Linear Stability Criterion Research Articles

Parallel firehose instability in electron-positron plasmas.

In a magnetized uniform plasma, firehose instability may arise as a result of pressure anisotropy of P_{||}>P_{⊥}, where P_{||} and P_{⊥} are the thermal pressure parallel and perpendicular to the magnetic field, respectively. In this paper, we examine the parallel firehose instability in electron-positron plasmas based on the particle simulations along with the linear fluid theory, which may give rise to the dispersion relation, instability criteria, and growth rate, etc., for comparisons with those calculated from the kinetic simulations. As for the firehose instability in electron-proton plasmas, the magnetic field grows rapidly and then decreases with oscillations. The nonlinear saturated state complies with the linear stability criterion, α=μ_{0}(P_{||}-P_{⊥})/B^{2}=1, derived from the fluid theory only for relatively smaller values of initial α or ω_{p}/ω_{c}, where ω_{p} and ω_{c} are the plasma and cyclotron frequencies, respectively. For relatively larger values of initial α and ω_{p}/ω_{c}, the saturated α values are smaller than 1 as a result of kinetic resonant effects. The dominant wave numbers are kc/ω_{p}<0.5 and the growth rates are in the range of 0.1-0.3ω_{c}, which are approximately consistent with the linear fluid theory for the same wavelengths. Both electrostatic and electromagnetic modes predicted by the linear fluid theory are identified in the kinetic simulations.

Available energy of magnetically confined plasmas

The concept of the available energy of a collisionless plasma is discussed in the context of magnetic confinement. The available energy quantifies how much of the plasma energy can be converted into fluctuations (including nonlinear ones) and is thus a measure of plasma stability, which can be used to derive linear and nonlinear stability criteria without solving an eigenvalue problem. In a magnetically confined plasma, the available energy is determined by the density and temperature profiles as well as the magnetic geometry. It also depends on what constraints limit the possible forms of plasma motion, such as the conservation of adiabatic invariants and the requirement that the transport be ambipolar. A general method based on Lagrange multipliers is devised to incorporate such constraints in the calculation of the available energy, and several particular cases are discussed for which it can be calculated explicitly. In particular, it is shown that it is impossible to confine a plasma in a Maxwellian ground state relative to perturbations with frequencies exceeding the ion bounce frequency.

Linear and Weakly Nonlinear Double-Diffusive Magnetoconvection in a Non-Newtonian Fluid Layer

The linear and weakly nonlinear stability of a doubly diffusive electrically conducting non-Newtonian couple stress fluid layer in the presence of a uniform vertical magnetic field is investigated. The system behaves like a triply diffusive one with the magnetic field as a third diffusing component. Several remarkable departures from those of doubly diffusive couple stress fluid systems are recognized and the implications of couple stresses on the same are explored. By performing the linear stability analysis, it has been recognized that (i) oscillatory convection occurs even if the diffusivity ratios are greater than unity, (ii) the bottom-heavy solute gradient in the presence of magnetic field, as well as magnetic field in the presence of bottom-heavy solute gradient, destabilize the system under certain conditions, (iii) disconnected closed oscillatory neutral curves occur for certain choices of physical parameters indicating the necessity of three critical values of thermal Rayleigh number to specify the linear stability criteria instead of the usual single value. A weakly nonlinear stability analysis has been performed using modified perturbation theory and the steady bifurcating equilibrium solution is found to be subcritical in some parameter space. Heat and mass transports are calculated in terms of Nusselt numbers and the influence of physical parameters on the same is discussed in detail.

H-Infinity Load Frequency Control of Networked Power Systems via an Event-Triggered Scheme

An event-triggered load frequency control problem for networked multiarea power systems is discussed, where the event-triggered control scheme has an adaptive threshold. This article is motivated by the consideration for a bandwidth-limited communication channel. To reduce the communication network bandwidth burden, this article develops an adaptive event-triggered control scheme to reduce the number of transmitted packets while keeping the desired control performance. From numerical simulations, we find that if the adaptive law is chosen inappropriately, then it is possible that the event-triggered scheme with an adaptive threshold is more conservative than one with a fixed threshold. Motivated by this, an improved adaptive law of the threshold for the event-triggered scheme is given such that the adaptive event-triggered scheme outperforms one with a constant threshold. By using the Lyapunov method and linear matrix inequality techniques, stability and stabilization criteria are obtained. Finally, numerical examples are given to illustrate the effectiveness of our results.

Absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow: a linear analysis

We present a study of absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow (EHD-Poiseuille). The electric field is imposed on two infinite flat plates filled with a non-conducting dielectric fluid with unipolar ion injection. Mathematically, the dispersion relation of the linearised problem is studied based on the asymptotic response of an impulse disturbance imposed on the base EHD-Poiseuille flow. Transverse, longitudinal and oblique rolls are investigated to identify the saddle point satisfying the pinching condition in the corresponding complex wavenumber space. It is found that when the ratio of Coulomb force to viscous force increases, the transverse rolls can transit from convective instability to absolute instability. The ratio of hydrodynamic mobility to electric mobility, which exerts negligible effect on the linear stability criterion when the cross-flow is small, has significant influence on the convective–absolute instability transition, especially when the ratio is small. As we change the value of the mobility ratio, a saddle point shift phenomenon occurs in the case of transverse rolls. The unstable longitudinal rolls are convectively unstable as long as there is a cross-flow, a result which is deduced from a one-mode Galerkin approximation. Longitudinal rolls have a larger growth rate than transverse rolls except for a small cross-flow. Finally, regarding the oblique rolls, a numerical search for the saddle point simultaneously in the complex streamwise and transverse wavenumber spaces always yields an absolute transverse wavenumber of zero, implying that oblique rolls give way to transverse rolls when the flow is unstable.

Nonlinear analysis of an improved continuum model considering mean-field velocity difference

Based on the velocity gradient model, an extended continuum model with consideration of the mean-field velocity difference is proposed in this paper. By using the linear stability theory, the linear stability criterion of the new model is gained, which proved that mean-field velocity difference has significant influence on stability of traffic flow. The KdV–Burgers equation is derived by using non-linear analysis method and the evolution of density wave near the neutral stability line is explored. Numerical simulations are carried out how mean-field velocity difference affect the stability of traffic flow, and energy consumption is also studied for this new macro model. At the same time, complicated traffic phenomena such as local cluster effects, shock waves and rarefaction waves can be reproduced in the new model by numerical simulation. Numerical results are consistent with the theoretical analysis, which indicates that the mean-field velocity difference not only suppresses traffic jam, but also depresses energy consumption.

An Extended Mean-Field Lattice Hydrodynamic Model With Consideration of the Average Effect of Multi-Lattice Interaction

In this paper, an extended mean-field lattice hydrodynamic model is proposed which considers the average effect of multi-lattices interaction. By stability analysis, the linear stability criterion of the proposed model is given. Through nonlinear analysis, we derive the mKdV equation to investigate the traffic jams described by traffic density wave. Then, the numerical simulations are carried out. The simulation results show that that the average effect of multi-lattices interaction plays the great important part in curbing traffic congestion. Increasing the impact of the mean-field multi-lattices interaction can curb the congestion effectively and make the traffic system more stable.

Stability and bifurcations of even periodic orbits in the Sitnikov problem

We study different families of even periodic solutions in the classical Sitnikov problem that emanate from the circular case as the eccentricity is increased. The families can be classified by the number N of full revolutions of the primaries and labelled by the number of zeroes p of the vertical coordinate of the massless body in half a period. We give a linear stability criterion of these branches depending on even N, based on the sign for the initial slope of the discriminant function for the associated Hill’s equation, in a computable interval of eccentricities. All families for $$N=2$$ are linearly stable for small and computable e. The results show a fundamental symmetry-driven difference between the even and odd N cases.

Effect of viscoelasticity on liquid sheet rupture

Thin liquid sheets are ubiquitous in many industrial processes, such as atomization and curtain coating. In the latter, the thickness of the deposited layer is limited by the breakup of the liquid curtain below a critical flow rate. The stability of a liquid sheet depends on the disturbance characteristics, sheet thickness and fluid properties. Experimental analyses have shown that thinner stable liquid curtains can be obtained with viscoelastic liquids; however, the underlining physical mechanisms associated with the increased stability are not fully understood. This work presents a numerical analysis of the effect of viscoelasticity on the stability of a thin liquid sheet. We first derive linear stability criteria for both planar and axisymmetric perturbations of Newtonian and Oldroyd-B liquids. The time evolution of planar and axisymmetric perturbations in an Oldroyd-B liquid sheet is evaluated using asymptotic expansion of the flow variables and a fully-implicit time integration scheme. The breakup time is calculated as a function of Deborah number and polymer to solvent viscosity ratio. The results show that the liquid rheological behavior does not change the linear stability criterion, however it has a strong effect on the growth rate of the disturbance and consequently on the breakup time.

Thermal Instability Analysis of an Elastico-Viscous Nanofluid Layer

The purpose of this paper is to study the thermal instability analysis of an elastico-viscous nanofluid layer heated from below. The Rivlin-Ericksen type fluid model is used to describe the rheological behavior of an elastico-viscous nanofluid. The linear stability criterion for the onset of both stationary and oscillatory convection is derived by applying the normal model analysis method. The presence of nanoparticles enhances the thermal conductivity of the fluid, and the model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The effect of the physical parameters of the system, namely the concentration Rayleigh number, Prandtl number, capacity ratio, Lewis number, and kinematic visco-elasticity coefficient, on the stability of the system is numerically investigated. In addition, sufficient conditions for the non-existence of oscillatory convection are reported.

Effect of the driver’s desire for smooth driving on the car-following model

To reduce the fuel consumption caused by the acceleration or deceleration, many drivers wish to drive cars as smoothly as possible, which means that driving behavior is influenced by the driver’s desire for smooth driving. In this paper, with the consideration of the driver’s desire for smooth driving, an improved car-following model is presented by incorporating the difference of the steady and history velocities into optimal velocity model. To show the effect of the factor discussed herein, the improved car-following model is investigated by the analytical and numerical methods. The linear stability criterion is obtained by the linear analysis method. And then the modified KdV equation is derived to describe the propagating behavior of traffic jams near the critical point. The amplitude of the traffic jam can be relieved by strengthening driver’s desire for smooth driving. The numerical simulations are explored to verify the validity of the analytical results and the effect of the driver’s desire for smooth driving. The analytical and numerical results demonstrate that the driver’s desire for smooth driving plays a positive role in improving traffic stability and flux.

KdV–Burgers equation in the modified continuum model considering the effect of friction and radius on a curved road

Considering the effect of friction and radius on a curved road, we propose an extended continuum model based on the full velocity difference model (FVDM, for short). Through the linear stability theory, the linear stability criterion of the continuum model is gained. By applying the nonlinear analysis, the KdV–Burgers equation is derived to describe the propagating behavior of traffic density wave near the neutral stability line. Numerical simulations show that traffic flow evolves into multiple local cluster, which corresponding to some particular traffic phenomenon with distinctive curved road factors. Numerical results indicate that the effect of friction and radius can suppress the traffic congestion and improve the stability of traffic flow efficiently.

Stability analysis of car-following model on straight and curved roads considering the preceding vehicle’s velocity feedback control

In order to reveal the impact of preceding vehicle’s velocity on traffic flow, an extended car-following model considering preceding vehicle’s velocity feedback control is proposed in this paper. The linear stability criterion of the new model is derived through control theory method and it shows that the feedback control signal impacts the stability of traffic flow. Numerical simulation results is in good agreement with the theoretical analysis, which prove that a smaller negative feedback control of the preceding vehicle’s velocity can enhance the stability of traffic flow, while a smaller positive feedback control of the preceding vehicle’s velocity can exacerbate traffic congestion. Moreover, the reaction coefficients of straight and curved road conditions also play an important role in the stability of traffic flow.

Stability criterion for solitons of the Zakharov–Kuznetsov-type equations

Early results concerning the linear stability of the solitons in equation of the KDV-type [1] are generalized to solitons describing by the Zakharov–Kuznetsov-type equation. The linear stability criterion for ground solitons in the Vakhitov–Kolokolov form is derived for such equations with arbitrary nonlinearity. For the power nonlinearity the instability criterion coincides with the condition of the Hamiltonian unboundedness from below. The latter represents the main feature for appearance of collapse in such systems.

Buckling strength topology optimization of 2D periodic materials based on linearized bifurcation analysis

Low density cellular materials may offer excellent mechanical properties and find wide applicability in lightweight design and infill structures for additive manufacturing, yet currently existing material structures are still far away from their theoretical limit in terms of compressive strength. To explore the existing potential, this paper presents a topology optimization framework for designing periodic cellular materials with maximized strength under compressive loading. Under this condition, the limiting factor of strength is the failure mechanism of buckling instability in the microstructure. In order to predict microstructural buckling, a simplified model based on homogenization theory, a linearized stability criterion and Floquet–Bloch theory is employed. Subsequently, a gradient-based topology optimization problem is formulated to maximize the buckling strength of the most critical failure mode. The framework is utilized to optimize square, triangular and hexagonal microstructures for three different macroscopic load conditions including biaxial, uniaxial and shear loading, and performance assessments are conducted by computation of associated failure surfaces in macroscopic stress space. In all cases, the optimized designs turn out to be first-order hierarchical type microstructures which offer major improvements of strength compared to the initial zero-order designs, however, the gains come at the cost of reductions in stiffness. Furthermore, it is illustrated how imposing geometric symmetry constraints can be exploited to control the shape of the failure surfaces.

Global stability of stochastic systems with Poisson distributed random time-delay

ABSTRACTThis paper gives global stability criteria for a controlled stochastic system with randomly distributed time-delay. The delay function is governed by a homogeneous Poisson process and is independent of the stochastic perturbation on the system. A delay-dependent gain controller is constructed and used to develop a stabilization criterion for the system. Based on the Lyapunov-Krasovskii functional method and stochastic analysis theory, notions of stochastic stability are examined in terms of linear matrix inequality and global stability criteria derived.

Three-dimensional finite amplitude electroconvection in dielectric liquids

Charge injection induced electroconvection in a dielectric liquid lying between two parallel plates is numerically simulated in three dimensions (3D) using a unified lattice Boltzmann method (LBM). Cellular flow patterns and their subcritical bifurcation phenomena of 3D electroconvection are numerically investigated for the first time. A unit conversion is also derived to connect the LBM system to the real physical system. The 3D LBM codes are validated by three carefully chosen cases and all results are found to be highly consistent with the analytical solutions or other numerical studies. For strong injection, the steady state roll, polygon, and square flow patterns are observed under different initial disturbances. Numerical results show that the hexagonal cell with the central region being empty of charge and centrally downward flow is preferred in symmetric systems under random initial disturbance. For weak injection, the numerical results show that the flow directly passes from the motionless state to turbulence once the system loses its linear stability. In addition, the numerically predicted linear and finite amplitude stability criteria of different flow patterns are discussed.

Stability Analysis With Considering the Transition Interval for PWM DC-DC Converters Based on Describing Function Method

The nonlinearity of the switching process in DC–DC converters can result in the inaccuracy and invalidation of traditional stability criterion based on linear modeling, which is very harmful in practice, especially for the DC-DC converters with high stability requirements. In this paper, the describing function method is adopted for the modeling of switching process, namely, pulse width modulation (PWM), in DC–DC converters, and the describing function of the PWM is derived in detail. Considering the nonlinear items in the obtained describing function of PWM, the selection of the parameters in these nonlinear items are first provided and proved in this paper. With the obtained describing function, the stability of the PWM DC–DC converter can be analyzed exactly. Taken a PWM boost converter as an example, the nonlinear model based on describing function and the linear model are established, respectively; furthermore, the stability analysis based on these two kinds of models are carried out. Comparing with the traditional linear stability criterion, the simulation and experimental results validate the effectiveness and accuracy of the stability analysis based on describing function method. Furthermore, the transition interval of the PWM DC–DC converter from a stable region to an unstable region can be determined exactly by the proposed stability analysis method, which is helpful to determine the stability margin in real engineering applications. Therefore, this paper provides a practical stability analysis method for PWM DC–DC converters.

The influence of continuous historical velocity difference information on micro-cooperative driving stability

In this paper, a new micro-cooperative driving car-following model is proposed to investigate the effect of continuous historical velocity difference information on traffic stability. The linear stability criterion of the new model is derived with linear stability theory and the results show that the unstable region in the headway-sensitivity space will be shrunk by taking the continuous historical velocity difference information into account. Through nonlinear analysis, the mKdV equation is derived to describe the traffic evolution behavior of the new model near the critical point. Via numerical simulations, the theoretical analysis results are verified and the results indicate that the continuous historical velocity difference information can enhance the stability of traffic flow in the micro-cooperative driving process.

An improved lattice hydrodynamic model considering the “backward looking” effect and the traffic interruption probability

By incorporating the “backward looking” effect and traffic interruption probability, we put forward an improved lattice model. Applying linear stability analysis, linear stability criterion is derived. The mKdV equation is deduced through nonlinear theory, which demonstrates that the solution of mKdV equation can describe traffic congestion. Furthermore, numerical simulation shows that the two factors can enhance traffic flow stability in the driving process.