Linear Stability Analysis Research Articles

Emergence of a hexagonal pattern in shear-thickening suspensions under orbital oscillations

A dense particle suspension under shear may lose its uniform state to large local density and stress fluctuations, which challenge the mean-field description of the system. Here, we explore the novel dynamics of a non-Brownian suspension under orbital oscillations, where localized density waves along the flow direction appear beyond an excitation frequency threshold and self-organize into a hexagonal pattern across the system. The spontaneous occurrence of the inhomogeneity pattern arises from a coupling between particle advection and the shear-thickening nature of the suspension. Through linear stability analysis, we show that they overcome the stabilizing effects of particle pressure at sufficient particle volume fraction and oscillation frequency. In addition, the long-standing density waves degenerate into random fluctuations when replacing the free surface with rigid confinement. It indicates that the shear-thickened state is intrinsically heterogeneous, and the boundary conditions are crucial for developing local disturbance.

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.

Linear stability analysis of MHD mixed convection flow of a radiating nanofluid in porous channel in presence of viscous dissipation

Purpose This paper aims to investigate the hydrodynamic instability properties of a mixed convection flow of nanofluid in a porous channel. Design/methodology/approach The treated single-phase nanofluid is a suspension consisting of water as the working fluid and alumina as a nanoparticle. The anisotropy of the porous medium and the effects of the inclination of the magnetic field are highlighted. The effects of viscous dissipation and thermal radiation are incorporated into the energy equation. The eigenvalue equation system resulting from the stability analysis is processed numerically by the spectral collocation method. Findings Analysis of the results in terms of growth rate reveals that increasing the volume fraction of nanoparticles increases the critical Reynolds number. Parameters such as the mechanical anisotropy parameter and Richardson number have a destabilizing effect. The Hartmann number, permeability parameter, magnetic field inclination, Prandtl number, wave number and thermal radiation parameter showed a stabilizing effect. The Eckert number has a negligible effect on the growth rate of the disturbances. Originality/value Linear stability analysis of Magnetohydrodynamics (MHD) mixed convection flow of a radiating nanofluid in porous channel in presence of viscous dissipation.

Elimination of lock-in phenomenon in vortex-induced vibration by passive modal control

Theoretical analysis and numerical results have shown that frequency lock-in in vortex-induced vibration (VIV) is caused by the instability of the structural mode rather than a resonant response to external excitations. However, there is a lack of experimental evidence supporting relevant theoretical research findings. This study investigates VIV suppression with a passive modal controller (PMC) for a circular cylinder at Reynolds numbers $Re = 60$ and $Re = 40$ , using experiments to distinguish the effects of stable and unstable wake modes. Comparative analysis before and after the implementation of the PMC reveals significant reduction in the vibration amplitude and the disappearance of the lock-in phenomenon at $Re = 60$ . The vibration frequency closely follows the vortex shedding frequency after control, while dynamic mode decomposition of the flow field indicates that the wake mode is dominant. For $Re = 40$ , the vibration is eliminated and the flow becomes steady. Additionally, the root loci of the coupled system are investigated before and after the PMC implementation via linear stability analysis. The results indicate that the PMC can alter the dynamic characteristics of the original system, causing the structural mode and PMC mode to couple when approaching the PMC frequency. Then, the interaction typically improves the stability of the structural mode. Finally, a parametric study is conducted in the experiment, as well as a linear stability analysis. The study provides experimental evidence that stability control of the structural mode is the key to suppressing VIV and eliminating the lock-in phenomenon.

Influence of the Dufour effect on striations formation in radio-frequency discharges

In recent years, interest in striation phenomena in radio frequency (rf) discharges has risen due to the availability of new experimental data and the implementation of new computational models. Depending on the conditions, different mechanisms of discharge striations are realized. These are the ionization instability, the instability due to the electron attachment to electronegative gases, or the instability due to thermoelectric transport. Although the first two mechanisms were modeled quite extensively in recent years, the understanding of the influence of the Dufour effect originating from plasma density gradients on the stability of radio frequency discharges in long tubes remains poor. In this paper, the influence of this mechanism on the longitudinal striations of radio frequency discharge is presented using a one-dimensional model of argon discharge driven with rf excitation under intermediate pressure conditions of 0.5 Torr. It is found that striation formation is sensitive to the value of the thermoelectric heat transport coefficient in the low electron temperature range. The critical value of this coefficient necessary for the instability onset is derived using the linear stability analysis.

Car-following model based on spatial expectation effect in connected vehicle environment: modeling, stability analysis and identification

In driving situations, if drivers have advanced access to downstream traffic conditions ahead of time, they may modify their driving behavior accordingly. This proactive approach can help reduce frequent acceleration and deceleration of vehicles and minimize traffic congestion. This paper considers the average headway of downstream vehicles to predict the traffic conditions of downstream roads. Based on the full velocity difference (FVD) model, we propose a car-following model that incorporates spatial expectation effect (SEFVD). Firstly, a linear stability analysis is performed on the SEFVD model, and the results indicate that the critical value of the SEFVD model is relatively small, and the stabilization region increases gradually with the number of vehicles downstream. Secondly, employing perturbation amplitudes on a virtual circular road, a numerical simulation of the SEFVD model is carried out and compared with the FVD model. The results demonstrate that the SEFVD model has the ability to suppress the formation of traffic oscillations and stabilize vehicle formation more quickly. Simulation results for the 11 cars starting from a traffic signal indicate that the delay time and kinematic wave speed of the SEFVD model are within reasonable limits. To evaluate the changes in energy use, we also carried out energy consumption experiments. The experimental findings show that the SEFVD model uses less energy compared to the FVD model. Finally, the parameters of the models are calibrated and verified by using NGSIM data. The total calibration error of the SEFVD model (m = 2) decreases by 42.97% compared to the FVD model, and the total validation error decreases by 29.3%. The results show that the proposed model can fit the actual data well, and the fitting error is smaller.

Thin film flow inside a uniformly heated/cooled inclined rotating cylinder

The present study investigates the flow of a thin film inside a uniformly heated/cooled cylinder rotating at various inclinations. The governing equations for mass, momentum, and energy are simplified using lubrication approximation and asymptotic analysis. The resulting spatiotemporal equation for film height evolution incorporates dimensionless parameters representing gravity, viscous drag, surface tension, and thermocapillary stress. Two-dimensional, steady-state solutions are derived, revealing that under the dominance of gravity over viscous drag, a liquid pool forms at the bottom of the horizontally rotating cylinder. This pool transforms into a more uniformly distributed thin film as the cylinder is rotated at slopes. Three-dimensional solutions show a ring structure for horizontally rotating cylinder, whereas it disappears for a cylinder rotating at slopes. Introducing infinitesimal axial perturbations to the steady solutions allows for the examination of their stability. The study finds that the thin film flow becomes unstable for a sloped cylinder. The gravitational force stabilizes (destabilizes) the flow for horizontal (vertical) cylinders. It is reported that thermocapillary stress has a stabilizing effect for a uniformly cooled cylinder. Additionally, the outcomes obtained through linear stability analysis have been corroborated through nonlinear computations.

Stability of penetrative convective currents in local thermal non-equilibrium

The aim of this paper is to investigate the onset of penetrative convection in a Darcy–Brinkman porous medium under the hypothesis of local thermal non-equilibrium. For the problem at stake, the strong form of the principle of exchange of stabilities has been proved, i.e. convective motions can occur only through secondary stationary motions. We perform linear and nonlinear stability analyses of the basic state, with particular regard to the behaviour of stability thresholds with respect to the relevant physical parameters characterizing the problem. The Chebyshev- τ method and the shooting method are employed and accurately implemented to solve the differential eigenvalue problems arising from linear and nonlinear analyses to determine critical Rayleigh numbers. Via numerical simulations, the stabilizing effect of the upper bounding plane temperature, of the Darcy number and of the interaction coefficient for the heat exchange, is demonstrated.

The application of the "inverse problem" method for constructing confining potentials that make N-soliton waveforms exact solutions in the Gross-Pitaevskii equation.

In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.

Tertiary wake mode in flows past elliptic cylinders

Linear stability analysis of the steady flow past elliptic cylinders of different aspect ratio (Ar) has been conducted for the flow Reynolds number (Re) in the range 30–200. A new unstable mode, which we refer as tertiary wake mode, has been discovered. Two other unstable modes (the primary wake mode and the secondary wake mode), already reported in the literature, are also found. The critical Re for the onset of instability of these modes and the corresponding Strouhal number (St) have been reported. Modes that have large growth rates tend to come close to the cylinder surface, and the extent to which they spread downstream in the wake is less compared to the weaker modes. The size of the vortical structures in a mode is inversely related to its St. The change in the characteristics of these modes with respect to change in Ar and Re, as well as their evolution leading to the fully developed flow, has been studied. Selective suppression of the unstable modes is effected using a slip-plate placed on the wake centerline. For Re = 150, it is shown that selective suppression of the unstable modes leads to different fully developed unsteady flows.

Comment on “Modulational instability and soliton trains in optical fiber media with real and imaginary Raman gains”

In their recent article, Nfor et al. [Optik - International Journal for Light and Electron Optics 285 (2023) 170951, https://doi.org/10.1016/j.ijleo.2023.170951] have studied, through a dissipative nonlinear Schrödinger equation, the modulational instability and soliton trains in optical fiber media with real and imaginary Raman gains. After finding a steady-state solution of this equation, they have applied the linear stability analysis to investigate its modulational instability. It appears that (i) their proposed steady-state solution is not really a continuous wave solution of their model equation, and (ii) their ansatz used for the modulation perturbation function violate all mathematical/physical rules, leading thus to an inappropriate results. Moreover, during their development on exact analytical solutions, authors have not established conditions under which those analytical solutions may exist, as one can easily verify in subsection 3.1. It is the aim of the present comment to correct all shortcomings about the modulational instability study carried out in their study. We start by presenting the correct steady-state solution of the considered dissipative nonlinear Schrödinger equation, and then investigate with details its modulational instability. We end this comment by establishing conditions under which equation (15) admits analytical solutions of form (16) when a6=0, and the present some other interesting solutions of this equation (16).

Dynamics and scaling of internally cooled convection

Our goal is to investigate fundamental properties of the system of internally cooled convection. The system consists of an upward thermal flux at the lower boundary, a mean temperature lapse-rate and a constant cooling term in the bulk with the bulk cooling in thermal equilibrium with the input heat flux. This simple model represents idealised dry convection in the atmospheric boundary layer, where the cooling mimics the radiative cooling to space notably through longwave radiation. We perform linear stability analysis of the model for different values of the mean stratification to derive the critical forcing above which the fluid is convectively unstable to small perturbations. The dynamic behaviour of the fluid system is described and the scaling of various important measured quantities such as the total vertical convective heat flux and the upward mass flux is measured. We introduce a lapse-rate dependent dimensionless Rayleigh-number Raγ that determines the behaviour of the system, finding that the convective heat-flux and mass-flux scale scale approximately as Raγ0.5 and Raγ0.7 respectively. The area-fraction of the domain that is occupied by upward and downward moving fluid and the skewness of the vertical velocity are studied to understand the asymmetry inherent in the system. We conclude with a short discussion on the relevance to atmospheric convection and the scope for further investigations of atmospheric convection using similar simplified approaches.

Mixed localized matter wave solitons in Bose–Einstein condensates with time-varying interatomic interaction and a time-varying complex harmonic trapping potential

We study the generation of mixed localized matter wave solitons in the quasi-one-dimensional Gross–Pitaevskii equation with a complex potential (model equation) that describes the dynamics of Bose–Einstein condensates trapped in an harmonic potential when the loss/gain of condensate atoms is taken into account. Performing a modified lens-type transformation, the integrability condition is derived and the model equation is converted to a Kundu-like nonlinear Schrödinger equation. Through the linear stability analysis, the phenomenon of the modulational instability is analyzed and the criterion of the modulational instability is established. Based on the generalized perturbation (n,N−n)−fold Darboux transformation, new mixed localized wave solutions are presented and used for analyzing the generation of mixed matter-wave solitons in Bose–Einstein condensates with two-body interatomic interactions. We show that mixed localized waves generated with these exact solutions can change from a strong interaction to a weak interaction by choosing the parameters such as the similarity parameters, the spectrum parameter, or the seed solution parameter. Our results show that the spectrum parameter is useful for generating interesting structures which are beneficial to understanding the complex physical phenomena in Bose–Einstein condensates with atomic gain/loss.

Gap solitons in quasi-1D Bose–Einstein condensate with three-body interactions under PT symmetry

The existence and stability of gap solitons in a quasi-one-dimensional Bose–Einstein condensate with three-body interactions loaded in a PT-symmetric potential are investigated numerically. Under the mean-field approximation, the dynamical behaviors of the system are described by a cubic-quintic Gross–Pitaevskii equation. Firstly, we obtained the band-gap structures via linearizing the GPE. The PT-symmetric of the system will be broken if the magnitude of the imaginary part of the external potential exceeds a critical value. Secondly, various of gap solitons were found by the Newton-Conjugate-Gradient method. Finally, the stability properties of gap solitons were investigated through the linear stability analysis and the direct long-time nonlinear dynamical evolution. The results indicated that the stability of on-site and off-site solitons was remarkably influenced by three-body interactions. There exists unstable on-site gap solitons when the PT-symmetric potential is taken into account. Additionally, the magnitude of the imaginary part of the potential also affects the profile and stability of the gap solitons. When it increases, the symmetry of the solitons will be broken.

Effect of gravity settling on the onset of thermal convection in a nanofluid-saturated porous medium layer

The onset of convection in a horizontal porous medium layer saturated with a nanofluid and heated from below is investigated via linear stability analysis and numerical simulation. The Darcy–Buongiorno model is used to describe the convective transport behaviour of the nanofluid and the settling effect of nanoparticles due to gravity is considered in addition to thermophoresis and Brownian diffusion. The linear stability analysis shows that the gravity settling is a substantial stabilizing mechanism restraining the destabilizing factors such as thermal buoyancy and thermophoresis. The stability threshold is determined by the relative strength of thermophoresis to gravity settling. It is found that the system is destabilized when the thermophoretic mobility prevails. As the nanoparticle size increases, the gravity settling effect is promoted and makes the system more stable. In particular, the onset of instability is dominated by the oscillatory mode once the nanoparticle concentration is in a stably stratified profile across the porous layer. When the Rayleigh–Darcy number $Ra_D$ exceeds the critical value, the spectrum of the growth rates of the unstable modes rises with increasing $Ra_D$ and $Rn$ (i.e. the concentration Rayleigh number), and eventually the unstable modes in the high-wavenumber region exhibit the same instability. The evolution of the convection is further examined by numerical simulation. The results verify the stability characteristics predicted by linear stability analysis. Moreover, the pattern of fingering convection of the nanofluid concentration is observed once the nanofluid concentration is unstably stratified and the density difference across the porous layer is large enough.

Stability analysis and numerical simulation of a car-following model considering safety potential field and V2X communication: A focus on influence weight of multiple vehicles

In a V2X environment, the target vehicle is capable of acquiring motion information from multiple vehicles ahead, and this information plays a crucial role in predicting the target vehicle's motion behavior. To better understand how leading vehicles affect the car-following behavior of the target vehicle, we have developed an improved car-following model that considers the effects of multiple leading vehicles. This model constructs distance-based and field-based models to describe the influence weights of different vehicles on the target vehicle. The stability conditions of the model were obtained through linear stability analysis, and the mKdV equation, which describes the evolution characteristics of traffic density waves in congested areas, was determined through nonlinear analysis. Numerical simulations were conducted to discuss the multi-vehicle effect λ, different vehicle influence weights (distance-based and field-based models) βj, and the number of leading vehicles considered q. The study found that in congested areas, initial perturbations evolve in the form of kink-antikink waves, moving rearward, with the amplitude of the headway curve decreasing as the multi-vehicle effect coefficient value increases. For different vehicle influence weights, the field-based model outperforms the distance-based model. Moreover, as the number of leading vehicles considered q increases, the stability of traffic flow gradually improves. The numerical results are consistent with the theoretical findings, and it is noted that the model successfully enhances vehicular movement efficiency, reduces congestion, and improves road safety. To minimize collision incidents, the improved model can be implemented as an active safety technology.

Spatiotemporal dynamics of modulation instability in Kerr nonlinear media with pure-quartic dispersion

We present the analytical and numerical analysis on spatiotemporal dynamics of modulation instability (MI) in Kerr nonlinear media with pure-quartic dispersion. We apply a linear stability method to arrive at a universal expression for pure-quartic instability gain, which can simultaneously describe the temporal, spatial, and spatiotemporal cases. The results show that the condition of such three fundamental instability occurrence in pure-quartic dispersion media (PDM) is the same as that in conventional quadratic dispersion media (QDM). Another interesting observation is that the maximum gain is also the same in both cases. For temporal instability, there exists a critical power below which the instability bandwidth becomes wider than the corresponding case in QDM or otherwise. For spatiotemporal instability, the instability region depending on critical frequencies is found to show some remarkable difference when compared with the latter case. The analytical prediction resulting from linear stability analysis is confirmed by the numerical method.

Explosive and semi-explosive transitions in parametrically perturbed systems

The phenomenon of transition between steady states, namely amplitude death and oscillation death, is identified in certain models of coupled parametrically driven nonlinear oscillators. Intriguingly, the nature of the transition is explosive. It is significant to note that this observation of explosive transition has been made in systems of identical coupled oscillators with repulsive mean-field coupling. In particular, we find that when the coupling strength increases, there are semi-explosive transitions in parametrically perturbed duffing oscillators and explosive transitions in parametrically perturbed van der Pol oscillators. We obtain analytical expressions for the critical bifurcation points through linear stability analysis. Detailed numerical studies have been performed to confirm the phenomenon.

Impact of Local Thermal Non-Equilibrium and Gravity Fluctuations on the Onset of a Darcy-Brinkman Porous Convection

The numerical investigation of the Darcy-Brinkman convective problem with gravity fluctuations in a Local Thermal Nonequilibrium (LTNE) model is conducted. Utilizing linear stability analysis, the convective problem is explored, and the numerical values of the Rayleigh and wave numbers for onset convection are computed through the one-term Galerkin approach. Three distinct types of gravity fluctuations (linear, parabolic, and exponential) are considered. The results show the Darcy number and gravity parameter delay the onset of convection. The porosity-scald conductivity ratio and interface heat transfer coefficient have a significant effect on the stability of the configuration. Graphical representations depict the effects of various parameters, highlighting the significant impacts of incorporating gravity fluctuations and non-equilibrium conditions in determining convection stability thresholds.

Nonlinear stability analysis of Rayleigh-Bénard problem for a Navier-Stokes-Voigt fluid

The linear and nonlinear stability analyses of thermosolutal convection in a non-Newtonian Navier-Stokes-Voigt fluid, considering Soret and Ekman damping effects, are conducted analytically. Instability thresholds are determined for thermosolutal convection within a viscoelastic fluid of the Kelvin-Voigt type, wherein a dissolved salt field exists. Two scenarios are examined: one where the fluid layer is heated from the bottom and concurrently salted from the bottom, and the other where the fluid layer is heated from the bottom and concurrently salted from the top. The governing partial differential equations system includes conservation laws of mass, momentum, energy, and salt concentration. Using the energy method, the disturbances to the fluid system are shown to decay exponentially. Analytical expressions are developed for the eigenvalue as a function of Soret, Lewis, Prandtl, Kelvin-Voigt, and Rayleigh friction numbers. The study illustrates the shift from a stationary mode of convection to an oscillatory mode and provides thresholds that indicate these transitions. It is found that the viscoelastic property of the fluid acts as a stabilizing agent for oscillatory mode convection. Rayleigh friction substantially controls the convection threshold. Upon comparing threshold values between linear and nonlinear theories, a subcritical instability region is observed in the heating bottom-salting bottom case (case-1), whereas such a region is absent in the heating bottom-salting top case (case-2).