*FLOW-3D* Validation: Electro-Hydrodynamics Applications

## Leaky-Dielectric Model

The Leaky-Dielectric model defined by MT (J.R. Melcher and G.I. Taylor, "Electrohydrodynamics: A Review of the Role of Interfacial Shear Stresses," Ann. Rev. Fl. Mech., 111, 1969) has been found useful for a variety of important applications including electro-spraying. In this model, a fluid is assumed to be weakly conducting to the extent that a bulk charge density does not exist, but a surface charge density can develop at fluid interfaces where there are discontinuities in conductivity and permittivity. Under these conditions, the application of an electric field having a component tangential to an interface may result in significant tangential stresses. The classic example studied by Melcher and Taylor consists of a shallow, conducting liquid in a container with electrodes at either end. At one end the electrode is extended over the liquid to form a tilted cover. This arrangement causes a tangential component of the electric field to be generated at the liquid surface, which in turn generates a tangential surface stress that drives a circulatory flow in the liquid.

## Dynamic Charge Density Model

A dynamic charge density model is available in ** FLOW-3D** as an enhancement to its other electro-hydrodynamic capabilities. This model does not require the Leaky-Dielectric approximation. Here, application to the Melcher-Taylor apparatus is shown to be in good agreement with theory and data. Actually, a simple one-dimensional theory for the horizontal velocity at the top surface of the liquid predicts values that are about a factor of two too large compared to measured data. Leaky-dialectric computations give values lying between data and theory.

Two things can explain the differences between data and theory.

- First, the conductivity of the corn oil used in the experiments was not large enough to fully satisfy the condition for a leaky-dielectric model; that is, the charge relaxation time was on the order of, rather than significantly smaller than, the characteristic time of the hydrodynamic flow.
- Second, the 1D theory does not account for flow losses at the ends of the container, which contribute to a slower overall flow velocity. Computations help to put these two differences into perspective, because we used a much larger conductivity than reported for corn oil to ensure the leaky dielectric assumption was satisfied.

## Computational Results

At smaller voltages (i.e., smaller flow velocities), end losses are less and the computational results are close to the 1D prediction. As the voltage is increased so are end losses and the leaky-dialectric computational results show a further decrease in flow velocity compared to the 1D theory, as would be expected.

All the computational results, however, are closer to the Leaky-Dielectric theory than to the experimental data because a relatively large conductivity that was used, k=8.05e-6S/m, which restricted charges to the free liquid surface as required in the Leaky-Dielectric assumption. Some confirmation of this can be obtained by comparing the computed and observed flow patterns. Computations (solid circles) show a much more concentrated flow from the surface turning downward at the left wall of the tank and producing a smaller eddy structure. In the experiments the flow near the left wall appears wider and the eddy is broader suggesting the existence of a larger region of net charge, which is consistent with a smaller conductivity in the experiments.

When the conductivity was reduced to k=1.0e-11S/m the approximate value reported by MT for the experiments, our computations give the bottom most data point at 10kV (solid triangle). In this case the computational result is a little below the experimental data. Because the precise value of conductivity was uncertain we raised the value to k=3.0e-11S/m, which is still quite small. Computational results with this value fall into agreement with the experimental data, i.e., the solid square plotted at 10kV.

Additional computations using the larger k=8.05e-6S/m and higher potentials of 12kV and 15kV produced results close to the theoretical curve (solid circles; at 15kV the point is off scale at a value of 116.6). Using the lower value of k=3.0e-11S/m with higher voltages of 12kV and 15kV produced results that remain closed to the experimental values (solid squares at 12kV and 15kV).

All computations used the same 2D grid of 78 by 23 cells and ran quickly, on the order of 1 to 2 minutes for each simulation.

We conclude from the excellent agreement between experimental observations and computations that the addition of fluid conductivity to *FLOW-3D*** **works well and improves upon the Leaky-Dielectric model because charges are not confined to liquid surfaces.