbers are as low as Re c (1) = 51 ± 1 and Re c (2) = 77 ± 1. When κ ≥ 10, it is shown by using as many as 46,000 velocity degrees of freedom that the detached eddy attaches to the drop when Re exceeds a critical value, Re ≥ Re c ∗ , which was heretofore unknown. Whereas only a single, large-primary-eddy is present inside the drop when Re c ∗ , a second but much smaller-secondary-eddy also forms inside the drop upon attachment. Two new correlations are developed that account for the effects of a tube wall and finite fluid inertia on drag for fluidized and falling droplets. Moreover, in contrast to related correlations of others for fluid spheres that are placed in an infinite expanse of ambient fluid, the new correlations are valid over the entire range of Reynolds numbers considered.

Effects of physical properties and geometry on shapes and stability of polarizable drops in external fields

Abstract Equilibrium shapes and stability of linearly and nonlinearly polarizable drops floating freely between the two faces of a parallel-plate capacitor and subjected to an electric or magnetic field are found by solving the Young-Laplace equation for surface shape and the Maxwell equations for field distribution. Accounting for nonlinear polarization at last brings theory and experiment into accord. Finite plate separation—as in any real experiment—is also found to play a big role.