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Fundamental Issues and Canonical Flows

On a Numerical Model for Free Surface Flows of a Conductive Liquid Under an Electrostatic Field

[+] Author and Article Information
Sajad Pooyan1

Department of Mechanical Engineering,  Ferdowsi University of Mashhad, Mashhad, Iransajad.pooyan@gmail.com

Mohammad Passandideh-Fard

Department of Mechanical Engineering,  Ferdowsi University of Mashhad, Mashhad, Iranmpfard@um.ac.ir

A perfectly conductive material is a material assumed to have infinite electric conductivity.

Horizontal and vertical lines passing through the cell center point.

1

Corresponding author.

J. Fluids Eng 134(9), 091205 (Aug 22, 2012) (12 pages) doi:10.1115/1.4007158 History: Received March 06, 2012; Revised May 19, 2012; Published August 22, 2012; Online August 22, 2012

In this paper, a numerical model is developed that can simulate the unsteady axisymmetric free-surface flow of a perfectly conductive liquid under an electrostatic field. The effect of the electrostatic field is modeled by a force distributed on the liquid free surface. Assuming the liquid as a perfect conductor makes it possible to reduce the general electromagnetic equations to electrostatic equations. The Navier–Stokes equations are solved to find the velocity and pressure fields. The free surface advection and reconstruction are performed based on the volume-of-fluid method using Youngs’ algorithm. To evaluate the effect of the electric field on the free surface, the electrostatic potential is first solved for the entire computational domain. Next, the electric field intensity and the surface density of the electric charge are calculated on the free surface after which the electric force can be determined. The computational method for treating this force is similar to that of the surface tension using the continuum surface force method. The developed model is validated by a comparison between the calculated results with those of the analytics as well as experiments for an electrowetting scenario.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

A schematic of the parameters used in the discretized Laplace equation (Eqs. 16 to 18). For cells containing a free surface, the points used to discretize the Laplace equation are positioned on the free surface.

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Figure 2

The points and directions used to calculate the electric field intensity on the free surface. The triangular markers are used to calculate the directional derivatives of the electric potential in r, z and a diagonal direction connecting the base point to the center of the cell i +2,j +2. The magnitude of the potential on these points is obtained using an interpolation of the potentials in the adjacent centerpoints (circles for cell i +2,j and squares for cell i +1,j).

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Figure 3

Electric potential difference between two concentric spheres. The radius of the outer sphere is twice as the inner sphere. The reference for the angle θ is also shown in the figure.

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Figure 4

A comparison between the analytical and numerical values of dimensionless electric field intensity magnitude for θ=0 using four different mesh resolutions. To nondimensionalize the values of the electric field intensity magnitude they are divided by their respective values on the surface of the inner sphere.

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Figure 5

The numerically calculated magnitude of electric field intensity on the surface of the inner sphere divided by its respective analytical value (CPR = 20)

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Figure 6

An approximation of the free surface using line segments in two adjacent cells. The ends of the line segments are interconnected on the boundary of the two cells.

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Figure 7

A comparison between the components of the free surface normal vectors calculated as described in Sec. 21 and their respective exact values. NX and NY are the horizontal and vertical components, respectively. A and N prefixes denote the exact and numerical values, respectively.

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Figure 8

The electric potential distribution around the conductive hemisphere between two infinite parallel plane electrodes maintained at 500 V potential difference

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Figure 9

The magnitude of the electric field intensity vector and its components on the surface of the hemisphere shown in Fig. 8. Numerical results, specified by symbols, are compared with the analytical data [33] represented by solid lines.

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Figure 10

A basic electrowetting setup. A conductive drop is placed on a metal electrode covered by a dielectric hydrophobic layer and a potential difference is applied between the drop and the electrode.

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Figure 11

The calculated droplet shape (solid white line) is compared with that of the experiments [6] for two cases where the electrowetting dimensionless number (η) was (a) 1 and (b) 0.5

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Figure 12

The effect of mesh size on (a) drop free surface shape and (b) drop final maximum radius and height for η=1 (Fig. 1). Mesh resolutions with 50, 75, 105 and 155 cells per initial drop radius are represented by green, red, orange and blue lines, in the figure.

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Figure 13

The electric potential distribution for the droplet displayed in part (a) of Fig. 1

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Figure 14

The electric charge surface density for the droplet displayed in part (a) of Fig. 1

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