On band gaps of nonlocal acoustic lattice metamaterials: a robust strain gradient model

We have proposed an “exact” strain gradient (SG) continuum model to properly predict the dispersive characteristics of diatomic lattice metamaterials with local and nonlocal interactions. The key enhancement is proposing a wavelength-dependent Taylor expansion to obtain a satisfactory accuracy when the wavelength gets close to the lattice spacing. Such a wavelength-dependent Taylor expansion is applied to the displacement field of the diatomic lattice, resulting in a novel SG model. For various kinds of diatomic lattices, the dispersion diagrams given by the proposed SG model always agree well with those given by the discrete model throughout the first Brillouin zone, manifesting the robustness of the present model. Based on this SG model, we have conducted the following discussions. (I) Both mass and stiffness ratios affect the band gap structures of diatomic lattice metamaterials, which is very helpful for the design of metamaterials. (II) The increase in the SG order can enhance the model performance if the modified Taylor expansion is adopted. Without doing so, the higher-order continuum model can suffer from a stronger instability issue and does not necessarily have a better accuracy. The proposed SG continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviors all over the first Brillouin zone. (III) The effects of the nonlocal interactions are analyzed. The nonlocal interactions reduce the workable range of the well-known long-wave approximation, causing more local extrema in the dispersive diagrams. The present model can serve as a satisfactory continuum theory when the wavelength gets close to the lattice spacing, i.e., when the long-wave approximation is no longer valid. For the convenience of band gap designs, we have also provided the design space from which one can easily obtain the proper mass and stiffness ratios corresponding to a requested band gap width.

Time Evolution of the Position Operators in a Bilayer Graphene

In this work, the researchers mainly focus on the trembling motion which is known as Zitterbewegung in a bilayer grapheme. This is effectively achieved by means of the long-wave approximation. That is, the Heisenberg representation is ultimately employed in order to derive the analytical expression concerning the expectation value related to the position operator along the longitudinal and transversal orientation, which describes the motion concerning the electronic wave packet inside the bilayer graphene. Parameters’ numbers are considered to explicate the packet of Gaussian wave, including the polarization of initial pseudo-spin as well as the wave number of the initial carrier number along with the localized wave packet’s width along the longitudinal as well as transversal orientation. Consequently, the researchers show that the obvious oscillation in position operator can be effectively controlled not only by what is known as the initial parameters concerning the wave packet. Rather, it can mainly be controlled by selecting the localized quantum state’s components. Furthermore, the interference’s analysis between the conduction as well as valence bands concerning quantum states is really emphasized as the ability of what can be described as the transient’s emergence, or in a sense, aperiodic temporal oscillations concerning the average value of position operator in the bilayer graphene.

Wave-induced Lagrangian drift in a porous seabed

The mean drift in a porous seabed caused by long surface waves in the overlying fluid is investigated theoretically. We use a Lagrangian formulation for the fluid and the porous bed. For the wave field we assume inviscid flow, and in the seabed, we apply Darcy’s law. Throughout the analysis, we assume that the long-wave approximation is valid. Since the pressure gradient is nonlinear in the Lagrangian formulation, the balance of forces in the porous bed now contains nonlinear terms that yield the mean horizontal Stokes drift. In addition, if the waves are spatially damped due to interaction with the underlying bed, there must be a nonlinear balance in the fluid layer between the mean surface gradient and the gradient of the radiation stress. This causes, through continuity of pressure, an additional force in the porous layer. The corresponding drift is larger than the Stokes drift if the depth of the porous bed is more than twice that of the fluid layer. The interaction between the fluid layer and the seabed can also cause the waves to become temporally attenuated. Again, through nonlinearity, this leads to a horizontal Stokes drift in the porous layer, but now damped in time. In the long-wave approximation only the horizontal component of the permeability in the porous medium appears, so our analysis is valid for a medium that has different permeabilities in the horizontal and vertical directions. It is suggested that the drift results may have an application to the transport of microplastics in the porous oceanic seabed.

Two-dimensional model of wave hydrodynamics with high accuracy dispersion relation. II. Fourth, sixth and eighth orders

Построена полностью нелинейная слабо дисперсионная модель волновой гидродинамики четвертого порядка длинноволновой аппроксимации. За скорость в модели взята усредненная по глубине горизонтальная составляющая скорости трехмерного течения. Учтена подвижность дна. Выполненная модификация модели обеспечивает шестой и восьмой порядки точности аппроксимации дисперсионного соотношения трехмерной модели потенциальных течений. In the numerical simulation of medium-length surface waves in the framework of nonlinear dispersive (NLD) models, an increased accuracy of reproducing the characteristics of the simulated processes is required. A number of works (Kirby (2016), e.g.) describe approaches to improve the known NLD-models. In particular, NLD-models of the fourth order of the long-wave approximation have been proposed and, based on a comparison of numerical results with experimental data, their high accuracy has been demonstrated (Ataie-Ashtiani and Najafi-Jilani (2007); Zhou and Teng (2010)). In these new models, the horizontal component of the velocity vector of the threedimensional (FNPF-) model of potential flows at a certain surface located between the bottom and the free boundary was chosen as the velocity vector. The result was a very cumbersome form of equations. In addition, the laws of conservation of mass and momentum do not hold for these models. The main result of this work is the derivation of a two-parameter fully nonlinear weakly dispersive (mSGN4) model of the fourth order of the long-wave approximation, which is a generalization of the well-known Serre-Green-Naghdi (SGN) second order model. In the derivation, the velocity averaged over the thickness of the liquid layer was used. The assumption about the potentiality of the three-dimensional flow was used only at the stage of closing the model. The movement of the bottom is taken into account. For the derived model, the law of conservation of mass is satisfied, and the law of conservation of total momentum is satisfied in the case of a horizontal stationary bottom. The equations of the mSGN4-model are invariant under the Galilean transformation and are presented in a compact form similar to the equations of gas dynamics. The dispersion relation of the mSGN4-model has the fourth order of accuracy in the long wave region and satisfactorily approximates the dispersion relation of the FNPF-model in the short wave region. Moreover, with a special choice of the values of the model parameters, an increased accuracy of approximating the dispersion relation of the FNPF-model at long waves (sixth or eighth order) is achieved. Analysis of the deviations of the values of the phase velocity of the mSGN4 model from the values of the “reference” speed of the FNPF model in the entire wavelength range showed that the most preferable is the mSGN4 model with the parameter values corresponding to the Pad’e approximant (2,4).

Heat And Mass Transfer With Radiation In A Convective Flow Between Vertical Wavy Channels Due To Travelling Thermal Waves

The objective of this paper is to investigate analytically the convective flow of heat and mass transfer in vertical wavy channels due to travelling thermal waves. Effect of radiation, temperature dependent heat source/sink, concentration dependent mass source are taken into account. To tackle the highly complex non-linear problem, the perturbation technique is applied with long wave approximation.

Effect of the evaporation action on the stability of a thin liquid layer in the Landau–Levich problem

The stability of the liquid layer in the Landau–Levich problem is theoretically investigated in the presence of the evaporation effect from the free surface. The free energy of a thin layer of an incompressible fluid is the sum of the dispersion (van der Waals) interaction and the specific electrical interaction caused by the presence of double electric layers at both interphase boundaries. The stability of such a system with respect to perturbations is studied in the framework of the long – wave approximation in the system of Navier-Stokes equations. A stability map is provided for different values of the evaporation parameter. It is established that the stability of the system increases with an increase in the dimensionless number of evaporation.

Two-dimensional model of wave hydrodynamics with high accuracy dispersion relation

Полностью нелинейная слабо дисперсионная модель волновой гидродинамики, учитывающая подвижность дна, модифицирована с целью повышения точности дисперсионного соотношения. Проведено сравнение с известными аналогичными моделями и выявлено различие в асимптотическом поведении их фазовых скоростей. Application of nonlinear dispersion wave hydrodynamics (NLD-) models for solving practical problems constantly stimulates the search for ways to expand their field of applicability and achieve a more accurate reproduction of the characteristics of the simulated processes. A productive step in this direction turned out to be the method proposed by Madsen & Sørensen (1992), which made it possible to increase the approximation order of the dispersion relation of the Peregrine model while preserving the third order of derivatives included in the original equations and the second order of long-wave approximation. Later, other approaches were proposed to achieve this goal, which had a noticeable effect on expanding the field of applicability of NLD-models (for example, Nwogu (1993), Beji & Nadaoka (1996)). In the present work, we set a similar goal - to improve the properties of the dispersion relation of the model (and, therefore, the phase velocity), providing the Pade approximation (2,2) of the dispersion relation of the 3D model of potential flows. In contrast to earlier works on this subject, where weakly non-linear models were considered, we proceed from the fully nonlinear weakly dispersive two-dimensional Serre - Green - Naghdi (SGN-) model. The novelty of the proposed method consists in modifying the formula for the non-hydrostatic part of the pressure, while the accuracy of the long-wave approximation is preserved. It is shown that in some special cases the obtained fully nonlinear model is close to the known models (for example, after appropriate simplification it coincides with the model from Beji & Nadaoka (1996)). A dispersion analysis was performed one of the results of which was the conclusion that for sufficiently long waves the approximation order of the dispersion relation of the 3D model increases from the second to the fourth and an improvement was also achieved for more short waves. The proposed modification of the SGN-model is invariant with respect to the Galilean transformation; the law of conservation of mass and the law of balance of the total momentum are satisfied. However, the law of conservation of total energy is not satisfied. Apparently all NLD-models with improved dispersion characteristics possess this negative quality.

Rayleigh–Taylor and Kelvin–Helmholtz instability studied in the frame of a dimension-reduced model

Introducing an extension of a recently derived dimension-reduced model for an infinitely deep inviscid and irrotational layer, a two-layer system is examined in the present paper. A second thin viscous layer is added on top of the original one-layer system. The set-up is a combination of a long-wave approximation (upper layer) and a deep-water approximation (lower layer). Linear stability analysis shows the emergency of Rayleigh–Taylor and Kelvin–Helmholtz instabilities. Finally, numerical solutions of the model reveal spatial and temporal pattern formation in the weakly nonlinear regime of both instabilities. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.