Multiscale Expansion Research Articles

Vortex Dynamo in an Obliquely Rotating Stratified Nanofluid by Small-Scale Non-Helical Forces

In this work, a large-scale instability of the hydrodynamic -effect in an obliquely rotating stratified nanofluid taking into account the effects of Brownian diffusion and particle flux under the influence of a temperature gradient (thermophoresis) is obtained. The instability is caused by the action of an external small-scale non-spiral force, which excites small-scale velocity oscillations with zero helicity and a low Reynolds number. Nonlinear equations for large-scale motions are obtained using the method of multiscale asymptotic expansions by a small parameter (Reynolds number). A linear large-scale instability of hydrodynamic -effect is investigated depending on the parameters of rotation , temperature stratification , and concentration of nanoparticles . A new effect of the generation of large-scale vortex structures in nanofluid at is associated with an increase in the concentration of nanoparticles is obtained. The maximum instability increment is reached at inclination angles for the Prandtl numbers , and for the Prandtl numbers at inclination angles . It has been found that the frequency changing of the parametric impact will make it possible to control and track the generation of large-scale vortex structures. It is shown that circularly polarized Beltrami vortices appear in nanofluid as the result of new large-scale instability development. In this paper, the saturation regime of large-scale instability in an obliquely rotating stratified nanofluid with an external small-scale non-spiral force is investigated. In the stationary regime was obtained a dynamic system of equations for large-scale perturbations of the velocity field. Numerical solutions of this system of equations are obtained, which show the existence of localized vortex structures in the form of nonlinear Beltrami waves and kinks. The velocity profile of kink tends to be constant at large Z values.

Kinetic Modeling of Multiple Scattering of Acoustic Waves in Randomly Heterogeneous Flows

We study the propagation of sound waves in a three-dimensional, infinite ambient flow with weak random fluctuations of the mean particle velocity and speed of sound. We more particularly address the regime where the acoustic wavelengths are comparable to the correlation lengths of the weak inhomogeneities--the so-called weak coupling limit. The analysis is carried on starting from the linearized Euler equations and the convected wave equation with variable density and speed of sound, which can be derived from the nonlinear Euler equations. We use a multi-scale expansion of the Wigner distribution of a velocity potential associated to the waves to derive a radiative transfer equation describing the evolution of the angularly resolved wave action in space/time phase space. The latter experiences convection, refraction and scattering when it propagates through the heterogeneous ambient flow, although the overall wave action is conserved. The convection and refraction phenomena are accounted for by the convective part of the transport equation and depend on the smooth variations of the ambient quantities. The scattering phenomenon is accounted for by the collisional part of the transport equation and depends on the cross-power spectral densities of the fluctuations of the ambient quantities at the wavelength scales. The refraction, phase shift, spectral broadening, and multiple scattering effects of the high-frequency regimes described in various previous publications are thus encompassed by the proposed model. The overall derivation is based on the interpretation of spatial-temporal Wigner transforms in terms of semiclassical operators in their standard quantization.

An Integrated Health Condition Detection Method for Rotating Machinery Using Refined Composite Multivariate Multiscale Amplitude-Aware Permutation Entropy

With a view to realizing the fault diagnosis of rotating machinery effectively, an integrated health condition detection approach for rotating machinery based on refined composite multivariate multiscale amplitude-aware permutation entropy (RCmvMAAPE), max-relevance and min-redundancy (mRmR), and whale optimization algorithm-based kernel extreme learning machine (WOA-KELM) is presented in this paper. The approach contains two crucial parts: health detection and fault recognition. In health detection stage, multivariate amplitude-aware permutation entropy (mvAAPE) is proposed to detect whether there is a fault in rotating machinery. Afterward, if it is detected that there is a fault, RCmvMAAPE is employed to extract the initial fault features that represent the fault states from the multivariate vibration signals. Based on the multivariate expansion and multiscale expansion of amplitude-aware permutation entropy, RCmvMAAPE enjoys the ability to effectively extract state information on multiple scales from multichannel series, thereby overcoming the defect of information loss in traditional methods. Then, mRmR is adopted to screen the sensitive features so as to form sensitive feature vectors, which are input into the WOA-KELM classifier for fault classification. Two typical rotating machinery cases are conducted to prove the effectiveness of the raised approach. The experimental results demonstrate that mvAAPE shows excellent performance in fault detection and can effectively detect the fault of rotating machinery. Meanwhile, the feature extraction method based on RCmvMAAPE and mRmR, as well as the classifier based on WOA-KELM, shows superior performance in feature extraction and fault recognition, respectively. Compared with other fault identification methods, the raised method enjoys better performance and the average fault recognition accuracy of the two typical cases in this paper can all reach above 98%.

Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

For each pair of positive parameters, we define a perforated domain by making a small hole of size in an open regular subset of (). The hole is situated at distance from the outer boundary of the domain. Thus, when both the size of the hole and its distance from tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain and we denote its solution by Our aim is to represent the map that takes to in terms of real analytic functions of defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of to suitable subsets of we prove a global representation formula that holds on the whole of Such a formula allows us to rigorously justify multiscale expansions, which we subsequently construct.

Wave Processes in Rotating Compressible Astrophysical Plasma Flows with Stable Stratification

The wave processes in a rotating layer of compressible astrophysical plasma with a stable stratification and a linear entropy profile are studied theoretically. The compressibility is taken into account in the anelastic approximation. In this approximation the acoustic waves are filtered out, the system contains terms with a potential temperature (entropy), and the continuity equation contains an initial stratified density profile. The Coriolis force in the magnetohydrodynamic equations for a compressible astrophysical plasma is considered in four different approximations: a traditional f-plane, a nontraditional f-plane (given the horizontal component of the Coriolis force), a traditional β-plane, and a nontraditional β-plane. Linear and nonlinear theories of wave processes have been developed for each Coriolis force approximation under consideration. New types of waves have been found, with the rotation, magnetic field, gravity, and compressibility serving as their restoring mechanisms. The compressibility effects are represented in the new dispersion equations by the Brunt–Vaisala frequency for compressible stratified flows dependent on both initial density and pressure profiles. All of the realizable types of three-wave interactions have been revealed through a qualitative analysis of the dispersion curves. A system of equations for the amplitudes of interacting waves and the growth rates of parametric instabilities have been derived by the multiscale expansion method.

The Schamel-Ostrovsky equation in nonlinear wave dynamics of cylindrical shells

On the basis of the smeared stiffener theory, the evolution of nonlinear axisymmetric longitudinal waves in a nonlinear elastic cylindrical shell reinforced by internal stringers is investigated. A non-classical nonlinear physical law is adopted for the shell material, which is characterized by a fractional degree of strain intensity. An analysis of the influence of an external elastic medium on a nonlinear wave process is carried out on the basis of the Winkler and shear-lag model. Using the asymptotic method of multiscale expansions, the Schamel-Ostrovsky quasi-hyperbolic equation for the component of longitudinal displacement was first derived. The Painlevé analysis of the equation showed the impossibility of exact solitary-wave solutions. The dispersion relation is analyzed, the maximum value of the phase velocity of infinitesimal disturbances is determined, above which stationary nonlinear waves can propagate. Based on the finite-difference approach and the Petviashvili method, a numerical simulation of the derived equation is carried out, profiles of stably propagating wave packets and solitary waves are constructed. A special dispersionless case, impossible for an unstiffened shell, has been identified, in which there are exact compactons, peakons and periodic solutions.

Wave Processes in Three-Dimensional Stratified Flows of a Rotating Plasma in the Boussinesq Approximation

Magnetohydrodynamic (MHD) waves in a stratified rotating plasma in a gravitational field are investigated in the Boussinesq approximation. A theory of flows on an f-plane, on a nontraditional f-plane (with regard to the horizontal component of the Coriolis force), on a β-plane, and on a nontraditional β‑plane is developed. In each case, linear solutions to systems of three-dimensional MHD equations in the Boussinesq approximation are obtained that describe magnetic gravito-inertial waves, magnetostrophic waves, and magnetic Rossby waves. All existing types of three-wave interactions are found with the use of dispersion equations. In the case of magnetic Rossby waves in the β-plane approximation, it is shown that the low-frequency mode of a magnetic Rossby wave in the Boussinesq approximation is equivalent to that in the shallow-water MHD approximation. By the multiscale expansion method, a system of amplitude equations for interacting waves and the increments of two types of instability occurring in the system, decay and amplification, are obtained. For each type of three-wave interactions, it is shown that there is a difference in the coefficients and differential operators in the system of three-wave interactions.

Solitary and periodic waves in collisionless plasmas: The Adlam-Allen model revisited.

We consider the Adlam-Allen (AA) system of partial differential equations, which, arguably, is the first model that was introduced to describe solitary waves in the context of propagation of hydrodynamic disturbances in collisionless plasmas. Here, we identify the solitary waves of the model by implementing a dynamical systems approach. The latter suggests that the model also possesses periodic wave solutions-which reduce to the solitary wave in the limiting case of an infinite period-as well as rational solutions that are obtained herein. In addition, employing a long-wave approximation via a relevant multiscale expansion method, we establish the asymptotic reduction of the AA system to the Korteweg-de Vries equation. Such a reduction is not only another justification for the above solitary wave dynamics, but may also offer additional insights for the emergence of other possible plasma waves. Direct numerical simulations are performed for the study of multiple solitary waves and their pairwise interactions. The stability of solitary waves is discussed in terms of potentially relevant criteria, while the robustness of spatially periodic wave solutions is touched upon via numerical experiments.

A novel second-order reduced homogenization approach for nonlinear thermo-mechanical problems of axisymmetric structures with periodic micro-configurations

A novel second-order reduced homogenization (SORH) approach is introduced for analyzing dynamic thermo-mechanical coupling problems in axisymmetric inelastic structures with periodic micro-configuration. The axisymmetric heterogeneous structures are periodically distributed in radial and axial directions and homogeneous distribution in circumferential directions. Firstly, the nonlinear coupled thermo-mechanical model is proposed, and the high-order nonlinear local problems, effective material parameters and the nonlinear homogenization equations are derived successively by the multiscale asymptotic expansion. Further, in order to reduce the large computational amount evaluated by the classical multiscale homogenization approach, the reduced-order nonlinear multiscale models and the corresponding finite-element algorithms are established in detail. The key features of the proposed approach are that an efficient reduced-model form based on transformation field analysis (TFA) to analyze nonlinear local cell problems is proposed and a nonlinear thermo-mechanical problem which considers the mutual coupling for the temperature and displacement fields is computed. In particular, a new SORH algorithm is proposed for investigating the axisymmetric inelastic structures. Finally, three typical numerical experiments are carried out, and the effectiveness and correctness of our presented algorithms in simulating and predicting the macroscopic behavior of the heterogeneous structures are confirmed.

Soliton pairs in two-dimensional nonlocal media.

We study the interaction of optical beams of different wavelengths, described by a two-component, two-dimensional (2D) nonlocal nonlinear Schrödinger (NLS) model. Using a multiscale expansion method the NLS model is asymptotically reduced to the completely integrable 2D Mel'nikov system, the soliton solutions of which are used to construct approximate dark-bright and antidark-bright soliton solutions of the original NLS model; the latter being unique to the nonlocal NLS system with no relevant counterparts in the local case. Direct numerical simulations show that, for sufficiently small amplitudes, both these types of soliton stripes do exist and propagate undistorted, in excellent agreement with the analytical predictions. Larger amplitude of these soliton stripes, when perturbed along the transverse direction, disintegrate either to filled vortex structures (the dark-bright solitons) or to radiation (the antidark-bright solitons).

Rectangular lattice Boltzmann method using multiple relaxation time collision operator in two and three dimensions

We present a lattice Boltzmann (lb) method using a rectangular, non-isotropic lattice based on d2q9 and d3q27 velocity sets in two and three dimensions. A second order multi-scale expansion ensures that the scheme correctly reproduces hydrodynamic behaviour. A novel set of basis vectors is introduced in order to allow independent adjustment of eigenvalues corresponding to second order moments as required in order to ensure correct hydrodynamic behaviour using the non-isotropic lattice. Errors are reduced compared to other rectangular grid implementations. Linear perturbation analysis indicates that our scheme has similar stability properties to the isotropic lb method. We investigate the error behaviour of our scheme by performing Taylor-Green vortex flow simulations and comparing our results to simulations using a square grid and also to analytical results. We demonstrate that our scheme is well suited to direct numerical simulation of wall bounded turbulent flows and compare to well known benchmark results.

Solitary waves and double layers in an inhomogeneous electronegative plasma with heavier negative ions

The linear dispersion relation and partial differential equations are derived for the electronegative plasma with heavier negative ions embedded in an ambient magnetic field. The appropriate transformation is used to study the nonlinear structures without introducing multi-scale expansion of various quantities. The nonlinear partial differential equation obtained sustains nonlinear structures like solitons and double layers. All the linear waves and the nonlinear structures like solitons and double layers are found to be modified predominantly under the influence of superthermal particles, density inhomogeneity, electron populations, and positive to negative heavy ion ratios.

An assessment of multiscale asymptotic expansion method for linear static problems of periodic composite structures

The decoupled multiscale asymptotic expansion method (MsAEM) is applicable to the analysis of composite structures with periodic microstructures. When using the MsAEM to solve linear static problems, unit-cell influence functions must be solved from the microscale (or unit cell) problem with microscopic periodic boundary conditions, and homogenized displacements and their derivatives must be solved from the macroscale (or homogenized) problem with macrostructural boundary conditions. This paper compares the influence functions solved by using the normalization condition, the Dirichlet homogeneous boundary condition, and the oversampling technique or the super unit cell approach. It is concluded that the oversampling technique is a better way of dealing with periodic boundary conditions compared to the normalization condition. Another contribution of this work is to propose a combination method in which the MsAEM is used inside the structure and the multiscale eigenelement method (MEM) is used dealing with the boundary layer problem, and it is shown that this combination method can achieve accurate microscopic results for structures with any boundary conditions. In addition, the paper reveals the effects of different order expansion terms and gives some practical suggestions on how to determine the expansion order. Numerical comparisons validate the present combination method and all conclusions.

Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid

We consider the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid with Navier slip-with-friction conditions at the solid boundary. The fluid-solid system occupies the whole plane. We prove the small-time global exact controllability of the position and velocity of the solid when the control takes the form of a distributed force supported in a compact subset (with nonvoid interior) of the fluid domain, away from the body. The strategy relies on the introduction of a small parameter: we consider fast and strong amplitude controls for which the “Navier-Stokes+rigid body” system behaves like a perturbation of the “Euler+rigid body” system. By the means of a multi-scale asymptotic expansion we construct a controlled solution to the “Navier-Stokes+rigid body” system thanks to some controlled solutions to “Euler+rigid body”-type systems and by using that the influence of the boundary layer on the solid motion turns out to be sufficiently small.

Direct and discretized perturbations revisited: A new error source interpretation, with application to moving boundary problem

It is well-known that qualitative and/or quantitative differences will be possibly induced, when applying direct (i.e., attacking directly the partial differential equations) or discretized (i.e., using a low-order Galerkin truncated model) perturbation methods to nonlinear structures using multi-scale expansions, especially for spatially continuous structures dominated by a quadratic nonlinearity (structures with initial curvature, i.e., buckled beam, shallow arch, sagged cable, etc). In this paper, a new mechanism associated with structure's non-zero (non-homogeneous) boundary motion and cubic nonlinearity, rather than quadratic one, will be shown to cause also differences if applying the two versions of multiple time scale method to the same nonlinear system. Inspired by the two different nonlinear sources above, i.e., quadratic and cubic, a fundamental interpretation of differences/errors between the two perturbation formulations is proposed, with the above two kinds of nonlinearity being included. Explicitly, the full-spectrum forced solution at lower-order and the following high-order cross-interactions with the structural modes are captured by the direct perturbation, but not by the discretized perturbation, if a finite mode truncation is used. Also two different correction schemes are proposed for eliminating the errors above, i.e., full basis Galerkin discretization and a rational drift function using structural Green's function, respectively.

On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot of attention in numerical analysis. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. To overcome this numerical burden various novel numerical methods with excellent efficiency were derived in recent years. The construction of each method thereby requests essentially the averaged model of the problem. However, the averaged model of each approach is found by different kinds of asymptotic approximation techniques reaching from the modulated Fourier expansion over the multiscale expansion by frequency up to the Chapman-Enskog expansion. In this work we give a first comparison of these recently introduced asymptotic series, reviewing their approximation validity to the KG in the asymptotic limit, their smoothness assumptions as well as their geometric properties, e.g., energy conservation and long-time behaviour of the remainder.

Numerical Treatment of the Nonconservative Product in a Multiscale Fluid Model for Plasmas in Thermal Nonequilibrium: Application to Solar Physics

This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model in order to properly investigate the influence on shock solutions of a nonconservative product present in the electron energy equation. Then, we derive jump conditions based on travelling wave solutions and propose an original numerical treatment in order to avoid non-physical shocks for the solution, that remains valid in the case of coarse-resolution simulations. A key element for the numerical scheme proposed is the presence of diffusion in the electron variables, consistent with the physically-sound scaling used in the model developed by Graille et al. following a multiscale Chapman-Enskog expansion method [M3AS, 19 (2009) 527--599]. The numerical strategy is eventually assessed in the framework of a solar physics test case. The computational method is able to capture the travelling wave solutions in both the highly- and coarsely-resolved cases.

Nonlinear Dynamo in Obliquely Rotating Stratified Electroconductive Fluid in an Uniformly Magnetic Field

In this paper, we investigated a new large-scale instability that arises in an obliquely rotating convective electrically conducting fluid in an external uniform magnetic field with a small-scale external force with zero helicity. This force excites small-scale velocity oscillations with a small Reynolds number. Using the method of multiscale asymptotic expansions, we obtain the nonlinear equations for vortex and magnetic disturbances in the third order of the Reynolds number. It is shown that the combined effects of the Coriolis force and the small external forces in a rotating conducting fluid possible large-scale instability. The linear stage of the magneto-vortex dynamo arising as a result of instabilities of -effect type is investigated. The mechanism of amplification of large-scale vortex disturbances due to the development of the hydrodynamic - effect taking into account the temperature stratification of the medium is studied. It was shown that a «weak» external magnetic field contributes to the generation of large-scale vortex and magnetic perturbations, while a «strong» external magnetic field suppresses the generation of magnetic-vortex perturbations. Numerical methods have been used to find stationary solutions of the equations of a nonlinear magneto-vortex dynamo in the form of localized chaotic structures in two cases when there is no external uniform magnetic field and when it is present.

Weakly Nonlinear Theory for Oscillatory Dynamics in a One-Dimensional PDE-ODE Model of Membrane Dynamics Coupled by a Bulk Diffusion Field

We study the dynamics of systems consisting of two spatially segregated ODE compartments coupled through a one-dimensional bulk diffusion field. For this coupled PDE-ODE system, we first employ a multi-scale asymptotic expansion to derive amplitude equations near codimension-one Hopf bifurcation points for both in-phase and anti-phase synchronization modes. The resulting normal form equations pertain to any vector nonlinearity restricted to the ODE compartments. In our first example, we apply our weakly nonlinear theory to a coupled PDE-ODE system with Sel'kov membrane kinetics, and show that the symmetric steady state undergoes supercritical Hopf bifurcations as the coupling strength and the diffusivity vary. We then consider the PDE diffusive coupling of two Lorenz oscillators. It is shown that this coupling mechanism can have a stabilizing effect, characterized by a significant increase in the Rayleigh number required for a Hopf bifurcation. Within the chaotic regime, we can distinguish between synchronous chaos, where both the left and right oscillators are in-phase, and chaotic states characterized by the absence of synchrony. Finally, we compute the largest Lyapunov exponent associated with a linearization around the synchronous manifold that only considers odd perturbations. This allows us to predict the transition to synchronous chaos as the coupling strength and the diffusivity increase.

A nonlocal variable coefficient KdV equation: Bäcklund transformation and nonlinear waves

A nonlocal two-layer fluid model is constructed through a simple symmetry reduction from the local one. Then, a general variable coefficients nonlocal KdV (VCKdV) equation with shifted space parity and delayed time reversal is derived from it by using multi-scale expansion method, with and without the so-called $$y-$$average method, respectively. A non-auto-Backlund transformation between the VCKdV equation and a constant coefficients KdV (CCKdV) equation is established. By using this transformation, various exact solutions of the VCKdV equation can be obtained from the seed solutions of the CCKdV equation. As some concrete examples, one solitary wave solution and two kinds of periodic wave solutions are given. Due to the inclusion of arbitrary functions in these solutions, they possess abundant dynamical behaviors with some of them analyzed graphically.