Peter G. Petropoulos

On the long-time behavior of unsplit perfectly matched layers

This paper shows how to eliminate an undesirable long-time linear growth of the electromagnetic field in a class of unsplit perfectly matched layers (PML) typically used as absorbing boundary conditions in computational electromagnetics codes. For the new PML equations, we give energy arguments that show the fields in the layer are bounded by a time-independent constant, hence they are long-time stable. Numerical experiments confirm the elimination of the linear growth, and the long-time boundedness of the fields.

Stability and phase error analysis of FD-TD in dispersive dielectrics

Four FD-TD extensions for the modeling of pulse propagation in Debye or Lorentz dispersive media are analyzed through studying the stability and phase error properties of the coupled difference equations corresponding to Maxwells equations and to the equations for the dispersion. For good overall accuracy the author shows that all schemes should be run at their Courant stability limit, and that the timestep should finely resolve the medium timescales. Particularly, for Debye schemes it should be at least /spl Delta/t=10/sup /spl minus/3//spl tau/, while for Lorentz schemes it should be /spl Delta/t=10/sup /spl minus/2//spl tau/, where /spl tau/ is a typical medium relaxation time. A numerical experiment with a Debye medium confirms this. The author has determined that two of the discretizations for Debye media are totally equivalent. In the Lorentz medium case the author establishes that the method that uses the polarization differential equation to model dispersion is stable for all wavenumbers, and that the method using the local-in-time constitutive relation is weakly unstable for modes with wavenumber k such that k/spl Delta/x>/spl pi//2. >

Phase error control for FD-TD methods of second and fourth order accuracy

For FD-TD methods(used to solve Maxwells equations) we determine the spatial resolution of the discretized domain in terms of the total computation time and the desired phase error. It is shown that the spatial step should vary as /spl Delta/x/spl sim/g[e/sub /spl phi/t/sub c/]/sup 1/s/ in order to maintain a prescribed phase error level e/sub /spl phi throughout the computation time t/sub c/, where s (=2 or 4) is the spatial order of accuracy of the scheme and g is a geometric factor. Significantly, we show that the rule of thumb of using 10-20 points per wavelength to determine the spatial cell size for the standard scheme is not optimal. Our results are verified by numerical simulations in two dimensions with the Yee (1966) scheme and a new fourth-order accurate FD-TD scheme. >

Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates

A scaling argument is used to derive reflectionless sponge layers to absorb outgoing time-harmonic waves in numerical solutions of the three-dimensional elliptic Maxwell equations in rectangular, c...

Linear stability of a two-fluid interface for electrohydrodynamic mixing in a channel

We study the electrohydrodynamic stability of the interface between two superposed viscous fluids in a channel subjected to a normal electric field. The two fluids can have different densities, viscosities, permittivities and conductivities. The interface allows surface charges, and there exists an electrical tangential shear stress at the interface owing to the finite conductivities of the two fluids. The long-wave linear stability analysis is performed within the generic Orr–Sommerfeld framework for both perfect and leaky dielectrics. In the framework of the long-wave linear stability analysis, the wave speed is expressed in terms of the ratio of viscosities, densities, permittivities and conductivities of the two fluids. For perfect dielectrics, the electric field always has a destabilizing effect, whereas for leaky dielectrics, the electric field can have either a destabilizing or a stabilizing effect depending on the ratios of permittivities and conductivities of the two fluids. In addition, the linear stability analysis for all wavenumbers is carried out numerically using the Chebyshev spectral method, and the various types of neutral stability curves (NSC) obtained are discussed.

Analysis of exponential time-differencing for FDTD in lossy dielectrics

We determine the stability condition and analyze the accuracy of the exponential and centered time-differencing schemes for finite-difference time-domain (FDTD) in an isotropic, homogeneous lossy dielectric with electric and magnetic conductivities /spl sigma/ and /spl sigma/*, respectively. We show that these schemes are equivalent and determine that, for accuracy, both schemes must be used with a time step that finely resolves the electric and magnetic conduction current relaxation time scales. The implications of these results for perfectly matched layer (PML)-type absorbing boundary conditions are discussed.

Dynamics and rupture of planar electrified liquid sheets

We investigate the stability of a thin two-dimensional liquid film when a uniform electric field is applied in a direction parallel to the initially flat bounding fluid interfaces. We consider the distinct physical effects of surface tension and electrically induced forces for an inviscid, incompressible nonconducting fluid. The film is assumed to be thin enough and the surface forces large enough that gravity can be ignored to leading order. Our aim is to analyze the nonlinear stability of the flow. We achieve this by deriving a set of nonlinear evolution equations for the local film thickness and local horizontal velocity. The equations are valid for waves which are long compared to the average film thickness and for symmetrical interfacial perturbations. The electric field effects enter nonlocally and the resulting system contains a combination of terms which are reminiscent of the Kortweg–de-Vries and the Benjamin–Ono equations. Periodic traveling waves are calculated and their behavior studied as the...

The effect of electric fields on the rupture of thin viscous films by van der Waals forces

We examine the stability of a thin two-dimensional incompressible liquid film when an electric field is applied in a direction parallel to the initially flat bounding fluid interfaces, and study the competition between surface tension, van der Waals, viscous, and electrically induced forces. The film is assumed to be sufficiently thin, and the surface tension and electrically induced forces are large enough that gravity can be ignored to the leading order. We analyze the nonlinear stability of the flow by deriving and numerically solving a set of nonlinear evolution equations for the local film thickness and the horizontal velocity. We find that the electric field forces enhance the stability of the flow and can remove rupture. If rupture occurs then the form of the singularity, to leading order, is that found in the absence of an electric field.

THE WAVE HIERARCHY FOR PROPAGATION IN RELAXING DIELECTRICS

Abstract We consider the propagation of arbitrary electromagnetic pulses in anomalously dispersive dielectrics characterized by M relaxation processes. A partial differential equation for the electric field in the dielectric is derived and analyzed. This single equation describes a hierarchy of M + 1 wave types, each type characterized by an attenuation coefficient and a wave speed. Our analysis identifies a “skin-depth” where the pulse response is described by a telegraphers equation with smoothing terms, travels with the wavefront speed, and decays exponentially. Past this shallow depth we show that the pulse response is described by a weakly dispersive advection-diffusion equation, travels with the sub-characteristic advection speed equal to the zero-frequency phase velocity in the dielectric, and decays algebraically. The analysis is verified with a numerical simulation. The relevance of our results to the development of numerical methods for such problems is discussed.

On the control and suppression of the Rayleigh-Taylor instability using electric fields

It is shown theoretically that an electric field can be used to control and suppress the classical Rayleigh-Taylor instability found in stratified flows when a heavy fluid lies above lighter fluid. Dielectric fluids of arbitrary viscosities and densities are considered and a theory is presented to show that a horizontal electric field (acting in the plane of the undisturbed liquid-liquid surface), causes growth rates and critical stability wavenumbers to be reduced thus shifting the instability to longer wavelengths. This facilitates complete stabilization in a given finite domain above a critical value of the electric field strength. Direct numerical simulations based on the Navier-Stokes equations coupled to the electrostatic fields are carried out and the linear theory is used to critically evaluate the codes before computing into the fully nonlinear stage. Excellent agreement is found between theory and simulations, both in unstable cases that compare growth rates and in stable cases that compare freq...