James Q. Feng

A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field

Axisymmetric steady flows driven by an electric field about a deformable fluid drop suspended in an immiscible fluid are studied within the framework of the leaky dielectric model. Deformations of the drop and the flow fields are determined by solving the nonlinear free-boundary problem composed of the Navier-Stokes system governing the flow field and Laplaces system governing the electric field. The solutions are obtained by using the Galerkin finite-element method with an elliptic mesh generation scheme. Under conditions of creeping flow and vanishingly small drop deformations, the results of finite-element computations recover the asymptotic results. When drop deformations become noticeable, the asymptotic results are often found to underestimate both the flow intensity and drop deformation. By tracking solution branches in parameter space with an arc-length continuation method, curves in parameter space of the drop deformation parameter D versus the square of the dimensionless field strength E usually exhibit a turning point when E reaches a critical value E c . Along such a family of drop shapes, steady solutions do not exist for E > E c . The nonlinear relationship revealed computationally between D and E 2 appears to be capable of providing insight into discrepancies reported in the literature between experiments and predictions based on the asymptotic theory. In some special cases with fluid conductivities closely matched, however, drop deformations are found to grow with E 2 indefinitely and no critical value E c is encountered by the corresponding solution branches. For most cases with realistic values of physical properties, the overall electrohydrodynamic behaviour is relatively insensitive to effects of finite-Reynolds-number flow. However, under extreme conditions when fluids of very low viscosities are involved, computational results illustrate a remarkable shape turnaround phenomenon: a drop with oblate deformation at low field strength can evolve into a prolate-like drop shape as the field strength is increased.

Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number

The steady axisymmetric flow in and around a deformable drop moving under the action of gravity along the axis of a vertical tube at intermediate Reynolds number is studied by solving the nonlinear free-boundary problem using a Galerkin finite-element method. For the case where the drop and suspending liquid have the same viscosity, the ratio of the densities is 6/5 or 5/6, and the radius of the tube is equal to twice the radius of a sphere having the drop volume, four significant results are apparent in the computations. First, we compute drops showing much more deformation, and in particular the development of considerably more non-convexity, than those found in previous calculations for non-zero Reynolds number. The degree of non-convexity typically grows with the Reynolds number. Secondly, external recirculation zones can be attached to or disjoint from the drop. We find when there is a single external recirculation zone, that is disjoint (as found by Dandy & Leal), it can attach to the drop as the Reynolds number is increased. As the Reynolds number further increases, this is immediately followed by division of the drop into two adjacent recirculating regions. Thirdly, we sometimes find two recirculation zones in the suspending liquid. Finally, the drag coefficient, axis ratio, and normalized interfacial and frontal areas of the drop can vary non-monotonically with the Weber number, exhibiting as many as four local extrema. The results are compared to previous theoretical and experimental work, and implications for drop motion and heat and mass transfer are discussed.

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.

A deformable liquid drop falling through a quiescent gas at terminal velocity

The steady axisymmetric flow internal and external to a deformable viscous liquid drop falling through a quiescent gas under the action of gravity is computed by solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted quadrilateral mesh. Considering typical values of the density and viscosity for common liquids and gases, numerical solutions are first computed for the liquid-to-gas density ratio ρ = 1000 and viscosity ratio μ from 50 to 1000. Visually noticeable drop deformation is shown to occur when the Weber number We ~ 5. For μ ≥ 100, drops of Reynolds number Re Re > 200 a flattened front and somewhat rounded rear, with that at Re = 200 exhibiting an almost fore–aft symmetric shape. As an indicator of drop deformation, the axis ratio (defined as drop width versus height) increases with increasing We and μ, but decreases with increasing Re . By tracking the solution branches around turning points using an arclength continuation algorithm, critical values of We for the ‘shape instability’ are determined typically within the range of 10 to 20, depending on the value of Re (for Re ≥ 100). The drop shape can change drastically from prolate- to oblate-like when μ Re ≤ 500). For example, for μ = 50 a drop at Re ≥ 200 exhibits a prolate shape when We We > 10. The various solutions computed at ρ = 1000 with the associated values of drag coefficient and drop shapes are found to be almost invariant at other values of ρ (e.g. from 500 to 1500) as long as the value of ρ/μ 2 is fixed, despite the fact that the internal circulation intensity changes according to the value of μ. The computed values of drag coefficient are shown to agree quite well with an empirical formula for rigid spheres with the radius of the sphere replaced by the radius of the cross-sectional area.

A spherical-cap bubble moving at terminal velocity in a viscous liquid

The nonlinear Navier-Stokes equations governing steady, laminar, axisymmetric flow past a deformable bubble are solved by the Galerkin finite-element method simultaneously with a set of elliptic partial differential equations governing boundary-fitted mesh. For Reynolds number 20 ≤ Re ≤ 500, numerical solutions of spherical-cap bubbles are obtained at capillary number Ca=1. Increasing Ca to 2 leads to a highly curved, cusp-like bubble rim that seems to correspond to skirt formation. The computed steady, axisymmetric spherical-cap bubbles with closed, laminar wakes compare reasonably with the available experimental results, especially for Re ≤100. By exploring the parameter space (for Re ≤200), a sufficient condition for steady axisymmetric solutions of bubbles with the spherical-cap shape is found to be roughly Ca>0.4. The basic characteristics of spherical-cap bubbles of Ca≥0.5, for a given Re ≥ 50, are found to be almost independent of the value of Ca (or Weber number We ≡ Re Ca). At a fixed Re ≥ 50, continuation by increasing Ca (or We) from a spherical bubble solution cannot lead to solutions of spherical-cap bubbles, but rather to a turning point at We slightly greater than 10 where the solution branch folds back to reduced values of Ca (or We). Yet continuation by reducing Ca (or We) from a spherical-cap bubble solution cannot arrive at a spherical bubble solution for Re ≥ 50, but rather at solutions with bubbles having more complicated shapes such as a sombrero, etc. Without thorough examinations of the solution stability, multiple steady axisymmetric solutions are shown to exist in the parameter space for a given set of parameters.

DIFFUSION-CONTROLLED QUASI-STATIONARY MASS TRANSFER FOR AN ISOLATED SPHERICAL PARTICLE IN AN UNBOUNDED MEDIUM

A consolidated mathematical formulation of the spherically symmetric mass transfer problem is presented, with the quasi-stationary approximating equations derived from a perturbation point of view for the leading-order effect. For the diffusion-controlled quasi-stationary process, a mathematically complete set of the exact analytical solutions is obtained in implicit forms to cover the entire parameter range. Furthermore, accurate explicit formulas for the particle radius as a function of time are also constructed semi-empirically for convenience in engineering practice. Both dissolution of a particle in a solvent and growth of it by precipitation in a supersaturated environment are considered in the present work.

Spraying fine fluid particles in insulating fluid systems by electrostatic polarization forces

It is experimentally demonstrated that electrostatic polarization forces can in principle be utilized in generating fine bubbles and droplets, despite the lack of charge carriers that have traditionally been thought to be necessary for successful electrostatic spraying. Under the condition that the permittivity of the dispersed phase is lower than that of the continuous phase, such as when gas bubbles are sprayed into insulating liquids, the spraying behavior is regular and easy to control. If the permittivity of the dispersed phase is higher than that of the continuous phase, such as when insulating liquids are sprayed into gases, the spraying behavior lacks regularity and further research is needed before pure polarization forces can find significant applications in practical processes.

A note on wave motions with a cylindrical fluid interface stabilized by rotation

Abstract The Rayleighs capillary instability of a cylindrical fluid interface can be suppressed by the centrifugal force due to coaxial rotation when the outer fluid is heavier than the inner one. Thus, wave motions on the rotationally stabilized cylindrical interfaces can occur in natural and technological processes, notwithstanding the lack of coherent treatment in the literature. This work provides a supplementary linear analysis of various wave motions in a three-dimensional framework for two-phase inviscid fluid systems with cylindrical interfaces stabilized by rotation. Many previous results found in the literature are recast in a general from. With gravity acting perpendicularly to the rotation axis, the interface disturbance is considered as an externally forced wave motion. The condition for the stability of the gravity-induced interface displacement in a two-phase rotating system is found to be exactly the same as that for the one-phase case such as an air column in a rotating liquid.