Linear Stability Analysis Research Articles

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).

Coherent Phase Change in Interstitial Solutions: A Hierarchy of Instabilities.

Metal hydrides or lithium ion battery electrodes can take the form of interstitial solid solutions with a miscibility gap. This work discusses theory approaches for locating, in temperature-composition space, coherent phase transformations during the charging/discharging of such systems and for identifying the associated transformation mechanisms. The focus is on the simplest scenario, where instabilities derive from the thermodynamics of the bulk phase alone, considering strain energy as the foremost consequence of coherency and admitting for stress relaxation at free surfaces. The extension of the approach to include capillarity is demonstrated by an example. The analysis rests on constrained equilibrium phase diagrams that are informed by geometry- and dimensionality-specific mechanical boundary conditions and on elastic instabilities-again geometry-specific-as implied by the theory of open-system elasticity. It is demonstrated that some scenarios afford the analysis of chemical stability to be based entirely on a linear stability analysis of the mechanical equilibrium, which provides closed-form solutions in a straightforward manner. Attention is on the impact of the system geometry (infinitely extended or of finite size) and on the chemical (closed or open system) and mechanical (incoherent or coherent) boundary conditions. Transformation mechanism maps are suggested for documenting the findings. The maps reveal a hierarchy of instabilities, which depend strongly on each of the above characteristics. Specifically, realistic, finite-sized systems differ qualitatively from idealized systems of infinite extension. Among the transformation mechanisms exposed by the analysis are a uniform switchover to the other phase when the open system reaches its chemical spinodal, practical coherent nucleation, as well as chemo-elastically coupled spontaneous buckling modes, which may take the form of either, single-phase or dual-phasestates.

Instability of Jeffrey Fluid Throughflow in a Porous Layer Induced by Heat Source and Soret Effect

Abstract In this study, we investigated the instability of thermosolutal convection of Jeffrey fluid in a porous layer with internal heating and the Soret effect. The layer is bounded by two fixed permeable parallel plates which are assumed to be isothermal and isosolutal. An existing initial flow in the vertical direction is passing the layer at a constant speed. The flow fields are adequately presented by PDEs and transformed into dimensionless forms. A small perturbation to the basic flow profiles with linear stability analysis results the problem in an eigenvalue problem. The Runge-Kutta method is used to derive the numerical value of the critical thermal Rayleigh number. The convective instability for asymptotic cases for Le = 1 and Pe = 0 are also examined as special cases. The analysis reveals that for a non-positive Soret parameter, the flow is stable for all Lewis numbers and independent of the heat source. But with a positive Soret parameter in the absence of a heat source, the fluid flow is stable for Le = 3 while the influence of a heat source destabilizes the flow for Le > 2. In high and low shear flows with increasing solutal gradient, the solutal Rayleigh number shows a highly destabilizing nature for all Le. Moreover, smaller relaxation and higher retardation time are the most unstable characteristics of the heat source system. In convective longitudinal rolls, the uni-cellular streamline patterns tend to become bi-cellular by the influence of positive Soret parameters and energy sources.

Two-dimensional line defect lattice solitons in nonlinear fractional Schrödinger equation

The existence, stability, and propagation dynamics of two-dimensional line defect lattice solitons (LDLSs) in a photorefractive crystal with focusing saturable nonlinearity under fractional diffraction effect are numerically investigated. It is shown that, the LDLSs stability domain narrows with increasing Lévy index in the semi-infinite bandgap either in the negative or positive defect lattice. While in the first bandgap, LDLSs are stable for the negative defect lattice and unstable for the positive defect lattice. Additionally, linear stability analysis indicates that the stability and power of LDLSs depend strongly on the Lévy index, defect depth, and propagation constant in different bandgaps. For LDLSs in the semi-infinite bandgap, the stability can be improved by decreasing either the Lévy index or the negative defect depth.

Complex pattern evolution of a two-dimensional space diffusion model of malware spread

In order to investigate the propagation mechanism of malware in cyber-physical systems (CPSs), the cross-diffusion in two-dimensional space is attempted to be introduced into a class of susceptible-infected (SI) malware propagation model depicted by partial differential equations (PDEs). Most of the traditional reaction-diffusion models of malware propagation only take into account the self-diffusion in one-dimensional space, but take less consideration of the cross-diffusion in two-dimensional space. This paper investigates the spatial diffusion behaviour of malware nodes spreading through physical devices. The formations of Turing patterns after homogeneous stationary instability triggered by Turing bifurcation are investigated by linear stability analysis and multiscale analysis methods. The conditions under the occurence of Hopf bifurcation and Turing bifurcation in the malware model are obtained. The amplitude equations are derived in the vicinity of the bifurcation point to explore the conditions for the formation of Turing patterns in two-dimensional space. And the corresponding patterns are obtained by varying the control parameters. It is shown that malicious virus nodes spread in different forms including hexagons, stripes and a mixture of the two. This paper will extend a new direction for the study of system security theory.

Intrinsic instability of lean hydrogen/ammonia premixed flames: Influence of Soret effect and pressure

The addition of hydrogen in ammonia/air mixtures can lead to the onset of intrinsic flame instabilities at conditions of technical relevance. The length and time scales of intrinsic instabilities can be estimated by means of linear stability analysis of planar premixed flames by evaluating the dispersion relation. In this work, we perform such linear stability analysis for hydrogen-enriched ammonia/air flames (50%H2-50%NH3 by volume) using direct numerical simulation with a detailed chemical kinetic mechanism. The impact of pressure and the inclusion of the Soret effect in the governing equations is assessed by comparing the resulting dispersion relation at atmospheric pressure and 10 atm. Our data indicate that both pressure and the Soret effects promote the onset of intrinsic instabilities. Comparisons with available numerical literature data as well as theoretical models are also discussed.

Stability analysis of electro-osmotic flow in a rotating microchannel

We investigate the linear stability analysis of rotating electro-osmotic flow in confined and unconfined configurations by appealing to the Debye–Hückel approximation. Pertaining to flow in confined and unconfined domains, the stability equations are solved using the Galerkin method to obtain the stability picture. Both qualitative and quantitative aspects of Ekman spirals are examined in stable and unstable scenarios within the unconfined domain. Within the confined domain, the variation of the real growth rate and the transition to instability are analysed using the modified Routh–Hurwitz criteria, employed for the first time in this context. The stability of the underlying flow, characterized by the number of roots with a positive real part, is determined by establishing a Routhian table. The inferences of this analysis show that the velocity plane produces intriguing closed Ekman spirals, which diminish in size with an increase in the rotation speed $\omega$ . The Ekman spirals in the stable region exhibit a distinct discontinuity, indicating the dissipation of disturbances over time. In the confined domain, the flow appears consistently stable for a set of involved parameters pertinent to this analysis, such as electrokinetic parameter $K=1.5$ and rotational parameter $\omega$ approximately up to $6$ . However, the flow instabilities become evident for $K=1.5$ and $\omega \geq 6$ .

Curvature controls beading in soft coated elastic cylinders: Finite wavemode instability and localized modulations

Axisymmetric beading instabilities in soft, elongated cylinders have been observed in a plethora of scenarios, ranging from cellular nanotunnels and nerves in biology to swollen cylinders and electrospun fibers in polymer physics. One of the common geometrical features that can be seen in these systems is the finite wavelength of the emerging pattern. However, modelling studies often predict that the instability has an infinite wavelength, which can be associated with localized necking or bulging. In this paper, we consider a soft elastic cylinder with a thin coating that resists bending, as described by the Helfrich free energy functional. The bending stiffness and natural mean curvature of the coating are two novel features whose competition against bulk elasticity and capillarity is investigated. For intermediate values of the bending stiffness, a linear stability analysis reveals that the mismatch between the current and natural mean curvature of the coating can lead to patterns emerging with a finite wavelength. This analysis creates a continuous bridge between the classical solutions of the shape equation obtained from the Helfrich functional and a curvature-controlled zero-wavemode instability, similar to the one induced by the competition between bulk elasticity and capillarity. A weakly non-linear analysis predicts that the criticality of the bifurcation depends on the controlling parameter, with both supercritical and subcritical bifurcations possible. When capillarity is introduced, the criticality of the bifurcation changes in a non-trivial way.

Instability in an anisotropic Brinkman layer due to downward or upward throughflow and purely due to internal heat generation

A linear stability analysis using normal mode technique is applied on an internally heated anisotropic Brinkman porous layer with vertical throughflow. Four internal heat source functions are considered to model the problem with same temperature at the bounding plates. The basic temperature profile due to pure internal heating is seen to be disturbed by throughflow (upward or downward) and the impact of through flow on these profiles is studied. Downward through flow, in particular, is seen to disturb the instability criterion for different profiles. Effect of anisotropy and modified Darcy number on onset of convection is also discussed.

An electric kickscooter multibody model: equations of motion and linear stability analysis

AbstractIn this work, a detailed multibody model of an electric kickscooter is presented. The model includes toroidal wheels as well as rear and front suspensions. The equations of motion are derived and linearized along the steady forward motion of the vehicle. Using an efficient linearization approach, suitable for complex multibody systems with holonomic and nonholonomic constraints, allows for obtaining the reduced linearized equations of motion as a function of the geometric, dynamic, wheels’, and suspensions’ parameters. The proposed electric kickscooter multibody model is validated with the stability results of a previously presented electric kickscooter benchmark. Since the resulting eigenvalues are parameterized regarding the design parameters, a detailed linear stability analysis of the system is performed. In particular, the influence on the stability of the toroidal geometry of the wheels, the elliptic cross-section of the toroidal wheels, the rider model, the steering axis inclination angle, the inertia tensor of the front frame, and the rear and front suspensions is analyzed. The model presented, together with the linearized equations of motion obtained in this work, enables a systematic analysis of the stability of these vehicles, which helps design new electric kickscooters with improved vehicle safety conditions and oriented to a wider range of potential users.

Convective instability of Ekman boundary layer flow of a Newtonian fluid over a stretchable rotating disk

ABSTRACT This paper examines a linear convective instability of an incompressible, Newtonian, viscous fluid rotating over a stretchy spinning disk in an Ekman boundary layer flow. First time in the literature, a numerical analysis of impacts of the stretching mechanism on the stability of the Ekman flow in the presence of the Coriolis force has been carried out, where the bottom disk is allowed to uniformly extend in radial direction. The von Kármán similarity transformations have been used to convert the governing equations of motion into a system of non-linear coupled ordinary differential equations (ODEs) in non-dimensional variables. Then velocity profiles of the flow have been obtained by numerically solving these dimensionless ODEs. The neutral stability curves were then obtained by performing a linear stability analysis for the convective mechanism by using the Chebyshev collocation method. Surface stretching has a globally stabilizing influence on both Type-I and Type-II instability modes of the Ekman flow, according to the stability curves. Then, the result established from the stability curves has been verified by calculating total kinetic energy change due to induced perturbations in the flow system. y.

A microscopic traffic flow model for explaining nonlinear traffic phenomena: Modeling, stability analysis and validation

As the core of reproducing the real nonlinear phenomena of traffic congestion, the investigation of car-following models, which take into account human factors, has consistently assumed a central role within the domain of transportation science. Nevertheless, it is noteworthy that prior studies have invariably employed a fixed value for driver reaction time to stimuli, whereas it is imperative to recognize the dynamic nature of this parameter, closely associated with the instantaneous vehicle speed. To address this issue, this paper constructs a dynamic sensitivity coefficient (DSC), and further develops a nonlinear human factoring car-following model to reveal the relation between the reaction time and the current speed. Firstly, the linear critical stability condition and the mKdV equation of the model are derived by the linear stability analysis and kink–antikink solution analytic method, respectively. Then, the numerical experiments are conducted to demonstrate that the proposed model is more practical and can reproduce the real driving behavior and phenomena. Finally, calibration and validation results exhibit the actual vehicle trajectory can be simulated well.

Lie group and stability analysis of tangent hyperbolic fluid flow driven by a shrinking sheet

This study elaborates on the mathematical modeling of tangent hyperbolic fluid flow driven by a shrinking sheet in a porous medium. The roles of Soret-Dufour and thermal radiation are considered to model the energy and concentration equations. The primary objective is to find an appropriate similarity transformation through Lie group analysis, which helps to produce the self-similar equations from the governing nonlinear partial systems. The similarity solutions are calculated numerically using the “bvp4c” solver in Matlab. Additionally, adequate suction is required to obtain the similarity solutions. As expected, dual solutions are found within a certain range of the suction parameter. The skin friction coefficient, heat, and mass transfer rates at the sheet are observed for pertinent flow constraints of physical interest. It reveals that the dual solutions bifurcate from a critical point, and beyond this point, no solution exists. Furthermore, the branches of dual solutions on the nondimensional velocity, temperature, and concentration profiles depict different characteristics. Thus, a linear temporal stability analysis has been employed to test the flow stability, revealing that the upper branch is physically reliable. The heat transfer rate increases by 7.69% and 9.55% in the upper and lower branches, respectively, for Soret parameter changes from 0.6 to 0.9. Meanwhile, the mass transfer rate decreases by 85.30% and 324.60% in the upper and lower branches, respectively. The consequences of Weissenberg and suction parameters show an opposite character on the velocity, temperature, and concentration profiles in the upper branch.

Mechanobiological tortuosity of blood vessels with stress-modulated growth and remodeling

The stability of blood vessels is essential for maintaining their functions, while severe blood vessel tortuosity leads to various clinical complications. The growth and remodeling of blood vessels, which are regulated by mechanical and biochemical cues, cause residual stresses that affect vessel stability. In this paper, we combine theory and simulations to study the mechanobiological behavior of blood vessels with stress-modulated growth and remodeling. Effects of the volumetric growth of the matrix and the turnover and reorientation of collagen fibers are accounted for. Linear stability analysis is first carried out to investigate the mechanobiological stability of blood vessels. By developing a finite element method that incorporates stress-modulated growth and remodeling, we validate the theoretical solution of the critical state and further capture the postbuckling evolution of blood vessels. Our results show that an increased internal pressure can lead to the thickening of the vessel wall, which stabilizes the blood vessels mechanically, whereas the effect of internal pressure on the mechanobiological buckling is nonmonotonic. Compared to the mechanical instability, the mechanobiological buckling results in a larger mode number and a shorter wavelength, as usually observed in varicose veins. Vessels during postbuckling may exhibit non-uniform wall thicknesses, in consistency with the experimental observation on tortuous aortas. These findings highlight the crucial role of mechanical remodeling in tissue morphogenesis and could deepen the understanding of mechanobiological mechanisms underlying the formation and development of blood vessel tortuosity.

Filtering induced explosive death in coupled FitzHugh–Nagumo neurons: Theory and experiment

Explosive emergent behavior governed by discontinuous and irreversible phase transition is in the center of recent research. In this paper we show that explosive amplitude death (AD) can be induced in a system of coupled oscillators if diffusion is accompanied by low-pass filtering in the mutual coupling path. Here we consider a ring network of diffusively coupled FitzHugh–Nagumo oscillators, and through theoretical and experimental investigations explore how the low-pass filter induces and controls the transition to explosive AD state. We derive the conditions of getting AD in the network through linear stability analysis. Using one and two parameter bifurcation analysis we find the explicit parametric zone of explosive death in the limiting case of two neurons, which reveals that the explosive death arises through the saddle–node bifurcation of limit cycle. In the strong coupling limit, we observe an interesting semi-explosive death scenario, where both continuous and discontinuous transitions are present. Finally, using electronic hardware circuits, we demonstrate the first experimental observation of explosive amplitude death that establishes the robustness of the scenario in a practical setup.

Linear stability and numerical analysis of vertical dense particulate flows in hydraulic conveying

Linear stability and numerical analysis of vertical dense particulate flows in hydraulic conveying

Investigating the role of edge neutrals in exciting tearing mode activity and achieving flat temperature profiles in LTX-β

We present observations, numerical simulations, and analysis from experiments in the Lithium Tokamak Experiment-Beta (LTX-β) in which the electron temperature profile (Te (r)) shifts from flat to peaked and a tearing mode is also destabilized when the average density (ne ave ) exceeds ∼1019 m−3. Flat Te (r) is obtained routinely in LTX-β, with a lithium coated, low-recycling first wall, once the external fueling is stopped and density decays [Boyle et al 2023 Nucl. Fusion 63 056020]. In the present experiment, flat Te profiles can be sustained while maintaining constant ne ave below a line averaged density threshold (ne ave th ) of ∼1019 m−3. Above ne ave th, Te (r) shifts from flat to peaked and a tearing mode is destabilized. Due to low recycling, the achieved ne ave can be controlled precisely by external fueling and hence, a certain threshold of the edge neutral inventory from the external fueling is experimentally manifested through ne ave th . The goal of the present work is to investigate the role of edge neutrals in determining Te (r) and MHD stability in the unique low-recycling regime of LTX-β. Our hypothesis is that the peaking of Te (r) beyond ne ave th is due ultimately to the edge cooling by the cold neutrals beyond a critical fueling flux. At lower fueling flux, flat Te (r) results in broader pressure profile and lower resistivity, which in turn stabilizes the tearing mode. This hypothesis is supported by edge neutral density estimation by DEGAS 2 code. Mode analysis by singular value decomposition confirms the tearing mode structure to be m/n = 2/1 (m and n being the poloidal and toroidal mode numbers). Linear tearing stability analysis with M3D-C1 predicts that plasmas with ne ave > 1019 are highly susceptible to a n = 1 tearing mode. ORBIT simulations, however, confirmed that the tearing modes do not contribute to the loss of fast ions from neutral beam injection. This study shows for the first time that the neutral inventory at the edge could be one of the deciding factors for the achievability of the unique operation regime of flat Te (r) and the excitation of tearing activity that could be disruptive for the plasmas.

Collective group drift in a partial-differential-equation-based opinion dynamics model with biased perception kernels.

In the age of technology, individuals accelerate their biased gathering of information, which in turn leads to a population becoming extreme and more polarized. Here we study a partial-differential-equationmodel for opinion dynamics that exhibits collective behavior subject to nonlocal interactions. We developed an interaction kernel function to represent biased information gathering. Through a linear stability analysis, we show that biased populations can still form opinionated groups. However, a population that is too heavily biased can no longer come to a consensus, that is, the initial homogeneous mixed state becomes stable. Numerical simulations with biased information gathering show the ability for groups to collectively drift towards one end of the opinion space. This means that a small bias in each individual will collectively lead to groups of individuals becoming extreme together. The characteristic time scale for a group's existence is captured from numerical experiments using the temporal correlation function. Supplementing this, we included a measure of how different each population is after regular time intervals using a form of the Manhattan and Euclidean distance metrics. We conclude by exploring how wall boundary conditions induce pattern formation initially on the most extreme sides of the domain.