The effect of surface-charge convection on the settling velocity of spherical drops in a uniform electric field

2016 ◽  
Vol 797 ◽  
pp. 536-548 ◽  
Author(s):  
Ehud Yariv ◽  
Yaniv Almog
Keyword(s):  
Difference Equation ◽  
Electric Field ◽  
Surface Charge ◽  
Settling Velocity ◽  
Numerical Solutions ◽  
Weak Field ◽  
Uniform Electric Field ◽  
Electrostatic Problem ◽  
Settling Speed ◽  
The Difference

The mechanism of surface-charge convection, quantified by the electric Reynolds number $Re$, renders the Melcher–Taylor electrohydrodynamic model inherently nonlinear, with the electrostatic problem coupled to the flow. Because of this nonlinear coupling, the settling speed of a drop under a uniform electric field differs from that in its absence. This difference was calculated by Xu & Homsy (J. Fluid Mech., vol. 564, 2006, pp. 395–414) assuming small $Re$. We here address the same problem using a different route, considering the case where the applied electric field is weak in the sense that the magnitude of the associated electrohydrodynamic velocity is small compared with the settling velocity. As convection is determined at leading order by the well-known flow associated with pure settling, the electrostatic problem becomes linear for arbitrary value of $Re$. The electrohydrodynamic correction to the settling speed is then provided as a linear functional of the electric-stress distribution associated with that problem. Calculation of the settling speed eventually amounts to the solution of a difference equation governing the respective coefficients in a spherical harmonics expansion of the electric potential. It is shown that, despite the present weak-field assumption, our model reproduces the small-$Re$ approximation of Xu and Homsy as a particular case. For finite $Re$, inspection of the difference equation reveals a singularity at the critical $Re$-value $4S(1+R)(1+M)/(1+S)M$, wherein $R$, $S$ and $M$ respectively denote the ratios of resistivity, permittivity and viscosity values in the suspending and drop phases, as defined by Melcher & Taylor (Annu. Rev. Fluid Mech., vol. 1, 1969, pp. 111–146). Straightforward numerical solutions of this equation for electric Reynolds numbers smaller than the critical value reveal a non-monotonic dependence of the settling speed upon the electric field magnitude, including a transition from velocity enhancement to velocity decrement.

scholarly journals Uniform electric-field-induced lateral migration of a sedimenting drop

2016 ◽  
Vol 792 ◽  
pp. 553-589 ◽  
Author(s):  
Aditya Bandopadhyay ◽  
Shubhadeep Mandal ◽  
N. K. Kishore ◽  
Suman Chakraborty
Keyword(s):  
Electric Field ◽  
Surface Charge ◽  
Tilt Angle ◽  
Shape Deformation ◽  
Uniform Electric Field ◽  
Lateral Motion ◽  
Drop Surface ◽  
Lateral Migration ◽  
Drop Velocity ◽  
Pressure Fields

We investigate the motion of a sedimenting drop in the presence of an electric field in an arbitrary direction, otherwise uniform, in the limit of small interface deformation and low-surface-charge convection. We analytically solve the electric potential in and around the leaky dielectric drop, and solve for the Stokesian velocity and pressure fields. We obtain the correction in drop velocity due to shape deformation and surface-charge convection considering small capillary number and small electric Reynolds number which signifies the importance of charge convection at the drop surface. We show that tilt angle, which quantifies the angle of inclination of the applied electric field with respect to the direction of gravity, has a significant effect on the magnitude and direction of the drop velocity. When the electric field is tilted with respect to the direction of gravity, we obtain a non-intuitive lateral motion of the drop in addition to the buoyancy-driven sedimentation. Both the charge convection and shape deformation yield this lateral migration of the drop. Our analysis indicates that depending on the magnitude of the tilt angle, conductivity and permittivity ratios, the direction of the sedimenting drop can be controlled effectively. Our experimental investigation further confirms the presence of lateral migration of the drop in the presence of a tilted electric field, which is in support of the essential findings from the analytical formalism.

Analytical Characteristic of Chromatography Device Using Dielectrophoresis Phenomenon

2011 ◽  
Vol 497 ◽  
pp. 87-92 ◽  
Author(s):  
Masaru Hakoda ◽  
Takashi Otaki
Keyword(s):  
Electric Field ◽  
Retention Time ◽  
Electrode Array ◽  
Yeast Cells ◽  
Uniform Electric Field ◽  
Analytical Characteristic ◽  
Sweep Frequency ◽  
Carrier Flow ◽  
Interdigitated Microelectrodes ◽  
The Difference

This paper reports the separation of cells using a dielectrophoretic (DEP) chromatography device. The device consists of a micro channel and an array of interdigitated microelectrodes on a glass substrate. The sample cells were fed pulse-wise into the carrier flow using a micro-injector. The cells in the sample received a non-uniform electric field made with an electrode array. The direction of DEP motion is towards the higher field when the cell is more polarizable than the medium (positive DEP), while the direction is towards the lower field when the cell is less polarizable than the medium (negative DEP). Therefore, the cell separation depends on the size and dielectric characteristic. The effects of carrier flow rate, frequency, applied voltage, and sweep frequency on the retention time of the sample in the device were examined. In this study, mouse-hybridoma 3-2H3 cells and yeast cells were used as the sample cell. The analytical characteristic of the DEP chromatography device was evaluated according to the difference of retention time by the electric field. As a result, the separation in the cells in the negative DEP using the DEP chromatography was found to be effective. In addition, the effect of the sweep frequency on the difference in the retention time of the mouse hybridoma 3-2H3 cells and the yeast cells was very large. Consequently, the effectiveness of the DEP chromatography device was proven.

MULTISCALED ASYMPTOTIC EXPANSIONS FOR THE ELECTRIC POTENTIAL: SURFACE CHARGE DENSITIES AND ELECTRIC FIELDS AT ROUNDED CORNERS

2007 ◽  
Vol 17 (06) ◽  
pp. 845-876 ◽  
Author(s):  
PATRICK CIARLET ◽  
SAMIR KADDOURI
Keyword(s):  
Electric Field ◽  
Charge Density ◽  
Surface Charge ◽  
Electric Fields ◽  
Asymptotic Expansions ◽  
Electric Potential ◽  
Curvature Radius ◽  
Geometrical Shape ◽  
Empirical Formulas ◽  
Electrostatic Problem

We are interested in computing the charge density and the electric field at the rounded tip of an electrode of small curvature. As a model, we focus on solving the electrostatic problem for the electric potential. For this problem, Peek's empirical formulas describe the relation between the electric field at the surface of the electrode and its curvature radius. However, it can be used only for electrodes with either a purely cylindrical, or a purely spherical, geometrical shape. Our aim is to justify rigorously these formulas, and to extend it to more general, two-dimensional, or three-dimensional axisymmetric, geometries. With the help of multiscaled asymptotic expansions, we establish an explicit formula for the electric potential in geometries that coincide with a cone at infinity. We also prove a formula for the surface charge density, which is very simple to compute with the Finite Element Method. In particular, the meshsize can be chosen independently of the curvature radius. We illustrate our mathematical results by numerical experiments.

Numerical Study of EHD-Enhanced Forced Convection Using Two-Way Coupling

2003 ◽  
Vol 125 (4) ◽  
pp. 760-764 ◽  
Author(s):  
M. Huang ◽  
F. C. Lai
Keyword(s):  
Heat Transfer ◽  
Electric Field ◽  
Heat Transfer Enhancement ◽  
Forced Convection ◽  
Electric Fields ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Transfer Enhancement ◽  
Wide Range ◽  
The Difference

Numerical results are presented for heat transfer enhancement using electric field in forced convection in a horizontal channel. The main objective of the present study is to verify the assumption that is commonly used in the numerical study of this kind of problem, which assumes that the electric field can modify the flow field but not vice versa (i.e., the so-called one-way coupling). To this end, numerical solutions are obtained for a wide range of governing parameters (V0=10, 12.5, 15 and 17.5 kV as well as ui=0.0759 to 1.2144 m/s) using both one-way and two-way couplings. The results obtained, in terms of the flow, temperature, and electric fields as well as the heat transfer enhancement, are thoroughly examined. Since the difference in the results obtained by two approaches is insignificant, it is concluded that the assumption of one-way coupling is valid for the problem considered.

scholarly journals Electrohydrodynamic settling of drop in uniform electric field: beyond Stokes flow regime

2019 ◽  
Vol 881 ◽  
pp. 498-523 ◽  
Author(s):  
Nalinikanta Behera ◽  
Shubhadeep Mandal ◽  
Suman Chakraborty
Keyword(s):  
Electric Field ◽  
Flow Regime ◽  
Stokes Flow ◽  
Capillary Number ◽  
Uniform Electric Field ◽  
Fluid Inertia ◽  
Drop Shape ◽  
Wide Range ◽  
Settling Speed ◽  
Leaky Dielectric

The electrohydrodynamics of a weakly conducting buoyant drop under the combined influence of gravity and a uniform electric field is studied computationally, focusing on the inertia-dominated regime. Numerical simulations are performed for both perfectly dielectric and leaky dielectric drops over a wide range of dimensionless parameters to explore the interplay of fluid inertia and electrical stress to govern the drop shape and charge convection. For perfectly dielectric drops, the fluid inertia alters the drop shape and the deformation behaviour of the drop follows a non-monotonic path. The drop shape at steady state exhibits the transition from oblate to prolate shape on increasing the electric field strength, in sharp contrast to the cases concerning the Stokes flow regime. Similar behaviour is also obtained for leaky dielectric drops for certain fluid properties. For leaky dielectric drops, the fluid inertia also affects the convective transport of charges at the drop surface and thereby alters the drop dynamics. Unlike the Stokes flow regime, where surface charge convection has little effect on the settling speed, the same modifies the drop settling speed quite significantly in the finite inertial regime depending on the combination of electrical conductivity ratio and permittivity ratio. For oblate drops at low capillary number, charge convection alters drop shape, while keeping the nature of deformation unaltered. However, for relatively large capillary number, the oblate drop transforms into a dimpled shape due to charge convection. For all cases, an interesting fact is noticed that under the combined action of electric and inertial forces, the resultant deformation is less than the summation of the deformations caused by individual effects, when inertial effects are strong. These results are likely to provide deep insights into the interplay of various nonlinearities towards altering electrohydrodynamic settling of drops and bubbles.

scholarly journals NONLOCAL EQUATIONS FOR THE ELECTROMAGNETIC FIELD IN THE INHOMOGENEOUS ACCELERATING STRUCTURES

2019 ◽  
pp. 71-76
Author(s):  
M.I. Ayzatsky
Keyword(s):  
Magnetic Field ◽  
Difference Equation ◽  
Electric Field ◽  
Boundary Conditions ◽  
Good Accuracy ◽  
Nonlocal Model ◽  
Nonlocal Equations ◽  
The Difference ◽  
Set Of Points ◽  
Infinite Set

The procedure for obtaining a difference equation, the solution of which is the components of the electric (or magnetic) field at the chosen set of points in the volume of the resonator chain, was developed. We started with the wave equation with boundary conditions and obtained the difference equation without boundary conditions. Boundary conditions were included into the coefficients of the difference equation for the electric field. As the obtained equation connects the field values in different points (in general case, on an infinite set of points), the proposed procedure represents a nonlocal model for field description. Solutions of the difference equation for the electric field were analyzed. It was shown that they coincide with good accuracy with the ones that were obtained by direct summing of relevant series.

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