23 results
Local dynamics during thinning and rupture of liquid sheets of power-law fluids
- Vishrut Garg, Sumeet S. Thete, Christopher R. Anthony, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 942 / 10 July 2022
- Published online by Cambridge University Press:
- 17 May 2022, A15
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Rupture of liquid sheets of power-law fluids surrounded by a gas is analysed under the competing influences of pressure due to van der Waals attraction, inertia, viscous stress and capillary pressure due to surface tension. Results of a combined theoretical and computational study are presented over the entire range of parameters governing the thinning of a power-law fluid of power-law exponent $0 < n \le 1$ ($n=1$: Newtonian fluid) and Ohnesorge number $0 \le Oh < \infty$, where $Oh \equiv \mu _0/\sqrt {\rho h_0 \sigma }$, and $\mu _0, \rho, h_0$ and $\sigma$ stand for the zero-deformation-rate viscosity, density, the initial sheet half-thickness and surface tension, respectively. The dynamics in the vicinity of the space–time singularity where the sheet ruptures is asymptotically self-similar, and thus the variation with time remaining until rupture $\tau \equiv t_R - t$, where $t_R$ is the time instant $t$ at which the sheet ruptures, of sheet half-thickness, lateral length scale and lateral velocity is determined analytically and confirmed by simulations. For sheets for which inertia is negligible ($Oh^{-1}=0$), two distinct viscous scaling regimes are found, one for $0.58 < n \le 1$ and the other for $n \le 0.58$. The thinning dynamics of inviscid sheets ($Oh = 0$) is identical to that of Newtonian ones. For real fluids for which neither viscosity nor inertia is negligible, it is shown that the aforementioned creeping and inertial flow regimes are transitory and the thinning of power-law sheets exhibits a remarkably richer set of scaling transitions compared with Newtonian sheets.
Electrohydrodynamics of lenticular drops and equatorial streaming
- Brayden W. Wagoner, Petia M. Vlahovska, Michael T. Harris, Osman A. Basaran
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- Journal of Fluid Mechanics / Volume 925 / 25 October 2021
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- 31 August 2021, A36
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Drops subjected to electric fields can deform into singular shapes exhibiting apparent sharp tips. At high field strengths, a perfectly conducting drop surrounded by a perfectly insulating exterior fluid deforms into a prolate-shaped drop with conical ends and can exist in hydrostatic equilibrium. On the conical ends, capillary stress, which is due to the out-of-plane curvature and is singular, balances electric normal stress which is also singular. If the two phases are not perfect conductors/insulators but are both leaky dielectrics and the drop is much more conducting and viscous than the exterior, electric tangential stress disrupts the hydrostatic force balance and leads to jet emission from the cone's apex. If, however, the physical situation is inverted so that a weakly conducting, slightly viscous drop is immersed in a highly conducting, more viscous exterior, the drop deforms into an oblate lens-like profile before eventually becoming unstable. In experiments, the equator of a lenticular drop superficially resembles a wedge prior to instability. Such a drop disintegrates by equatorial streaming by ejecting a thin liquid sheet from its equator. We show theoretically by performing a local analysis that a lenticular drop's equatorial profile can be a wedge only if an approximate form of the surface charge transport equation – continuity of normal current condition – is used. Moreover, we demonstrate via numerical simulation that such wedge-shaped drops do not become unstable and therefore cannot emit equatorial sheets. We then show by transient simulations how equatorial streaming can occur when charge transport along the interface is analysed without approximation.
Oscillations of a ring-constrained charged drop
- Brayden W. Wagoner, Vishrut Garg, Michael T. Harris, Doraiswami Ramkrishna, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 921 / 25 August 2021
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- 30 June 2021, A19
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Free drops of uncharged and charged inviscid, conducting fluids subjected to small-amplitude perturbations undergo linear oscillations (Rayleigh, Proc. R. Soc. London, vol. 29, no. 196–199, 1879, pp. 71–97; Rayleigh, Philos. Mag., vol. 14, no. 87, 1882, pp. 184–186). There exist a countably infinite number of oscillation modes, $n=2, 3, \ldots$, each of which has a characteristic frequency and mode shape. Presence of charge ($Q$) lowers modal frequencies and leads to instability when $Q>Q_R$ (Rayleigh limit). The $n=0$ and $n=1$ modes are disallowed because they violate volume conservation and cause centre of mass (COM) motion. Thus, the first mode to become unstable is the $n=2$ prolate–oblate mode. For free drops, there is a one-to-one correspondence between mode number and shape (Legendre polynomial $P_n$). Recent research has shifted to studying oscillations of spherical drops constrained by solid rings. Pinning the drop introduces a new low-frequency mode of oscillation ($n=1$), one associated primarily with COM translation of the constrained drop. We analyse theoretically the effect of charge on oscillations of constrained drops. Using normal modes and solving a linear operator eigenvalue problem, we determine the frequency of each oscillation mode. Results demonstrate that for ring-constrained charged drops (RCCDs), the association between mode number and shape is lost. For certain pinning locations, oscillations exhibit eigenvalue veering as $Q$ increases. While slightly charged RCCDs pinned at zeros of $P_2$ have a first mode that involves COM motion and a second mode that entails prolate–oblate oscillations, the modes flip as $Q$ increases. Thereafter, prolate–oblate oscillations of RCCDs adopt the role of being the first mode because they exhibit the lowest vibration frequency. At the Rayleigh limit, the first eigenmode – prolate–oblate oscillations – loses stability while the second – involving COM motion – remains stable.
Pinch-off of a surfactant-covered jet
- Hansol Wee, Brayden W. Wagoner, Vishrut Garg, Pritish M. Kamat, Osman A. Basaran
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- Journal of Fluid Mechanics / Volume 908 / 10 February 2021
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- 11 December 2020, A38
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Surfactants at fluid interfaces not only lower and cause gradients in surface tension but can induce additional surface rheological effects in response to dilatational and shear deformations. Surface tension and surface viscosities are both functions of surfactant concentration. Measurement of surface tension and determination of its effects on interfacial flows are now well established. Measurement of surface viscosities, however, is notoriously difficult. Consequently, quantitative characterization of their effects in interfacial flows has proven challenging. One reason behind this difficulty is that, with most existing methods of measurement, it is often impossible to isolate the effects of surface viscous stresses from those due to Marangoni stresses. Here, a combined asymptotic and numerical analysis is presented of the pinch-off of a surfactant-covered Newtonian liquid jet. Similarity solutions obtained from slender-jet theory and numerical solutions are presented for jets with and without surface rheological effects. Near pinch-off, it is demonstrated that Marangoni stresses become negligible compared to other forces. The rate of jet thinning is shown to be significantly lowered by surface viscous effects. From analysis of the dynamics near the pinch-off singularity, a simple analytical formula is derived for inferring surface viscosities. Three-dimensional, axisymmetric simulations confirm the validity of the asymptotic analyses but also demonstrate that a thinning jet traverses a number of intermediate regimes before eventually entering the final asymptotic regime.
Electric-field-induced transitions from spherical to discocyte and lens-shaped drops
- Brayden W. Wagoner, Petia M. Vlahovska, Michael T. Harris, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 904 / 10 December 2020
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- 12 October 2020, R4
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When a poorly conducting drop that is surrounded by a more conducting exterior fluid is subjected to an electric field, the drop can deform into an oblate shape at low field strengths. Such drops become unstable at high field strengths and display two types of dynamics, dimpling and equatorial streaming, the physics of which is currently not understood. If the drop is more viscous, dimples form and grow at the poles of the drop and eventually the discocyte-shaped drop breaks up to form a torus. If the exterior fluid is more viscous, the drop deforms into a lens and sheds rings from the equator that subsequently break into a number of smaller droplets. A theoretical explanation as to why dimple- and lens-shaped drops occur, and the mechanisms for the onset of these instabilities, are provided by determining steady-state solutions by simulation and inferring their stability from bifurcation analysis. For large drop viscosities, electric shear stress is shown to play a dominant role and to result in dimpling. For small drop viscosities, equatorial normal stresses (electric, hydrodynamic and capillary) become unbounded and lead to the lens shape.
Bubble coalescence in low-viscosity power-law fluids
- Pritish M. Kamat, Christopher R. Anthony, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 902 / 10 November 2020
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- 04 September 2020, A8
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As two spherical gas bubbles of radii $\tilde {R}$ are brought together inside a liquid of density $\tilde {\rho }$, viscosity $\tilde {\mu }$ and surface tension $\tilde {\sigma }$, the liquid sheet separating them drains, thins and ultimately ruptures. The instant and location at which the bubbles make contact, and whereby a circular hole of vanishingly small radius is formed in the thin sheet, represent the occurrence of a finite-time singularity. The large curvature near the edge of the sheet where the hole has just formed, and where the two bubbles are now connected via a microscopic gas bridge, drives liquid to flow radially outward, causing the sheet to retract and the radius of the hole $\tilde {R}_{min}$ to increase with time. Recent work in this area has uncovered self-similarity and universal scaling regimes when two bubbles coalesce in a Newtonian fluid. Motivated by applications in which the exterior is a deformation-rate-thinning, power-law fluid, recent studies on bubble coalescence in Newtonian fluids are extended to coalescence in power-law fluids. In such fluids, viscosity decreases with deformation rate $\dot {\tilde {\gamma }}$ raised to the $n - 1$ power where $0 < n \le 1$ ($n = 1$ for a Newtonian fluid). Attention is focused here on power-law fluids that are slightly viscous at zero deformation rate, i.e. when the Ohnesorge number $Oh = \tilde {\mu }_{0}/(\tilde {\rho } \tilde {R} \tilde {\sigma })^{1/2}$ is small ($Oh \ll 1$) and where $\tilde {\mu }_0$ is the zero-deformation-rate viscosity. A combination of thin-film theory and three-dimensional, axisymmetric computations is used to probe the dynamics in the aftermath of the singularity. Heretofore unexplored regimes are uncovered, and criteria are developed for transitions between different regimes. The existence of a truly inviscid regime, predicted long ago by Keller (Phys. Fluids, vol. 26, 1983, pp. 3451–3453) and which comes into play as a purely geometrical limit of the free-surface shape, is also reported. New insights are presented on the much studied Newtonian limit beyond the initial regime reported by Munro et al. (J. Fluid Mech., vol. 773, 2015, R3). The paper concludes with a phase diagram in $(n, \tilde {R}_{min}/\tilde {R})$-space, where the index $n$ characterizes the fluid and $\tilde {R}_{min}/\tilde {R}$ the extent of coalescence, that highlights the various regimes and transitions between them.
Surfactant-driven escape from endpinching during contraction of nearly inviscid filaments
- Pritish M. Kamat, Brayden W. Wagoner, Alfonso A. Castrejón-Pita, José R. Castrejón-Pita, Christopher R. Anthony, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 899 / 25 September 2020
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- 24 July 2020, A28
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Highly stretched liquid drops, or filaments, surrounded by a gas are routinely encountered in nature and industry. Such filaments can exhibit complex and unexpected dynamics as they contract under the action of surface tension. Instead of simply retracting to a sphere of the same volume, low-viscosity filaments exceeding a critical aspect ratio undergo localized pinch-off at their two ends resulting in a sequence of daughter droplets – a phenomenon called endpinching – which is an archetype breakup mode that is distinct from the classical Rayleigh–Plateau instability seen in jet breakup. It has been shown that endpinching can be precluded in filaments of intermediate viscosity, with the so-called escape from endpinching being understood heretofore only qualitatively as being caused by a viscous mechanism. Here, we show that a similar escape can also occur in nearly inviscid filaments when surfactants are present at the free surface of a recoiling filament. The fluid dynamics of the escape phenomenon is probed by numerical simulations. The computational results are used to show that the escape is driven by the action of Marangoni stress. Despite the apparently distinct physical origins of escape in moderately viscous surfactant-free filaments and that in nearly inviscid but surfactant-covered filaments, it is demonstrated that the genesis of all escape events can be attributed to a single cause – the generation of vorticity at curved interfaces. By analysing vorticity dynamics and the balance of vorticity in recoiling filaments, the manner in which surface tension gradients and concomitant Marangoni stresses can lead to escape from endpinching is clarified.
Inertial impedance of coalescence during collision of liquid drops
- Krishnaraj Sambath, Vishrut Garg, Sumeet S. Thete, Hariprasad J. Subramani, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 876 / 10 October 2019
- Published online by Cambridge University Press:
- 01 August 2019, pp. 449-480
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The fluid dynamics of the collision and coalescence of liquid drops has intrigued scientists and engineers for more than a century owing to its ubiquitousness in nature, e.g. raindrop coalescence, and industrial applications, e.g. breaking of emulsions in the oil and gas industry. The complexity of the underlying dynamics, which includes occurrence of hydrodynamic singularities, has required study of the problem at different scales – macroscopic, mesoscopic and molecular – using stochastic and deterministic methods. In this work, a multi-scale, deterministic method is adopted to simulate the approach, collision, and eventual coalescence of two drops where the drops as well as the ambient fluid are incompressible, Newtonian fluids. The free boundary problem governing the dynamics consists of the Navier–Stokes system and associated initial and boundary conditions that have been augmented to account for the effects of disjoining pressure as the separation between the drops becomes of the order of a few hundred nanometres. This free boundary problem is solved by a Galerkin finite element-based algorithm. The interplay of inertial, viscous, capillary and van der Waals forces on the coalescence dynamics is investigated. It is shown that, in certain situations, because of inertia two drops that are driven together can first bounce before ultimately coalescing. This bounce delays coalescence and can result in the computed value of the film drainage time departing significantly from that predicted from existing scaling theories.
Self-similar rupture of thin films of power-law fluids on a substrate
- Vishrut Garg, Pritish M. Kamat, Christopher R. Anthony, Sumeet S. Thete, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 826 / 10 September 2017
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- 04 August 2017, pp. 455-483
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Thinning and rupture of a thin film of a power-law fluid on a solid substrate under the balance between destabilizing van der Waals pressure and stabilizing capillary pressure is analysed. In a power-law fluid, viscosity is not constant but is proportional to the deformation rate raised to the $n-1$ power, where $0<n\leqslant 1$ is the power-law exponent ($n=1$ for a Newtonian fluid). In the first part of the paper, use is made of the slenderness of the film and the lubrication approximation is applied to the equations of motion to derive a spatially one-dimensional nonlinear evolution equation for film thickness. The variation with time remaining until rupture of the film thickness, the lateral length scale, fluid velocity and viscosity is determined analytically and confirmed by numerical simulations for both line rupture and point rupture. The self-similarity of the numerically computed film profiles in the vicinity of the location where the film thickness is a minimum is demonstrated by rescaling of the transient profiles with the scales deduced from theory. It is then shown that, in contrast to films of Newtonian fluids undergoing rupture for which inertia is always negligible, inertia can become important during thinning of films of power-law fluids in certain situations. The critical conditions for which inertia becomes important and the lubrication approximation is no longer valid are determined analytically. In the second part of the paper, thinning and rupture of thin films of power-law fluids in situations when inertia is important are simulated by solving numerically the spatially two-dimensional, transient Cauchy momentum and continuity equations. It is shown that as such films continue to thin, a change of scaling occurs from a regime in which van der Waals, capillary and viscous forces are important to one where the dominant balance of forces is between van der Waals, capillary and inertial forces while viscous force is negligible.
Thin-sheet flow between coalescing bubbles
- James P. Munro, Christopher R. Anthony, Osman A. Basaran, John R. Lister
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- Journal of Fluid Mechanics / Volume 773 / 25 June 2015
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- 20 May 2015, R3
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When two spherical bubbles touch, a hole is formed in the fluid sheet between them, and capillary pressure acting on its tightly curved edge drives an outward radial flow which widens the hole joining the bubbles. Recent images of the early stages of this process (Paulsen et al., Nat. Commun., vol. 5, 2014) show that the radius of the hole $r_{\!E}$ at time $t$ grows proportional to $t^{1/2}$, and that the rate is dependent on the fluid viscosity. Here, we explain this behaviour in terms of similarity solutions to a third-order system of radial extensional-flow equations for the thickness and velocity of the sheet of fluid between the bubbles, and determine the growth rate as a function of the Ohnesorge number $\mathit{Oh}$. The initially quadratic sheet profile allows the ratio of viscous and inertial effects to be independent of time. We show that the sheet is slender for $r_{\!E}\ll a$ if $\mathit{Oh}\gg 1$, where $a$ is the bubble radius, but only slender for $r_{\!E}\ll \mathit{Oh}^{2}a$ if $\mathit{Oh}\ll 1$ due to a compressional boundary layer of length $L\propto \mathit{Oh}\,r_{\!E}$, after which there is a change in the structure but not the speed of the retracting sheet. For $\mathit{Oh}\ll 1$, the detailed analysis justifies a simple momentum-balance argument, which gives the analytic prediction $r_{\!E}\sim (32a{\it\gamma}/3{\it\rho})^{1/4}t^{1/2}$, where ${\it\gamma}$ is the surface tension and ${\it\rho}$ is the density.
Breakup of electrified jets
- ROBERT T. COLLINS, MICHAEL T. HARRIS, OSMAN A. BASARAN
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- Journal of Fluid Mechanics / Volume 588 / 10 October 2007
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- 24 September 2007, pp. 75-129
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Breakup of electrified jets is important in applications as diverse as electrospraying, electroseparations and electrospray mass spectrometry. Breakup of a perfectly conducting, incompressible Newtonian liquid jet surrounded by a passive insulating gas that is stressed by a radial electric field is studied by a temporal analysis. An initially quiescent jet is subjected to axially periodic shape perturbations and the ensuing dynamics are followed numerically until pinch-off by both a three-dimensional but axisymmetric (two-dimensional) and a one-dimensional slender-jet algorithm. Results computed with these algorithms are verified against predictions from linear theory for short times. Breakup times, ratios of the sizes of the primary to satellite drops formed at pinch-off, and the Coulombic stability of these drops are reported over a wide range of electrical Bond numbers, NE (ratio of electric to surface tension force), Ohnesorge numbers, NOh (ratio of viscous to surface tension force), and disturbance wavenumbers, k. Effect of surface charge on interface overturning is investigated. Furthermore, the influence of electrostatic stresses on the dynamics of pinch-off and the mechanisms of satellite drop formation is also addressed.
Nonlinear oscillations of viscous liquid drops
- Osman A. Basaran
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- Journal of Fluid Mechanics / Volume 241 / August 1992
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- 26 April 2006, pp. 169-198
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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 Newton's 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
- Fred K. Wohlhuter, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 235 / February 1992
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- 26 April 2006, pp. 481-510
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Axisymmetric equilibrium shapes and stability of linearly 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 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, κ, drop shapes develop conical tips as drop deformation increases. However, three types of behaviour are found, depending on the value of κ. When κ < κ1, the drop deformation grows without bound as field strength rises. On the other hand, when κ > κ2 > κ1, 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 κ1 < κ < κ2, 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.
Dynamics of drop formation from a capillary in the presence of an electric field
- Xiaoguang Zhang, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 326 / 10 November 1996
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- 26 April 2006, pp. 239-263
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This paper reports an experimental study of the effects of an externally applied electric field on the dynamics of drop formation in the dripping mode from a vertical metal capillary. The fluid issuing out of the capillary is a viscous liquid, the surrounding ambient fluid is air, and the electric field is generated by establishing a potential difference between the capillary and a horizontal, circular electrode of large radius placed downstream of the capillary outlet. By means of an ultra-high-speed video system that is capable of recording up to 12000 frames per second, special attention is paid to the dynamics of the liquid thread that connects the primary drop that is about to detach and fall from the capillary to the rest of the conical liquid mass that is hanging from it. The experiments show that as the strength of the electric field increases, the volume of the primary drop decreases whereas the maximum length attained by the thread increases. The reduction in the volume of primary drops and the increase in the length of threads occur because the effective electromechanical surface tension of the fluid interface falls as the field strength rises. For the highly conducting drops of aqueous NaCl solutions studied in this work, the increase in thread length is due solely to the rising importance of normal electric stress relative to the falling importance of surface tension. However, as the conductivity of the drop liquid decreases, the thread length is further increased on account of the stabilizing influence exerted by the increasing electric shear stress that acts on the charged liquid–gas interface. Two new phenomena are also reported that have profound implications for electrohydrodynamics and practical applications. First, it is shown that whereas the liquid thread always ruptures at its downstream end in the absence of an applied electric field or when the field strength is low, it ruptures at its upstream end when the field strength is sufficiently high. Since satellite drops are produced directly from the thread once both of its ends have ruptured, the change in the mechanism of breakup with field strength influences the dynamics and fate of satellite drops. Second, it is demonstrated that the generation of satellites, which are often undesirable in applications, can be suppressed by the judicious application of an electric field. This is accomplished by using a field of moderate strength to induce charges of the opposite sign on the nearby surfaces of the satellite drop and the liquid that remains pendant from the tube following thread rupture. At high field strengths, induced charge effects are too weak to compete with net charge effects: the satellite is repelled by the pendant drop and falls under gravity as a distinct entity.
Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall
- James Q. Feng, Osman A. Basaran
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- Journal:
- Journal of Fluid Mechanics / Volume 275 / 25 September 1994
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- 26 April 2006, pp. 351-378
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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 Rec. Therefore, along a family of bubble shapes, solutions do not exist when Re > Rec. 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.
Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field
- Osman A. Basaran, Fred K. Wohlhuter
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- Journal:
- Journal of Fluid Mechanics / Volume 244 / November 1992
- Published online by Cambridge University Press:
- 26 April 2006, pp. 1-16
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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. Motivated by the desire to explain certain experiments with ferrofluids, a constitutive relation often used to describe the variation of polarization with applied field strength is adopted here to close the set of equations that govern the distribution of electric field. Specifically, the nonlinear polarization, P, is described by a Langevin equation of the form P = α[coth (τE) −1/(τE)], where E is the electric field strength. As expected, the results show that nonlinearly polarizable drops behave similarly to linearly polarizable drops at low field strengths when drop deformations are small. However, it is demonstrated that at higher values of the field strength when drop deformations are substantial, nonlinearly polarizable supported drops whose contact lines are fixed, as well as ones whose contact angles are prescribed, display hysteresis in drop deformation over a wide range of values of the Langevin parameters α and τ. Indeed, properly accounting for the nonlinearity of the polarization improves the quantitative agreement between theory and the experiments of Bacri et al. (1982) and Bacri & Salin (1982, 1983). Detailed examination of the electric fields inside nonlinearly polarizable supported drops reveals that they are very non-uniform, in contrast to the nearly uniform fields usually found inside linearly polarizable drops.
Dynamics and breakup of a contracting liquid filament
- PATRICK K. NOTZ, OSMAN A. BASARAN
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- Journal:
- Journal of Fluid Mechanics / Volume 512 / 10 August 2004
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- 23 July 2004, pp. 223-256
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Contraction of a filament of an incompressible Newtonian liquid in a passive ambient fluid is studied computationally to provide insights into the dynamics of satellite drops created during drop formation. This free boundary problem, which is composed of the Navier–Stokes system and the associated initial and boundary conditions that govern the evolution in time of the filament shape and the velocity and pressure fields within it, is solved by the method of lines incorporating the finite element method for spatial discretization. The finite element algorithm developed here utilizes an adaptive elliptic mesh generation technique that is capable of tracking the dynamics of the filament up to the incipience of pinch-off without the use of remeshing. The correctness of the algorithm is verified by demonstrating that its predictions accord with (a) previously published results of Basaran (1992) on the analysis of finite-amplitude oscillations of viscous drops, (b) simulations of the dynamics of contracting filaments carried out with the well-benchmarked algorithm of Wilkes et al. (1999), and (c) scaling laws governing interface rupture and transitions that can occur from one scaling law to another as pinch-off is approached. In dimensionless form, just two parameters govern the problem: the dimensionless half-length $L_o$ and the Ohnesorge number $Oh$ which measures the relative importance of viscous force to capillary force. Regions of the parameter space are identified where filaments (a) contract to a sphere without breaking into multiple droplets, (b) break via the so-called endpinching mechanism where daughter drops pinch-off from the ends of the main filament, and (c) break after undergoing a series of complex oscillations. Predictions made with the new algorithm are also compared to those made with a model based on the slender-jet approximation. A region of the parameter space is found where the slender-jet approximation fares poorly, and its cause is elucidated by examination of the vorticity dynamics and flow fields within contracting filaments.
Hysteretic response of supported drops during forced oscillations
- EDWARD D. WILKES, OSMAN A. BASARAN
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- Journal:
- Journal of Fluid Mechanics / Volume 393 / 25 August 1999
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- 25 August 1999, pp. 333-356
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Viscous liquid drops undergoing forced oscillations have been shown to exhibit hysteretic deformation under certain conditions both in experiments and by solution of simplified model equations that can only provide a qualitative description of their true response. The first hysteretic deformation results for oscillating pendant drops obtained by solving the full transient, nonlinear Navier–Stokes system are presented herein using a sweep procedure in which either the forcing amplitude A or frequency Ω is first increased and then decreased over a given range. The results show the emergence of turning-point bifurcations in the parameter space of drop deformation versus the swept parameter. For example, when a sweep is carried out by varying Ω while holding A fixed, the first turning point occurs at Ω ≡ Ωu as Ω is being increased and the second one occurs at Ω ≡ Ωl < Ωu as Ω is being decreased. The two turning points shift further from each other and toward lower values of the swept parameter as Reynolds number Re is increased. These turning points mark the ends of a hysteresis range within which the drop may attain either of two stable steady oscillatory states – limit cycles – as identified by two distinct solution branches. In the hysteresis range, one solution branch, referred to as the upper solution branch, is characterized by drops having larger maximum deformations compared to those on the other branch, referred to as the lower solution branch. Over the range Ωl [les ] Ω [les ] Ωu, the sweep procedure enables detection of the upper solution branch which cannot be found if initially static drops are set into oscillation as in previous studies of forced oscillations of supported and captive drops, or liquid bridges. The locations of the turning points and the associated jumps in drop response amplitudes observed at them are studied over the parameter ranges 0.05 [les ] A [les ] 0.125, 20 [les ] Re [les ] 40, and gravitational Bond number 0 [les ] G [les ] 1. Critical forcing amplitudes for onset of hysteresis are also determined for these Re values. The new findings have important ramifications in several practical applications. First, that Ωu − Ωl increases as Re increases overcomes the limitation which is inherent to the current practice of inferring the surface tension and/or viscosity of a bridge/drop liquid from measurement of its resonance frequencies (Chen & Tsamopoulos 1993; Mollot et al. 1993). Moreover, that the value of A for onset of hysteresis can be as low as 5% of the drop radius, or lower, has important implications for other free-surface flows such as coating flows.
Multiphase Electrodispersion Precipitation of Zirconia Powders
- Michael T. Harris, Warren G. Sisson, Timothy C. Scott, Osman A. Basaran, Charles H. Byers, W. Ren, Thomas T. Meek
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- Journal:
- MRS Online Proceedings Library Archive / Volume 346 / 1994
- Published online by Cambridge University Press:
- 21 February 2011, 171
- Print publication:
- 1994
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The multiphase electrodispersion precipitation of zirconia powders has been done in the electric dispersion reactor (EDR). This paper presents the first results obtained where the bench-scale EDR unit was operated in the continuous mode to synthesize 130 ± 2 g of ZrO2 powder in approximately 12 h. An aqueous solution of zirconyl nitrate was dispersed and precipitated in a 2-ethyl-l-hexanol continuous phase containing 0.012 M to 0.12 M ammonia. A gravity settler was used to remove soft agglomerates of the ZrO2 particles from the organic solvent. Electric bed filtration was employed to remove the fines from the solvent, which was then recycled.
The particle-size distribution was varied by changing the electric field strength. At high field strengths (approximately 20 kV/cm), the particle sizes ranged from approximately 0.1 to 5 μm. The dried powder had a consistency of talcum powder. Microwave and conventional heating experiments showed that the powders were sinterable. The BET surface area of the powders ranged from approximately 20 to 90 m2/g.
Porous Spherical Shells and Microspheres by Electrodispersion Precipitation
- Michael T. Harris, Warren G. Sisson, Susan M. Hayes, Sophie J. Bobrowski, Osman A. Basaran
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- Journal:
- MRS Online Proceedings Library Archive / Volume 372 / 1994
- Published online by Cambridge University Press:
- 15 February 2011, 43
- Print publication:
- 1994
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Pulsed electric fields have been used to enhance the dispersion of aqueous metal (Zr and Al) salt solutions from a nozzle and into a nonconducting liquid continuous phase that is immiscible with the aqueous phase. The diameter of the resulting microdroplets ranged in size from approximately 0.1 to 10 μm. Precipitation of hydrous metal oxides occurred as ammonia, which was dissolved in varying amounts in the continuous phase, diffused into the aqueous microdroplets. Spherical shells were formed at higher ammonia concentrations and microspheres were produced at lower ammonia concentrations. Upon drying, dimples appeared in the particles that were synthesized at higher ammonia concentrations. The latter result accords with the well known fact that under certain conditions spherical shells collapse when a fluid is extracted from the core of the particle. No dimples were observed in the microspheres that were produced at lower ammonia concentrations. Analog X-ray dot maps for aluminum and zirconium were done to determine the spatial distribution of each metal in the particles.