Nonuniform Media Research Articles

Symbolic computation, Bäcklund transformation and analytic N-soliton-like solution for a nonlinear Schrödinger equation with nonuniformity term from certain space/laboratory plasmas

With symbolic computation, a bilinear Bäcklund transformation is presented for a nonlinear Schrödinger equation with nonuniformity term from certain space/laboratory plasmas, and correspondingly the one-soliton-like solution is derived from the Bäcklund transformation. Simultaneously, the N -soliton-like solution in double Wronskian form is also given. Besides, the authors verify that the ( N − 1 ) - and N -soliton-like solutions satisfy the Bäcklund transformation. The results obtained in this paper might be valuable for the study of the nonuniform media.

The Kadomtsev-Petviashvili equation with self-consistent sources in nonuniform media

We derive the nonisospectral Kadomtsev-Petviashvili equation with self-consistent sources and obtain N-soliton solutions of the equation by both Hirota’s method and the Wronskian technique. We discuss one-soliton characteristics, two-soliton scattering in nonuniform media, and source effects.

ELECTROMAGNETIC SOURCE TRANSFORMATIONS AND SCALARIZATION IN STRATIFIED GYROTROPIC MEDIA

It is known that with restrictions on the type of the constitutive equations, Maxwell’s equations in non-uniform media can sometimes be reduced to two 2nd order differential equations for 2 scalar quantities only. These results have previously been obtained in two quite different ways, either by a “scalarization of the sources”, where the relevant scalar quantities are essentially vector potential components, and the derivation was limited to isotropic media, or alternatively by using the “scalar Hertz potentials”, and this method has been applied to more general media. In this paper, it is shown that both methods are equivalent for gyrotropic media. We show that the scalarization can be obtained by a combination of transformations between electric and magnetic sources and gauge transformations. It is shown that the method based on the vector potential, which previously used a non-traditional definition of the vector potentials, can also be obtained using the traditional definition provided a proper gauge condition is applied, and this method is then extended from isotropic to gyrotropic media. It is shown that the 2 basic scalar Hertz potentials occurring in the second method are invariant under the source scalarization transformations of the first method and therefore are the natural potentials for obtaining scalarization. Finally, it is shown that both methods are also equivalent with a much older third method based on Hertz vectors.

The influence of dimensionality on simulations of mass recovery from nonuniform dense non-aqueous phase liquid (DNAPL) source zones

The influence of model dimensionality on predictions of mass recovery from dense non-aqueous phase liquid (DNAPL) source zones in nonuniform permeability fields was investigated using a modified version of the modular three-dimensional transport simulator (MT3DMS). Thirty-two initial two- (2D) and three-dimensional (3D) tetrachloroethene–DNAPL source zone architectures, taken from a recent modeling study, were used as initial conditions for this analysis. Commonly employed source zone metrics were analyzed to determine differences between 2D and 3D predictions: (i) down-gradient flux-averaged contaminant concentration, (ii) reductions in contaminant mass flux through a down-gradient boundary, (iii) source zone ganglia-to-pool (GTP) ratio, and (iv) time required to achieve a remediation objective. 3D flux-averaged contaminant concentrations were approximately 3.5 times lower than concentrations simulated in 2D. This difference was attributed to dilution of the contaminant concentrations down gradient of the source zone. Contaminant flux reduction predictions for a given mass recovery were generally 5% higher in 3D simulations than in 2D simulations. The GTP ratio declined over time as mass was recovered in both 2D and 3D simulations. Although the source longevity (i.e., time required to achieve 99.99% mass recovery) differed between individual 2D and 3D realizations, the mean source longevity for the 2D and 3D simulation ensembles was within 2%. 2D simulations tended to over-predict the time required to achieve lower mass recovery levels (e.g. 50% mass recovery) due to a smaller contaminated area exposed to uncontaminated water. These findings suggest that ensemble averages of 2D numerical simulations of DNAPL migration, entrapment, dissolution, and mass recovery in statistically homogenous, nonuniform media may provide reasonable approximations to average behavior obtained using simulations conducted in fully three-dimensional domains.

Conductivity of structurally anisotropic 2D composites

The character of electric conduction in a two-dimensional structurally anisotropic composite containing elliptic inclusions with a large ratio of the semiaxis. Dielectric and ideally conducting inclusions were considered. It was demonstrated that the approximation linear in concentration of inclusions fails beginning from rather small concentrations. The region of intermediate concentrations was studied using approximate methods based on the analogy between electric conductivity and the diffusion of a single particle in a nonuniform medium. The conductivity of the model system under study was analyzed over a wide concentration range with the use of the effective medium theory; the results are in close agreement with those obtained by qualitative methods. The behavior of conductivity near the percolation threshold (metal-to-dielectric phase transition) was examined using the similarity. The conductivity of thin composite films with inclusions in the form of carbon nanotubes was examined, as an example.

Fronts in randomly advected and heterogeneous media and nonuniversality of Burgers turbulence: Theory and numerics

A recently established mathematical equivalence-between weakly perturbed Huygens fronts (e.g., flames in weak turbulence or geometrical-optics wave fronts in slightly nonuniform media) and the inviscid limit of white-noise-driven Burgers turbulence-motivates theoretical and numerical estimates of Burgers-turbulence properties for specific types of white-in-time forcing. Existing mathematical relations between Burgers turbulence and the statistical mechanics of directed polymers, allowing use of the replica method, are exploited to obtain systematic upper bounds on the Burgers energy density, corresponding to the ground-state binding energy of the directed polymer and the speedup of the Huygens front. The results are complementary to previous studies of both Burgers turbulence and directed polymers, which have focused on universal scaling properties instead of forcing-dependent parameters. The upper-bound formula can be heuristically understood in terms of renormalization of a different kind from that previously used in combustion models, and also shows that the burning velocity of an idealized turbulent flame does not diverge with increasing Reynolds number at fixed turbulence intensity, a conclusion that applies even to strong turbulence. Numerical simulations of the one-dimensional inviscid Burgers equation using a Lagrangian finite-element method confirm that the theoretical upper bounds are sharp within about 15% for various forcing spectra (corresponding to various two-dimensional random media). These computations provide a quantitative test of the replica method. The inferred nonuniversality (spectrum dependence) of the front speedup is of direct importance for combustion modeling.

Spinwave propagation and coupling in magnonic waveguides

In this study, we start with the Landau-Lifshitz-Gilbert equation to describe the spinwaves in magnetic medium using the linearization approach. First, the reflection and refraction behaviors of spinwaves on an interface of a nonuniform magnetic medium is investigated. The study is then extended to spinwave propagation in lossless planar magnonic waveguides. Spinwave modes have been characterized based on the magnetic properties of the waveguides. Coupling between spinwaves in the structure of double magnonic waveguides is found to be possible due to the dipole interaction of the spinwaves. The spinwave coupling and power transfer associated with it is totally dependent on the waveguide structure and properties and the spinwave modes.

Numerical simulation for the 3D seepage flow with fractional derivatives in porous media

In this paper, the numerical simulation of the 3D seepage flow with fractional derivatives in porous media is considered under two special cases: non-continued seepage flow in uniform media (NCSF-UM) and continued seepage flow in non-uniform media (CSF-NUM). A fractional alternating direction implicit scheme (FADIS) for the NCSF-UM and a modified Douglas scheme (MDS) for the CSF-NUM are proposed. The stability, consistency and convergence of both FADIS and MDS in a bounded domain are discussed. A method for improving the speed of convergence by Richardson extrapolation for the MDS is also presented. Finally, numerical results are presented to support our theoretical analysis.

Transport equation for gas-dynamic perturbations in a spatially nonuniform self-igniting medium

An approach is developed to describe the formation and propagation of small gas-dynamic perturbations in a spatially nonuniform self-igniting medium for arbitrary chemical-reaction kinetics. An asymptotic equation describing the variation in the amplitude of the gas-dynamic perturbation along the characteristic is derived, and the conditions of its applicability are defined. Zel’dovich’s spontaneous flame wave is shown to be a natural zeroth approximation to the solution of the equations of gas dynamics. A method is proposed to determine the point in space at which a shock wave is formed.

Recursive formula for the peak delay time with travel distance in von Kármán type non-uniform random media on the basis of the Markov approximation

SUMMARY Direct waves of microearthquakes in the high-frequency range (>1 Hz) strongly reflect the random inhomogeneities near their ray paths. This study conducts numerical simulations of envelope broadening of impulsively radiated wavelet assuming spatially non-uniform distri- bution of random inhomogeneities. We assume plural von Karman type power spectral density functions (PSDF) for random inhomogeneity to clarify how the non-uniformly distributed ran- dom media affect the frequency dependence of envelope broadening. We employ the stochastic ray path method based on the Markov approximation for the mutual coherence function. This method is appropriate to simulate multiple forward scattering during the wave propagation. We mainly examine the travel distance and frequency dependence of the peak delay time in relation to the parameters characterizing the PSDFs. The peak delay time, which is defined as the time lag from the direct-wave onset to the maximum amplitude arrival of its envelope, is the best parameter reflecting the accumulated scattering effect in random media and is quite insensitive to the intrinsic attenuation. According to the numerical simulations in various non-uniform random media, we find some remarkable features in travel distance and frequency dependence, which cannot be found in uniform random media. For example, the frequency dependence in uniform random media is uniquely determined by the spectral gradient of PSDF for arbitrary travel distance; however, that in non-uniform media gradually changes as travel distance in- creases if the waves have experienced a change of spectral gradient in PSDF. Considering the results of our simulation, we propose a simple recursive formula to calculate the peak delay time in non-uniform random media. This recursive formula can predict the simulation results appropriately and relate the peak delay times to two parameters quantifying the von Karman type PSDF in short wavelengths. It will become a mathematical base for the inversion of peak delay times of bandpass filtered traces to estimate the spatial distribution of random inhomogeneity spectra.

The influence of the environment on the propagation of protostellar outflows

The properties of bipolar outflows depend on the structure in the environment as well as the nature of the jet. To help distinguish between the two, we investigate here the properties pertaining to the ambient medium. We execute axisymmetric hydrodynamic simulations, injecting continuous atomic jets into molecular media with density gradients (protostellar cores) and density discontinuities (thick swept-up sheets). We determine the distribution of outflowing mass with radial velocity (the mass spectrum) to quantify our approach and to compare to observationally determined values. We uncover a sequence from clump entrainment in the flanks to bow shock sweeping as the density profile steepens. We also find that the dense, highly supersonic outflows remain collimated but can become turbulent after passing through a shell. The mass spectra vary substantially in time, especially at radial speeds exceeding 15 km s -1 . The mass spectra also vary according to the conditions: both envelope-type density distributions and the passage through dense sheets generate considerably steeper mass spectra than a uniform medium. The simulations suggest that observed outflows penetrate highly non-uniform media.

Mass transfer in nonuniform media

The lattice model was used to derive equations describing diffusion in a nonuniform medium in the absence of local equilibrium at nonzero temperature gradients. Equations are obtained that extend a diffusion equation with fractional derivative in time and a nonlinear diffusion equation, which was previously obtained within the framework of a generalized thermodynamic approach, to cover diffusion in a medium with a nonuniform temperature field.

Singular perturbation analysis for convection-diffusion-reaction equation on nonuniform media

In this paper we study the a one-dimensional reaction-diffusion-convection (R-D-C) equation with weak spatial non-uniformities to the diffusion and reaction rates. The convection is represented by a uniform flow field. Singular perturbation analysis will be used to compute the speed of the front as well as the asymptotic solution.

Spatially autoresonant stimulated Raman scattering in nonuniform plasmas

New solutions to the coupled three-wave equations in a nonuniform plasma medium are presented that include both space and time dependence of the waves. By including the dominant nonlinear frequency shift of the material wave, it is shown that if the driving waves are sufficiently strong (in relation to the medium gradient), a nonlinearly phase-locked solution develops that is characteristic of autoresonance. In this case, the material (electrostatic) wave develops into a front starting at the linear resonance point and moving with the wave group velocity in a manner such that the intensity increases linearly with the propagation distance. The forms of the other two (electromagnetic) waves follow naturally from the Manley–Rowe relations.

Phase growth in bistable systems with impurities

A system of coupled chaotic bistable maps on a lattice with randomly distributed impurities is investigated as a model for studying the phenomenon of phase growth in nonuniform media. The statistical properties of the system are characterized by means of the average size of spatial domains of equivalent spin variables that define the phases. It is found that the rate at which phase domains grow becomes smaller when impurities are present and that the average size of the resulting domains in the inhomogeneous state of the system decreases when the density of impurities is increased. The phase diagram showing regions where homogeneous, heterogeneous, and chessboard patterns occur on the space of parameters of the system is obtained. A critical boundary that separates the regime of slow growth of domains from the regime of fast growth in the heterogeneous region of the phase diagram is calculated. The transition between these two growth regimes is explained in terms of the stability properties of the local phase configurations. Our results show that the inclusion of spatial inhomogeneities can be used as a control mechanism for the size and growth velocity of phase domains forming in spatiotemporal systems.

Supernova remnant evolution in uniform and non-uniform media

Aims. In this work numerical simulations showing the time evolution of supernova remnants (SNRs) in uniform and non-uniform interstellar medium (ISM) are presented. Methods. We use a hydrodynamic model including a kinematic calculation of the interstellar magnetic field. Important parameters influencing SNR evolution include the ejecta mass and energy of the remnant, as well as the ISM density and adiabatic index. Results. By varying these parameters we constructed an analytical expression giving the return time of the SNR reverse shock to the origin, in terms of these parameters. We also found that the reverse shock spends half of its time moving outward and the other half returning to the origin. Also computed is SNR evolution in non-uniform media where the blast wave moves from one medium into either a less or more dense medium. As the SNR moves into a medium of higher density a reflection wave is created at the interface between the two media which is driven back toward the center. This drives mass via a nonspherical flow away from the discontinuity. As this wave moves inward it also drags some of the ISM field lines (if the field is parallel with the interface) with it and heats the inside of the SNR resulting in larger temperatures in this region. When a SNR explodes in a medium with a high density and the blast wave propagates into a medium with a lower density, a cavity is being blown away changing the geometry of the high density region. Also, once the forward shock moves into the medium of less density a second reverse shock will start to evolve in this region.

Formation of traveling waves in nematics due to material parameter ramps

We propose a simple one-dimensional linear model to describe the formation of localized traveling waves due to material parameter ramps in nematic liquid crystals. We assume that due to some external perturbation the material parameters (conductivity, dielectric constants, viscosity, elasticity) of the liquid crystal become slowly varying functions of position. Temperature gradient in localized regions induced by a laser beam could be one such perturbation. We obtain a 4x4 system of electrohydrodynamic equations [partial derivative equations (PDE) with respect to time and position]. At first we assume that all parameters change from their nonperturbed value by the same ramp function. Then, to reveal the parameters which slowly change are predominant in the formation of localized traveling waves, we obtain four more systems of equations. Each time we assume that just one of the material parameters changes while the others are held constant. Accordingly, by reducing 4x4 systems of equations into one equation and using a WKB-like approach for nonuniform media, we obtain the relevant dispersion relations. We show that ramps of elasticity and dielectric parameters play the dominant role in the formation of localized traveling waves in nonuniform nematic media.

On the stochastic simulation procedure of estimating critical hydraulic gradient for gas storage in unlined rock caverns

We investigated some aspects of spatial variability and their effect on critical hydraulic gradient which is essential for gas containment of underground storage caverns. Monte Carlo technique can be effectively applied to obtain an approximate solution to the two-dimensional steady flow of a stochastically defined non-uniform medium. For the stochastic simulation we generated hydraulic conductivity field on the selected grid resolution using HYDRO_GEN with estimated (based on actual data) In-K statistics with mean, variance, anisotropic integral scales. In this study, among various covariance functions, a Gaussian covariance function (GCF) was used. To find the critical value of the hydraulic gradient, probability density functions (PDFs) using 1000 outputs at an interested cell were developed. The results obtained in this study were compared with previous results for an exponential covariance function (ECF). It was found that in a stationary In K field the uncertainty of hydraulic head and gradient depend not only on the variance and integral scale of the In K field but also on the shape of its covariance function. From these results we can conclude that the critical range of hydraulic gradient is significantly affected by the type of covariance function. Thus, when critical hydraulic gradient is to be determined one should consider shape of covariance function as well as statistical parameters such as mean, variance and correlation scale.

On the origin of asymmetries in bilateral supernova remnants

Aims. We investigate whether the morphology of bilateral supernova remnants (BSNRs) observed in the radio band is determined mainly either by a non-uniform interstellar medium (ISM) or by a non-uniform ambient magnetic field. Methods. We perform 3D MHD simulations of a spherical SNR shock propagating through a magnetized ISM. Two cases of shock propagation are considered: 1) through a gradient of ambient density with a uniform ambient magnetic field; 2) through a homogeneous medium with a gradient of ambient magnetic field strength. From the simulations, we synthesize the synchrotron radio emission, making different assumptions about the details of acceleration and injection of relativistic electrons. Results. We find that asymmetric BSNRs are produced if the line-of-sight is not aligned with the gradient of ambient plasma density or with the gradient of ambient magnetic field strength. We derive useful parameters to quantify the degree of asymmetry of the remnants that may provide a powerful diagnostic of the microphysics of strong shock waves through the comparison between models and observations. Conclusions. BSNRs with two radio limbs of different brightness can be explained if a gradient of ambient density or, most likely, of ambient magnetic field strength is perpendicular to the radio limbs. BSNRs with converging similar radio arcs can be explained if the gradient runs between the two arcs.

Three-dimensional visualization of ductile fracture in an Al–Si alloy by high-resolution synchrotron X-ray microtomography

A direct three-dimensional visualization of the ductile fracture process under monotonic loading, in an Al–Si–Mg alloy specimen containing a fatigue pre-crack, has been conducted by means of the high-resolution X-ray computed tomography. Different morphology features of the crack viewed on different cross sections within the bulk of the specimen are detected. The isolated voids are distinguished from the microcracks. The evolution of the crack and voids as a function of the applied load are monitored and analyzed three-dimensionally. The advantages of the technique over traditional cross sectional observations are demonstrated, which facilitates to understand the fracture mechanisms in ductile materials, and to elucidate the fracture/microstructure interaction in non-uniform mediums.