Wave Number Of Perturbation Research Articles

Thin liquid film stability in the presence of bottom topography and surfactant

We consider gravity-driven fluid flow down a wavy inclined surface in the presence of surfactant. The periodicity of the bottom topography allows us to leverage Floquet theory to determine the form of the solution to the linearized governing partial differential equations. The result is that perturbations from steady state are wavelike, and a dispersion relation is identified which relates the wavenumber of an initial perturbation, κ, to its complex frequency, ω. The real part of ω provides a criterion for determining linear flow stability. We observe that the addition of surfactant generally has a stabilizing effect on the flow, but has a destabilizing effect for small wavenumbers. These results are compared and validated against numerical simulations of the nonlinear system. The linear and nonlinear analyses show good agreement, except at small wavenumbers, where the linear results could not be replicated.

Effects of mass diffusion on Rayleigh–Taylor instability under a large gravity

Rayleigh–Taylor instabilities (RTI) play an important role in the evolution of inertial confinement fusion (ICF) processes, while analytical prediction of the RTI growth rate often fails to reach an agreement with the experimental and simulation results. Accurate analytical prediction of RTI growth is of great significance to the success of ICF schemes. In this paper, we study the effects of mass diffusion and exponential density distribution on RTI under a large gravity by solving the Rayleigh equation with a linear approximation to the density distribution of the mixing layer. The width of the mixing layer is assigned by evaluating the length scale of concentration diffusion and gravitational sedimentation. The latter term is missing in the former isobaric diffusion treatment and is supposed to change the structure of the mixing layer under the gravity. While both effects tend to dampen the instability growth, mass diffusion dominates the damping of perturbations of larger wavenumber and exponential density distribution dominates those of smaller wavenumber, resulting in a non-monotonicity of the density suppression factor of the instability growth rate over perturbation wavenumbers.

Fluctuation-driven dynamics of liquid nano-threads with external hydrodynamic perturbations

Instability and rupture dynamics of a liquid nano-thread, subjected to external hydrodynamic perturbations, are captured by a stochastic lubrication equation (SLE) incorporating thermal fluctuations via Gaussian white noise. Linear instability analysis of the SLE is conducted to derive the spectra and distribution functions of thermal capillary waves influenced by external perturbations and thermal fluctuations. The SLE is also solved numerically using a second-order finite difference method with a correlated noise model. Both theoretical and numerical solutions, validated through molecular dynamics, indicate that surface tension forces due to specific external perturbations overcome the random effects of thermal fluctuations, determining both the thermal capillary waves and the evolution of perturbation growth. The results also show two distinct regimes: (i) the hydrodynamic regime, where external perturbations dominate, leading to uniform ruptures, and (ii) the thermal-fluctuation regime, where external perturbations are surpassed by thermal fluctuations, resulting in non-uniform ruptures. The transition between these regimes, modelled by a criterion developed from linear instability theory, exhibits a strong dependence on the amplitudes and wavenumbers of the external perturbations.

Modulational instability, modulated wave, and optical solitons for a generalized highly dispersive cubic-quintic-septic-nonic medium with self-frequency shift and self-steepening nonlinear terms

In this research, we delve into a generalized highly dispersive (HD) nonlinear Schrödinger equation, enriched with cubic-quintic-septic-nonic (CQSN) nonlinearities. The core of our investigation revolves around the perturbation of plane waves, aiming to understand their stability characteristics in such a complex medium. We investigate the influence of various factors such as the amplitude of the plane wave, perturbed wave number, nonic nonlinear term, and fourth-order dispersion term. Our findings indicate that increasing the amplitude of the plane wave widens the modulation instability (MI) bands and amplifies the MI growth rate. In contrast, increasing the nonic nonlinear term has opposing effects, narrowing the MI bands and diminishing the amplitude of the MI growth rate. Increasing the fourth-order dispersion term does not affect the amplitude of the MI growth rate but narrows the MI bands. The observed pattern of increasing and then decreasing MI intensity with rising K can be attributed to the complex interplay among phase matching conditions, dispersion effects, and nonlinear saturation. Initially, higher K enhances phase matching and boosts MI growth. However, as K increases further, the combined influence of dispersion and nonlinear effects can diminish the effectiveness of phase matching, resulting in a reduction in MI intensity. A significant portion of our work is dedicated to identifying and analyzing modulated rational, polynomial Jacobi elliptic function solutions, and the emergence of optical solitons within this framework. These findings provide new insights into the nonlinear dynamics underpinning the generalized HDNLSE, enriched with CQSN nonlinearities, offering valuable contributions to the theoretical understanding of such phenomena.

Weakly nonlocal matter-wave droplets and soliton trains engineering in a Bose-Einstein condensate

We propose a comprehensive analysis of the modulational instability (MI) phenomenon in one-dimensional Bose-Einstein condensate (BEC) in the context of mutually symmetric components in the binary mixture. The proposed model considers weak nonlocal cubic mean-field and quadratic beyond mean-field interactions arising from quantum fluctuations. The instability regions are analyzed using linear stability analysis of a continuous wave, considering maximum perturbation wavenumber and growth rate. The regions for quantum droplets (QDs) and solitons are identified by varying the nonlocality parameter. Numerical simulations confirm analytical predictions, revealing that nonlocality influences the formation of various QD profiles and instability time. Furthermore, solitonic patterns exhibit nonlocality-dependent behavior. Our study opens new doors to a better characterization of QDs in binary mixtures in the presence of higher-order beyond mean-field effects.

Rogue and solitary waves of a system of coupled nonlinear Schrödinger equations in a left-handed transmission line with second-neighbors coupling

The effects of the second-order neighbor interaction have been effective on the coherent localized waves and modulation instability spectrum. A coupled nonlinear Schrödinger equation is derived in the nonlinear left-handed electrical lattice, and the linear stability is used to formulate the modulation instability spectrum expression. Unstable modes are displayed to show symmetric lobes and increasing bandwidths under the influence of the neighbor coupling. The Benjamin-Feir instability has prospered in the network, and for weak perturbed wave numbers, the unstable modes increase. The numerical simulation is used to develop localized waves, including rogue waves with one crest and two humps, Akhmediev breathers, and other modulated structures. We notice that, for strong values of the perturbed wave number, an exponential growth of the continuous wave arises to confirm the fact that the nonlinear left-handed electrical lattice with coupling strength is opened to coherent localized waves. The long-lived nature of the obtained structures has also been demonstrated for specific times of propagation.

Stability analysis of viscous multi-layer shear flows with interfacial slip

Abstract One of the most fundamental interfacial instabilities in ideal, immiscible, incompressible multifluid flows is the celebrated Kelvin–Helmholtz (KH) instability. It predicts short-wave instabilities that, in the absence of other mollifying physical mechanisms (e.g. surface tension, viscosity), render the nonlinear problem ill-posed and lead to finite-time singularities. The crucial driving mechanism is the jump in tangential velocity across the liquid–liquid interface, i.e. interfacial slip, that can occur since viscosity is absent. The purpose of the present work is to analyse analogous instabilities for viscous flows at small or moderate Reynolds numbers as opposed to the infinite Reynolds numbers that underpin KH instabilities. The problem is physically motivated by both experiments and simulations. The fundamental model considered consists of two superposed viscous, incompressible, immiscible fluid layers sheared in a plane Couette flow configuration, with slip present at the deforming liquid–liquid interface. The origin of slip in viscous flows has been observed in experiments and molecular dynamics simulations, and can be modelled by employing a Navier-slip boundary condition at the liquid–liquid interface. The emerging novel instabilities are studied in detail here. The linear stability of the system is addressed asymptotically for long- and short-waves, and for arbitrary wavenumbers using a combination of analytical and numerical calculations. Slip is found to be capable of destabilising perturbations of all wavelengths. In regimes where the flow is stable to perturbations of all wavelengths in the absence of slip, its presence can induce a Turing-type instability by destabilization of a small band of finite wavenumber perturbations. In the case where the underlying layer is asymptotically thin, the results are found to agree with the linear properties of a weakly non-linear asymptotic model that is also derived here. The weakly nonlinear model extends previous work by the authors that had a thin overlying layer that produces a different evolution equation.

Stability of low-pressure turbine boundary layers under variable Reynolds number and pressure gradient

The free-stream turbulence induced transition occurring under typical low-pressure turbine flow conditions is investigated by comparing linear stability theory with wind tunnel measurements acquired over a flat plate subjected to high turbulence intensity. The analysis was carried out, accounting for three different Reynolds numbers and four different adverse pressure gradients. First, a non-similarity-based boundary layer (BL) solver was used to compute base flows and validated against pressure taps and particle image velocimetry (PIV) measurements. Successively, the optimal disturbances and their spatial transient growth were calculated by coupling classical linear stability theory and a direct-adjoint optimization procedure on all flow conditions considered. Linear stability results were compared with experimental particle image velocimetry measurements on both wall-normal and wall-parallel planes. Finally, the sensitivity of the disturbance spatial transient growth to the spanwise wavenumber of perturbations, the receptivity position, and the location where disturbance energy is maximized were investigated via the built numerical model. Overall, the optimal perturbations computed by linear stability theory show good agreement with the streaky structures surveyed in experiments. Interestingly, the energy growth of disturbances was found to be maximum for all the flow conditions examined, when perturbations entered the boundary layer close to the position where minimum pressure occurs.

Non-modal stability analysis of magneto-hydrodynamic flow in a single pipe

A complete understanding of the stability of fluid flows under varying magnetic field profiles is imperative for achieving control of plasma and operating fluids in the blankets of future fusion reactors. In this context, the primary objective of this study is to investigate the influence of varying magnetic profiles on the flow regime of a generic fluid, which is representative of both thermonuclear plasma and conductive fluids within a nuclear fusion reactor. To this aim in this work non-modal stability theory is adopted to perform stability analysis of a magneto-hydrodynamic (MHD) flow in an infinite circular pipe in order to study the effects of the magnetic field on the fluid dynamics of the pipe flow. In particular, the effects on the general stability of two magnetic field profiles are compared with the reference case of a pipe Poiseuille flow without magnetic field. Firstly, the classic modal stability technique is employed to study asymptotical stability. Then, non-modal stability analysis is applied to magneto-hydrodynamic pipe flow to study the system's response for a finite time immediately after a perturbation. Fourier–Chebyshev Petrov–Galerkin spectral method is used to compute the eigenvalues and pseudospectra of the weak formulation associated with the linearised system. Investigations on the dependence of spectra and transient growths on the specific magnetic profiles are conducted for different values of perturbation wave numbers. The obtained results show that in general the magnetic field has an effect of stabilization on the system, which depends on the specific magnetic profile considered. In addition, the non-modal stability analysis reveals that the inclusion of the magnetic field mitigates the effects of perturbations also in the short term, a phenomenon that cannot be seen using only modal stability analysis.

Thermally Driven Convection Generated by Reaction Fronts in Viscous Fluids

Reaction fronts propagating in liquids separate reacted from unreacted fluid. These reactions may release heat, increasing the temperature of the propagating medium. As fronts propagate, they will induce density changes leading to convection. Exothermic fronts that propagate upward increase the temperature of the reacted fluid located underneath the front. For positive expansion coefficients, the warmer fluid will tend to rise due to buoyancy. In the opposite case, for fronts propagating downward with the warmer fluid on top, an unexpected thermally driven instability can also take place. In this work, we carry out a linear stability analysis introducing perturbations of fixed wavelength. We obtain a dispersion relation between the perturbation wave number and its growth rate. For either direction of propagation, we find that the front is stable for very short wavelengths, but is unstable for large enough wavelengths. We carry out a numerical solution of a cubic reaction–diffusion–advection equation coupled to Navier–Stokes hydrodynamics in a two-dimensional rectangular domain. We find transitions between the non-axisymmetric and axisymmetric fronts increasing with the width of the domain.

Influence of parametric forcing on Marangoni instability

We study a thin, laterally confined heated liquid layer subjected to mechanical parametric forcing without gravity. In the absence of parametric forcing, the liquid layer exhibits the Marangoni instability, provided the temperature difference across the layer exceeds a threshold. This threshold varies with the perturbation wavenumber, according to a curve with two minima, which correspond to long- and short-wave instability modes. The most unstable mode depends on the lateral confinement of the liquid layer. In wide containers, the long-wave mode is typically observed, and this can lead to the formation of dry spots. We focus on this mode, as the short-wave mode is found to be unaffected by parametric forcing. We use linear stability analysis and nonlinear computations based on a reduced-order model to investigate how parametric forcing can prevent the formation of dry spots. At low forcing frequencies, the liquid film can be rendered linearly stable within a finite range of forcing amplitudes, which decreases with increasing frequency and ultimately disappears at a cutoff frequency. Outside this range, the flow becomes unstable to either the Marangoni instability (for small amplitudes) or the Faraday instability (for large amplitudes). At high frequencies, beyond the cutoff frequency, linear stabilization through parametric forcing is not possible. However, a nonlinear saturation mechanism, occurring at forcing amplitudes below the Faraday instability threshold, can greatly reduce the film surface deformation and therefore prevent dry spots. Although dry spots can also be avoided at larger forcing amplitudes, this comes at the expense of generating large-amplitude Faraday waves.

Solitonic rogue and modulated wave patterns in the monoatomic chain with anharmonic potential

Modulation instability and rogue wave structures have been investigated in this work. This study is an extension of the work in Abbagari (2023), where a nonlinear Schrödinger equation with higher-order dispersion is derived to show only the development of the modulated waves bounded to bright soliton as a nonlinear exhibition of modulation instability. Here, the coupled nonlinear Schrödinger equation is derived by using the multi-scale scheme. An overview of the analytical calculations of the perturbed plane wave is carried out to show the effects of the nonlinear chain parameters on the modulation instability growth rate and bandwidths. The interest of this study lies equally in the nonlinear modes of excitation, where solitonic waves are generated under certain conditions in lower and upper frequency bands. On the other hand, relevant results have been developed to show the features of the type I and type II rogue waves of the Manakov system. Such investigations are obtained under the variation of the interaction potential parameters and the free parameter of the similarity method. Via a numerical simulation, rogue wave structures have been generated as a consequence of the long-time evolution of the perturbed plane wave. At a specific time of propagation, another localized object has been obtained to show the Akhmediev breathers and Kuznetsov-Ma solitons clusters under a strong perturbed wave number. These results have opened up new features, and many applications could follow in the future.

Solitonic rogue waves induced by the modulation instability in a split-ring-resonator-based left-handed coplanar waveguide

Modulation instability and nonlinear excitation modes in lower and upper frequency bands are investigated in the nonlinear left-handed coplanar waveguide, where nonlinear capacitance (voltage-dependent) is used. The multi-scale method is employed to derive a system of two coupled nonlinear Schödinger equations governing the evolution of the kink and bright solitons of these modes. A modulation instability is investigated to show the effects of the left-handed and right-handed influences on the modulation instability gain and the intensity of the plane wave. In the lower and upper frequency modes, the breathing structures have strong values for the left-handed component. At the end, the numerical simulation of the continuous wave is used to show the development of the modulated waves and localized objects. Another relevant localized structure has been obtained to display the second-order breathers and Akhmediev breathers of types A and B under the variation of the perturbed wave number. Both linear inductances and perturbed wave numbers are tools for controlling the amplitude of the localized structures, and modulation instability is sensitive to these parameters.

High-amplitude effect on Richtmyer–Meshkov instability at a single-mode heavy–light interface

An experimental study is conducted to explore the high-amplitude effect on Richtmyer–Meshkov instability (RMI) at a single-mode heavy–light interface. A wide range of scaled initial amplitude (ka0, where k and a0 are perturbation wavenumber and initial amplitude, respectively) is considered. Qualitatively, nonstandard (standard) indirect phase inversion occurs in experiments with high (low and moderate) ka0. The nonstandard indirect phase inversion exhibits a complex process, and the interface mixing width does not reduce to near zero. Quantitatively, the linear model poorly (accurately) predicts the post-phase-inversion linear amplitude growth rate when ka0 is high (low and moderate). Additionally, a representative theoretical reduction factor fortuitously evaluates the high-amplitude effect on the post-phase-inversion linear amplitude growth rate well. The high-amplitude effect significantly alters the nonlinear evolution law, which differs from the case of RMI at a light–heavy interface. None of the considered nonlinear models can accurately predict the amplitude evolution under all ka0 conditions, regardless of whether their expressions are related to ka0 or not. Based on the current experimental results, an empirical nonlinear model is proposed to describe RMI at a single-mode heavy–light interface across a wide range of ka0 conditions.

Current modified higher-order Schrödinger equation of broader bandwidth capillary-gravity waves

A higher-order nonlinear Schrödinger equation of capillary-gravity waves for broader bandwidth on infinite depth of water including the effect of depth uniform current is established. The derivation is made from Zakharov's integral equation by extending the narrow bandwidth restriction to make it more suited for application to problems involving actual sea waves. On the basis of this equation, a stability analysis is made for uniform Stokes waves. After obtaining an instability condition, instability regions in the perturbed wave number space are displayed that are in good agreement with the exact numerical findings. It is found that the modifications in the stability characteristics at the fourth-order term are due to the interaction between the frequency-dispersion term and the mean flow. It is seen that the growth rate of sideband instability decreases due to the effects of both surface tension and depth uniform following currents. Significant deviations of the instability regions are observed between narrow-banded and broader-banded results. In addition, we have depicted the instability growth rate for the case of pure capillary waves.

Effects of ablation velocity on ablative Rayleigh–Taylor instability

The influence of the ablation velocity Va on the evolution of single-mode ablative Rayleigh–Taylor instability from the linear to the deeply nonlinear phases is investigated via two-dimensional numerical simulations. Linear growth rates from simulations agree well with the asymptotic theory except for larger discrepancies in the intermediate Froude number regime. The weakly nonlinear growth behavior of the bubble amplitude is found dependent on a critical perturbation wavenumber in a broad Froude number regime. For a linearly stable mode, its nonlinear excitation threshold is higher for larger Va and thus harder to be exceeded. For short-wavelength modes taking significant ablation effects, the bubble penetration velocity is found to reaccelerate after the first saturation and eventually saturate at a larger value with larger Va, due to stronger vortex-acceleration effects and more significant increase in g.

Dynamical behaviour of solitons and modulation instability analysis of a nonautonomous [formula omitted]-dimensional nonlinear Schrödinger equation

This study uses a semi-inverse variational technique to investigate the unknown dynamical behaviour of optical solitons in the solutions of a nonautonomous (1+1)-dimensional nonlinear Schrödinger (NLS) equation. This method produces the bright, anti-bright, and bell-shaped solitons that depend on the delicate balance among the time-dependent spatiotemporal dispersion (STD), group velocity dispersion (GVD), and self-phase modulation (SPM). It is obtained that the soliton solution exists only for non-zero SPM, although the soliton velocity and frequency do not depend on SPM. The soliton amplitude and intensity depends explicitly on time-dependent dispersion and self-phase modulation of the medium. It is found that the focusing medium has a remarkable impact on the intensity of the soliton in comparison to the defocusing, linear, and quadratic nonlinearity. The modulation instability (MI) analysis of the NLS equation has also been investigated using standard linear stability analysis. The explicit dispersion relation has been derived, and MI gain is discussed using different STD, GVD, and SPM coefficients as test functions of time t. The MI gain varies rapidly with the perturbation wave number at a fixed time. The 3D and contour plot of MI gain show that the stability of the NLS equation can be managed significantly by using the time-dependent spatiotemporal, group velocity dispersion, self-phase modulation, wave number, and initial incidence power.

The impact of convection on morphological instability of a planar crystallization front

A linear morphological stability analysis of a planar solid-liquid phase interface describing the solidification processes of a binary melt with convection is carried out. The developed theory includes conductive and convective heat and mass transfer mechanisms near the phase interface and generalizes previously known theories of morphological stability. The amplification rate as a function of wavenumber of perturbations and neutral stability curve that divides the stability/instability parametric domains are obtained. It is shown that these domains are highly dependent of convection intensity, which represents a stabilizing factor for solidification processes. A criterion of concentration supercooling in the steady-state solidification conditions with convection is found. The obtained dispersion relation and neutral stability curve define various crystallization scenarios such as (i) morphological instability and concentration supercooling appearing to the formation of a two-phase mushy layer, (ii) morphological stability and concentration supercooling leading to the existence of a slurry layer, (iii) morphological stability without concentration supercooling when the planar solidification front is stable, and (iv) morphological instability without concentration supercooling forming the mesoscopically rough phase interface.

Modulational electrostatic wave–wave interactions in plasma fluids modeled by asymmetric coupled nonlinear Schrödinger (CNLS) equations

The interaction between two co-propagating electrostatic wavepackets characterized by arbitrary carrier wavenumber is considered. A one-dimensional (1D) non-magnetized plasma model is adopted, consisting of a cold inertial ion fluid evolving against a thermalized (Maxwell–Boltzmann distributed) electron background. A multiple-scale perturbation method is employed to reduce the original model equations to a pair of coupled nonlinear Schrödinger (CNLS) equations governing the dynamics of the wavepacket amplitudes (envelopes). The CNLS equations are in general asymmetric for arbitrary carrier wavenumbers. Similar CNLS systems have been derived in the past in various physical contexts, and were found to support soliton, breather, and rogue wave solutions, among others. A detailed stability analysis reveals that modulational instability (MI) is possible in a wide range of values in the parameter space. The instability window and the corresponding growth rate are determined, considering different case studies, and their dependence on the carrier and the perturbation wavenumber is investigated from first principles. Wave–wave coupling is shown to favor MI occurrence by extending its range of occurrence and by enhancing its growth rate. Our findings generalize previously known results usually associated with symmetric NLS equations in nonlinear optics, though taking into account the difference between the different envelope wavenumbers and thus group velocities.

Electrohydrodynamic capillary instability of Rivlin–Ericksen viscoelastic fluid film with mass and heat transfer

AbstractThis paper presents an analysis of viscous fluids and a Rivlin–Ericksen (R‐E) viscoelastic fluid interface under the influence of heat and mass transfer, while both fluids are exposed to an axial electric field. The fluids are restricted within an annular region that is enclosed by two rigid cylinders. The outer section of the annular region holds the R‐E viscoelastic fluid, while the inner section is filled with the viscous fluid. To ascertain the correlation between perturbation growth and wavenumber, the theory of potential flow on viscoelastic–viscous fluids is applied, and the result is represented as a second‐order polynomial. This correlation is numerically solved using the Newton–Raphson method. Variables of viscous flow, such as electric field strength, heat transfer coefficient, viscoelasticity, viscosity, and so forth, are numerically studied. With an increase in electric field strength, the perturbation growth decays and expands for the particular combinations of permittivity and conductivity ratio, showing the dual effect of the axial electric field.