Long-wavelength Approximation Research Articles

Solitary waves propagation described by Korteweg-de Vries equation in the granular chain with initial prestress

The paper work relates to Nesterenko’s problem to further study the solitary wave when the strong external force acts on the granular chain. We also study the problem under the long-wavelength approximation and find that such kind of solitary wave in system with the initial prestress can be described by the Korteweg-de Vries (KdV) equation. It is found that the results of analytical and numerical are in an excellent agreement. Furthermore, we study the scattering of the KdV solitary wave in different granular materials both in theoretical and numerical methods. It is found that the numbers and the amplitudes of both the reflected and the transmitted waves depend not only on the amplitude of the incident solitary wave but also on the variations of both sides of the discontinuity such as the mass, Young’s modulus or radius of the grains.

Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip

The present study concerned with the impact of velocity slip on MHD peristaltic flow through a porous medium with heat and mass transfer is investigated. The relevant equations of flow with heat and mass transfer have been developed. Analytic solution is carried out under long-wavelength and small Reynolds number approximations. The expressions for the stream function, temperature and concentration and the heat transfer coefficient are obtained. Numerical results are graphically discussed for various values of physical parameters of interest. The velocity and temperature field increase with an increase in the velocity slip parameter and permeability parameter while it decreases with an increase in the Hartmann number.

Upconverted photoluminescence induced by radiative coupling between excitons

We propose an unconventional scheme of photoluminescence in a semiconductor thin film, where the nonlocal correlation between an excitonic wave and a light wave prominently enhances the exciton-light coupling beyond the long-wavelength approximation (the so-called excitonic superradiance regime). On the basis of the developed method extending input-output theory, we elucidate atypical photoluminescence effects due to the strong wave-wave correlation. In particular, the up-converted photoluminescence based on the coherent superposition of excitons is found to be highly efficient, i.e., it can be realized by weak pumping without auxiliary systems such as cavities or photonic antennas.

Using electric fields to induce patterning in leaky dielectric fluids in a rod-annular geometry

The stability and axisymmetric deformation of two immiscible, viscous, perfect or leaky dielectric fluids confined in the annulus between two concentric cylinders are studied in the presence of radial electric fields. The fields are set up by imposing a constant voltage potential difference between the inner and outer cylinders. We derive a set of equations for the interface in the long-wavelength approximation which retains the essential physics of the system and allows for interfacial deformations to be as large as the annular gap hence accounting for possible touchdown at the inner or outer electrode. The effects of the electric parameters are evaluated initially by performing a linear stability analysis which shows excellent agreement with the linear theory of the full axisymmetric problem in the appropriate long wavelength regime. The nonlinear interfacial dynamics are investigated by carrying out direct numerical simulations of the derived long wave models, both in the absence and presence of electric fields. For non-electrified thin layer flows (i.e. one of the layers thin relative to the other) the long-time dynamics agree with the lubrication approximation results found in literature. When the liquid layers have comparable thickness our results demonstrate the existence of both finite time and infinite time singularities (asymptotic touching solutions) in the system. It is shown that a two-side touching solution is possible for both the non-electrified and perfect dielectric cases, while only one-side touching is found in the case of leaky dielectric liquids, where the flattened interface shape resembles the pattern solutions found in literature. Meanwhile the finite-time singular solution agrees qualitatively with the experiments of~\cite{Reynolds1965}.

Impact of velocity slip and joule heating on MHD peristaltic flow through a porous medium with chemical reaction

The present study investigates the effect of joule heating and velocity slip on MHD peristaltic flow in a porous medium with chemical reaction. The relevant equations on the fluid flow have been developed. Analytic solution is carried out under long-wavelength and low-Reynolds number approximations. Exact solution is evaluated for the stream function, which is used to find the velocity of the fluid flow, temperature, concentration. Numerical computations have been performed for the influence of various emerging parameters on the flow characteristics velocity, temperature, concentration are shown and discussed with the help of graphs. Also, the expressions for skin friction coefficient, Nusselt number and Sherwood number at the channel wall are obtained and analyzed. It is found that a generative chemical reaction is greater than the destructive chemical reaction on the concentration. The size of the trapping bolus increases with increasing velocity slip parameterβ.

Analysis of Entropy Generation in the Flow of Peristaltic Nanofluids in Channels With Compliant Walls

Entropy generation during peristaltic flow of nanofluids in a non-uniform two dimensional channel with compliant walls has been studied. The mathematical modelling of the governing flow problem is obtained under the approximation of long wavelength and zero Reynolds number (creeping flow regime). The resulting non-linear partial differential equations are solved with the help of a perturbation method. The analytic and numerical results of different parameters are demonstrated mathematically and graphically. The present analysis provides a theoretical model to estimate the characteristics of several Newtonian and non-Newtonian fluid flows, such as peristaltic transport of blood.

Slip effects and endoscopy analysis on blood flow of particle-fluid suspension induced by peristaltic wave

This study describes the endoscopy and slip effects on blood flow of particulate fluid suspension induced by peristaltic wave through a non-uniform annulus. An approximation of long wavelength and low Reynolds is used to model the governing equation of continuity and momentum equation for fluid phase and particulate phase. The influence of all the pertinent parameters has been obtained such as slip parameter (β), Jeffrey fluid parameter (λ1), particle volume fraction (C) and volume flow rate Q(z,t), velocity constant (v0) and the inner radius (ϵ) against pressure gradient, pressure rise, friction forces, velocity of fluid phase and particulate phase (uf,up). Numerical results are carried out for pressure rise and friction forces on the inner and outer tube.

Optical properties of plasmonic nanopillars in extended quasi-static limits

An extended method based on the long-wavelength approximations is introduced to investigate the optical properties of finite-size metal nanopillars for different incident light polarizations. In this proposed method, the optical depolarization factors are the key parameters and strongly depend on the position of the considered point dipole inside the irregular-shaped nanoparticles. Interestingly, depending on the polarization, it is shown that the conventional point dipole positioned at the center of a nonspherical nanoparticle seldom is the proper choice as its simple electromagnetic model. The adequate off-center point, though, is determined and physically associated with the electric-field distribution inside the nanoparticle, which is supported by numerical computations. Remarkably, the proposed extended MLWA could extend the quasi-static limits and, consequently, more accurately predict the optical properties of the finite-size plasmonic nanopillar.

Site-city interaction: theoretical, numerical and experimental crossed-analysis

The collective excitation of city structures by a seismic wavefield and the subsequent multiple Structure-Soil-Structure Interactions (SSSIs) between the buildings are usually disregarded in conventional seismology and earthquake engineering practice. The objective here is to qualify and quantify these complex multiple SSSIs through the design of an elementary study case, which serves as a benchmark for theoretical, numerical and experimental crossed-analysis. The experimental specimen consists of an idealized site-city setup with up to 37 anisotropic resonant structures arranged at the top surface of an elastic layer and in co-resonance with it. The experimental data from shaking table measurements is compared with the theoretical and numerical results provided respectively by an equivalent city-impedance model derived analytically from homogenization in the long-wavelength approximation and a model based on boundary elements. The signatures of the site-city interactions are identified in the frequency, time and space domain, and in particular consist of a frequency-dependent free/rigid switch in the surface condition at the city resonance, beatings in the records and the depolarization of the wavefield. A parametric study on the city density shows that multiple SSSIs among the city structures (five are sufficient) can have significant effects on both the seismic response of its implantation site and that of the buildings. Key parameters are provided to assess site-city interactions in the low seismic frequency range: They involve the mass and rigidity of the city compared to those of the soil and the damping of the building.

Symmetry in the problem on wave-flow modes of a thin viscous-fluid layer

It is shown that a suitable transformation in the transverse-coordinate shift makes the set of equations describing the flow of viscous fluid films in the long-wavelength approximation invariant with respect to the reflection transformation. When constructing the flow models with the Galerkin spectral method, this symmetry in the equation for the longitudinal velocity stimulates its representation with the help of a set of functions symmetric with respect to the transverse coordinate.

Generalized velocity-gauge form of the light-matter interaction Hamiltonian beyond the dipole approximation

The exact velocity-gauge minimal-coupling Hamiltonian describing the laser-matter interaction is transformed into another form by means of a series of gauge transformations. The Hamiltonian corresponding to this point of view is valid for an arbitrary time- and space-dependent laser field, also known as a nondipole field. In effect, the Hamiltonian represents a generalization of the original velocity-gauge minimal-coupling Hamiltonian in the sense that the particle's (classical) velocity in the laser propagation direction is also explicitly accounted for by a new operator term. Imposing the so-called long-wavelength approximation (LWA) on the field, i.e., assuming the laser wavelength being much larger than the extent of the atomic system, the spatial dependence of the field can be neglected and the interaction Hamiltonian reduces to a simpler form. Nevertheless, the resulting LWA Hamiltonian includes the effect of the magnetic-field component of the laser, which is in clear contrast with the usual dipole approximation Hamiltonian derived by imposing the LWA directly on the initial velocity-gauge minimal-coupling Hamiltonian. As such, the weak-field condition necessary to justify neglecting the magnetic field, and the LWA condition, can be considered independently in this formalism, making it an attractive alternative for a broad range of applications in strong-field physics. We demonstrate that, from a numerical perspective, this form of the light-matter interaction is advantageous compared to its standard velocity-gauge counterpart as it gives rise to faster convergence properties when describing ionization dynamics in superintense fields beyond the dipole approximation.

Localized surface plasmon resonance and the size effects of magneto-optic rods

Localized surface plasmon resonance of cylindrical magneto optical particles provides an important mechanism for the formation of chiral edge states in two-dimensional magneto-optical photonic crystals. These states are an electromagnetic analogy of the so-called chiral edge state's (CESs) in a quantum Hall system where the power transmission is unidirectional due to particular topological properties of the bands. Just like their electronic counterpart, the number of optical CESs in the band gap opened by an applied magnetic field is determined by the sum of the Chern numbers of the lower bands. For a two-dimensional photonic crystal composed of ferrite rods magnetized along their axis, the coupling of the localized surface plasmon resonance states of each rod results in a narrow flat band-gap, which contains one-way edge modes arising from the circulation of the energy flow around each rod excited by the resonance with broken time-reversal symmetry. So far the interpretation of the resonance-related chiral edge states are based on the long-wavelength approximation of the localized surface plasmon resonance of an individual magneto-optical particle. Though the results agree with the experimental results qualitatively, an obvious quantitative deviation is still obvious. In this work we apply the scattering theory to analyze the resonance condition and the features of both the far-field and the near-field at resonance for cylindrical magneto-optical particles. Our calculation shows that the splitting of scattering peaks of different orders will occur due to the magneto-optical effect. Such a split is observed between an (+n)-peak and an (-n) peak, as a sign of the broken time-reversal symmetry, and also between peaks of lower-order and higher-order. Another important feature is the simultaneous occurring of the far-field resonance and the near-field resonance, where the latter is characterized by a peak of energy-flow circulation around the particle. Based on this model the effects of particle size on the resonance peaks are discussed. It is shown that the resonance peaks are moved and broadened with the particle size increasing. The results explain the obvious deviation of the position of the resonance band-gap from the predicted frequency according to the previous long-wavelength approximation. Furthermore, the calculation of a particle of moderately-large size (nearly one-tenth of the incident wavelength) demonstrates the appearance of higher-order modes up to n=4 circling around the particle surface. This implies that these higher-order modes may also make non-trivial contribution to the formation of the flat band-gap observed in a photonic crystal of ferrite-rods and affect the behaviours of the chiral-edge state existing in such a gap. Particularly, it may be helpful in realizing the multimodes of chiral edge states in magneto-optical photonic crystals.

Peristaltic Flow of Couple Stress Fluid in a Non-Uniform Rectangular Duct Having Compliant Walls

The present study investigates the peristaltic flow of couple stress fluid in a non-uniform rectangular duct with compliant walls. Mathematical modeling is based upon the laws of mass and linear momentum. Analytic solutions are carried out by the eigen function expansion method under long-wavelength and low-Reynolds number approximations. The features of the flow characteristics are analyzed by plotting the graphs of various values of physical parameters of interest. Trapping bolus scheme is also presented through streamlines.

Nonlocal dispersion anomalies of Dyakonov-like surface waves at hyperbolic media interfaces

Dyakonov-like surface waves (DSWs) propagating obliquely on an anisotropic nanostructure have been theoretically proved in a few cases including 2D photonic crystals and metal-insulator (MI) layered metamaterials. Up to now, the long-wavelength approximation has been employed in order to obtain effective parameters to be introduced in the Dyakonov equation, which is largely restricted to material inhomogeneities of a few nanometers when including metallic elements. Here, we explore DSWs propagating obliquely at the interface between an insulator and a hyperbolic metamaterial, the latter consisting of a 1D MI bandgap grating using realistic slab sizes. We found unexpected favorable conditions for the existence of such surface waves. The finite element method is used to investigate the peculiarities of this new family of DSWs.

Trapping study of nanofluids in an annulus with cilia

The main purpose of this article is to present a mathematical model of ciliary motion in an annulus. In this analysis, two dimensional flow of a viscous nanofluid is observed in an annulus with ciliated tips. The current theoretical model may be supposed as mathematical illustration to the movement of ciliary motion in the presence of an endoscopic tube (or catheter tube). The inner tube is rigid, while the outer tube takes a metachronal wave. The features of ciliary structures are determined by the dominance of viscous effects over inertial effects using the long-wavelength approximation. Exact solutions have been established for both velocity and temperature profiles, which include nanoparticle effects. The features of the ciliary motion are analyzed by plotting graphs and discussed in detail.

Shallow-water conservation laws

The derivation of basic conservation laws in the shallow-water theory from the multidimensional integral laws of conservation of mass and total momentum describing the plane-parallel flow of an ideal incompressible fluid above a horizontal bottom is proposed. The restrictions on flow parameters arising in this case have the integral form and are much weaker in comparison with the requirement of flow potentiality and the condition of long-wavelength approximation. The last fact substantiates the use of the shallow-water model for the mathematical modeling of a much wider class of wave flows, the parameters of which are not related directly to the restrictions of the long-wavelength approximation.

Wave flows initiated by vertical lifting of a rectangular beam from shallow water

The problem of flows initiated by vertical lifting of a rectangular beam partially submerged in shallow water filling a rectangular prismatic channel with a horizontal bottom is studied in the long-wavelength approximation. The width of the beam is equal to the channel width, and its upper and lower planes are parallel to the channel bottom. In the first stage of the flow, the lower surface of the low beam is completely submerged in the liquid, which is lifted after it by hydrostatic pressure. Conditions for the well-posedness of this problem are obtained, and solutions describing the liquid flow in the region adjacent to the bottom surface of the beam and in outer regions with a free upper boundary are constructed for different laws of lifting of the beam.

Casimir Effect in Hemisphere Capped Tubes

In this paper we investigate the vacuum densities for a massive scalar field with general curvature coupling in background of a (2+1)-dimensional spacetime corresponding to a cylindrical tube with a hemispherical cap. A complete set of mode functions is constructed and the positive-frequency Wightman function is evaluated for both the cylindrical and hemispherical subspaces. On the base of this, the vacuum expectation values of the field squared and energy-momentum tensor are investigated. The mean field squared and the normal stress are finite on the boundary separating two subspaces, whereas the energy density and the parallel stress diverge as the inverse power of the distance from the boundary. For a conformally coupled field, the vacuum energy density is negative on the cylindrical part of the space. On the hemisphere, it is negative near the top and positive close to the boundary. In the case of minimal coupling the energy density on the cup is negative. On the tube it is positive near the boundary and negative at large distances. Though the geometries of the subspaces are different, the Casimir pressures on the separate sides of the boundary are equal and the net Casimir force vanishes. The results obtained may be applied to capped carbon nanotubes described by an effective field theory in the long-wavelength approximation.

Theoretical evaluation of left-handed material parameters with cylindrical inclusions

We studied the electric and magnetic responses from the infinite chain of circular metallic cylinders periodically engrossed in a dielectric matrix (wax). The study used the effective medium theory developed for the long-wavelength approximations of complex effective constants. The analytical results show that the presented metamaterial exhibits the phenomenon with negative effective parameters. Moreover, increase in volume fraction increases the negativity of the effective parameters. With an increase in gigahertz (GHz) frequencies, the effective dielectric constant approaches zero, and a minor increase in the negativity of the effective magnetic constant is observed.

Solitary waves in asymmetric electron–positron–ion plasmas

By solving the coupled equations of the electromagnetic field and electrostatic potential, we investigate solitary waves in an asymmetric electron–positron plasma and/or electron–positron–ion plasmas with delicate features. It is found that the solutions of the coupled equations can capture multipeak structures of solitary waves in the case of cold plasma, which are left out by using the long-wavelength approximation. By considering the effect of ion motion with respect to non-relativistic and ultra-relativistic temperature plasmas, we find that the ions’ mobility can lead to larger-amplitude solitary waves; especially, this becomes more obvious for a high-temperature plasma. The effects of asymmetric temperature between electrons and positrons and the ion fraction on the solitary waves are also studied and presented. It is shown that the amplitudes of solitary waves decrease with positron temperature in asymmetric temperature electron–positron plasmas and decrease also with ion concentration.