Nonlinear Stability Research Articles

Nonlinear stability of entropy waves for the Euler equations

Nonlinear stability of entropy waves for the Euler equations

Effects of throughflow and gravity modulation on triple-diffusive convection in a sparsely packed porous medium

This article investigates the impact of four different gravity modulations on triple-diffusive convection in a porous layer with throughflow. The study conducts a weakly nonlinear stability analysis. Additionally, cubic Ginzburg–Landau equation is derived to observe the effects of throughflow, gravity modulation, and concentration. The solution of Ginzburg–Landau equation is used to calculate heat transfer and mass transport in terms of the Nusselt and Sherwood numbers, respectively. Furthermore, the mean Nusselt and Sherwood numbers are calculated. The study investigates the influence of Lewis numbers, solutal Rayleigh numbers, and Peclet numbers on the stability and instability of the system graphically for the four types of gravity modulations. The results show that the Sawtooth modulation destabilizes the system, while the square waveform modulation stabilizes it. The mean Nusselt number decreases with both the first Lewis number Le 1 and the second Lewis number Le 2, as well as the mean Sherwood number, which decreases with the first solutal Rayleigh number Rs 1 and the second solutal Rayleigh number Rs 2. This indicates system stability. Furthermore, it is noted that the heat and mass transport for triangular modulation fall between those of sawtooth modulation and square modulation.

Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

The stability characteristics of a thin film of viscoelastic (Walters’ B′ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number kc which include the viscoelastic parameter Γ, angle of inclination of the plane θ, Reynolds number Re and Weber number We. The weakly nonlinear stability analysis that is based on the second Landau constant J2, reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is 3(1+3Γ)Re−3cotθ−4ReWek2 = 0. It is found that all the four distinct flow zones of this problem exist in Re-k-, θ-k- and Γ-k-plane after the critical value of Rec,θc and Γc, respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of Γ irrespective of the values of Re and θ, as for example, a solution of polyisobutylene in cetane, compared with the viscous (Γ=0) film flow case. Finally, we scrutinize the effect of Γ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.

Multi-objective shape-section optimization of free-form latticed shells using the RBF-NSGA-II algorithm

In this study, a novel RBF-NSGA-II optimization framework based on the response surface methodology (RSM) and the constrained NSGA-II algorithm is proposed to simultaneously improve the static, nonlinear stability and economic performance of free-form latticed shells. The bending stiffness of joints is innovatively considered one of the design variables for shape-member section coupled optimization. The augmented radius basis functions (RBFs) are adopted to develop the metamodels of both the objectives and constraints. The improved minimum distance selection method (TMDSM) is used to select the optimal solution from the Pareto optimal solution set. Moreover, three alternative solutions are presented as supplements to meet various engineering requirements. The proposed RBF-NSGA-II algorithm is validated to have high accuracy for multi-objective optimization problems with high nonlinearity. Compared with those of the initial structures, the strain energy and steel consumption of the optimal structure decreased by 82.23 % and 17.97 %, respectively, while the buckling load considering both the geometric nonlinearity and material nonlinearity increased by 1.2 times. The involvement of the joint stiffness is demonstrated to have a great influence on the multi-objective optimization results. It is necessary to consider the joint stiffness as one of the design variables during shape-member section coupled optimization.

Subdiffusive processes in BWRs

The objective of this work is the analysis of subdiffusive processes in BWR reactors. Nuclear reactors are highly heterogeneous systems where the phenomena at the reactor scale are not possible to describe with constitutive standard laws (normal diffusion) of energy, momentum, and mass. However, there are methods to achieve it such as the volumetric average to upscale the reactor and incorporate up-scaled constitutive laws. However, it can also be proposed constitutive laws of fractional order to consider anomalous diffusion. For the neutronic processes, the neutron diffusion theory is non-appropriated to the reactor scale, because this theory is applied to the homogeneous reactor. Given the heterogeneity of these systems in describing the dynamic behaviour of BWRs with point models, it represents a challenge to the incorporation of the heterogeneous effects whose main characteristic is that the transport phenomena are subdiffusive with relaxation times (i.e., non-instantaneous) by nature. The point models called reducer-order models are important for stability analysis and to develop control strategies. In linear or nonlinear stability analysis and system control, they are carried out with reduced order models that are zero dimension, i.e., point models that depend exclusively on time. This is because existing methodologies propose mathematical expressions in the time or frequency domain for stability analysis or to establish control strategies. The challenge is for ROM models to be able to consider subdiffusive effects on neutron motion. This is achieved by various routes from the P1 theory that includes the temporal term with relaxation effects (relaxation time) or by proposing a more general constitutive equation of fractional order for the neutron current vector, as is presented in this work. Moreover, the Fractional Order Models behavior must be as a transport theory based model, as well as in those nuclear and thermohydraulic codes which are very useful but complex, for which is sometimes it is impossible to obtain analytical expressions. To incorporate these effects in this work we consider point models of fractional order, which represent the subdiffusive phenomenon when the fractional order (known as anomalous diffusion coefficient) is greater than 0 and less than 1.

Nonlinear Stability and Asymptotic Behavior of Periodic Wave Trains in Reaction–Diffusion Systems Against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations

We present a nonlinear stability theory for periodic wave trains in reaction–diffusion systems, which relies on pure L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the semigroup generated by the linearization about the wave train and introduce a spatio-temporal phase modulation to capture the most critical dynamics, which is governed by a viscous Burgers’ equation. We then aim to close a nonlinear stability argument by iterative estimates on the corresponding Duhamel formulation, where, hampered by the lack of localization, we must rely on diffusive smoothing to render decay of the semigroup. However, this decay is not strong enough to control all terms in the Duhamel formulation. We address this difficulty by applying the Cole–Hopf transform to eliminate the critical Burgers’-type nonlinearities. Ultimately, we establish nonlinear stability of diffusively spectrally stable wave trains against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations. Moreover, we show that the perturbed solution converges to a modulated wave train, whose phase and wavenumber are approximated by solutions to the associated viscous Hamilton–Jacobi and Burgers’ equation, respectively.

Nonlinear stability of ES-FG porous sandwich cylindrical shells subjected to axial compression in thermal environment

Stability analysis of cylindrical shells subjected to axial compression is a common problem in engineering applications. The phenomenon of instability can easily appear with thin cylindrical shells. Therefore, to increase the load-carrying capacity of this type of structure, the cylinders are increased in thickness by adding a porous core layer and reinforcing orthogonal stiffeners. In this study, the nonlinear buckling and postbuckling behavior of functionally graded (FG) porous sandwich cylindrical shells with orthogonal eccentrically stiffeners (ES) are investigated via analytical approach. The FG porous sandwich cylindrical shells are rested on Pasternak elastic foundations and subjected to axial compression load in a thermal environment. The reinforcement of the shells is achieved through closely spaced stringers and rings, with the material properties of both the shell and stiffeners graded continuously in the thickness direction. The governing equations are derived based on the Donnell’s shell theory with von Karman geometrical nonlinearity and the smeared stiffeners technique. The Galerkin procedure is employed to ascertain the critical load and post-buckling response, employing a three-term deflection solution. The study analyzed the effects of parameters such as porosity coefficient, material, temperature, dimensional parameters, stiffener, and foundation characteristics.

A Multiple Scale Approach to Nonlinear Vibrations and Stability of Doubly Curved Shells/Panels Under External and Parametric Excitations

This study focuses on nonlinear vibrations and stability of doubly curved shells/panels. The thin-walled shell structure is assumed to be acted upon by external loads as well as parametric excitation. Using the first-order shear deformation Fülgge theory assumptions, governing equations of motion are derived for a general doubly curved geometry. Nonlinear components which are compatible with the Von-Karman kinematic assumptions are retained in the formulation and the rest of nonlinear components are neglected due to the fact that strains are assumed to be small and rotations to be moderate. The formulation can be readily reduced to any geometry including cylindrical shells and spherical panels, as benchmark examples in this study. Employing the Galerkin discretization approach and applying a modification to the Volmir assumption, a reduced order form of the original system is determined, which is shown to be a great representative of the original system with much more computational advantages. Using the method of multiple scales, the reduced order nonlinear governing equations are solved to discover the primary resonance, forced vibration characteristics and stability margins of thin-walled shells/panels under the action of parametric excitation. Several interesting findings on nonlinear behavior of cylindrical shells and spherical panels (as benchmark geometries) are presented while giving a demonstration of the accuracy and computational advantages of the proposed model.

Higher order mKdV breathers: nonlinear stability

We are interested in stability results for breather solutions of the 5th, 7th and 9th order mKdV equations. We show that these higher order mKdV breathers are stable in $H^2(\R)$, in the same way as \emph{classical} mKdV breathers. We also show that breather solutions of the 5th, 7th and 9th order mKdV equations satisfy the same stationary fourth order nonlinear elliptic equation as the mKdV breather, independently of the order, 5th, 7th or 9th, considered.

Transition Threshold for the 3D Couette Flow in a Finite Channel

In this paper, we study nonlinear stability of the 3D plane Couette flow ( y , 0 , 0 ) (y,0,0) at high Reynolds number R e {Re} in a finite channel T × [ − 1 , 1 ] × T \mathbb {T}\times [-1,1]\times \mathbb {T} . It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity v 0 v_0 satisfies ‖ v 0 − ( y , 0 , 0 ) ‖ H 2 ≤ c 0 R e − 1 \|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1} for some c 0 > 0 c_0>0 independent of R e Re , then the solution of the 3D Navier-Stokes equations is global in time and does not transit away from the Couette flow in the L ∞ L^\infty sense, and rapidly converges to a streak solution for t ≫ R e 1 3 t\gg Re^{\frac 13} due to the mixing-enhanced dissipation effect. This result confirms the threshold result obtained by Chapman via an asymptotic analysis(JFM 2002). The most key ingredient of the proof is the resolvent estimates for the full linearized 3D Navier-Stokes system around the flow ( V ( y , z ) , 0 , 0 ) (V(y,z), 0,0) , where V ( y , z ) V(y,z) is a small perturbation(but independent of R e Re ) of y y .

On the existence and stability of stationary solutions for the compressible micropolar fluids

We consider the existence, uniqueness and nonlinear stability of stationary solutions for the three-dimensional compressible micropolar fluids with the external force of general form. More precisely, with the weighted L2 and L∞ estimates on solutions to the linearized problem, we show the existence and uniqueness of stationary solutions in some suitable function space by the contraction mapping principle. The stability of the steady flow is studied by an elementary energy method.

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.

Novel convergence to forced traveling waves for a nonlocal dispersal population model in a shifting environment

Some new types of the nonlinear stability of forced traveling waves for a nonlocal dispersal population model in a shifting environment are given in this paper. Firstly, when the speed of the shifting habitat edge c is larger than the spreading speed of the species c⁎, by applying the comparison principle, the Fourier transform and the L2-weighted energy estimate, we show that all forced traveling waves with the speed c>c⁎ are exponentially stable in the form of e−μt, for some μ>0. In order to extend the stability results to c=c⁎, we take another weight function and construct the different weighted Sobolev space. Then by using the Fourier transform and the L1-weighted energy estimate, we prove that all forced traveling waves with the speed c>c⁎ are exponentially stable in the form of t−12e−νt, for some ν>0, while all forced traveling waves with the speed c=c⁎ are algebraically stable in the form of t−12, which are quite different from the previous stability results.

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).

The existence and nonlinear stability of stationary solutions to outflow problem for a reduced gravity two and a half layer model

The main concern of this paper is to study large-time behavior of solutions for an outflow problem to the reduced gravity two and a half layer model in a half space. We establish the existence of stationary solutions and further obtain the large time asymptotic stability of small-amplitude stationary solutions provided that the initial perturbation is sufficiently small. Moreover, we obtain the convergence rate of the solution toward the stationary solution in some weighted Sobolev spaces only for the supersonic case. The proof is based on the basic energy method. A key point for the proof of convergence rates is to capture the positivity of the temporal energy dissipation functional and boundary terms with suitable space weight functions either algebraic or exponential depending on whether or not the incoming far-field velocity satisfy some additional conditions.

Nonlinear forced vibration and stability of an axially moving beam with a free internal hinge

Nonlinear forced vibration and stability of an axially moving beam with a free internal hinge

About the onset of the Hopf bifurcation for convective flows in horizontal annuli

Experimental and numerical results cannot yet settle whether, between horizontal coaxial cylinders, if the curvature is large, the first transition for convection is an exchange of stability or rather an Hopf bifurcation. We directly show that if the curvature tends to infinity, no periodic linear perturbation exists when the Rayleigh number is equal to the critical one for nonlinear stability.

Nonlinear dust-acoustic wave dynamics in nonthermal Saturnian E-ring with negative ion moderation

The nonlinear stability dynamics of dust-acoustic waves (DAWs) near the plasma-active satellite of Saturn, Enceladus, is herein theoretically investigated in the perturbative framework. Enceladus being the source of the materials of Saturn's one of the most diffuse and wide E-ring, has a very active plasma environment rich in diversified collective dynamics of freedom. This theoretical model formalism is developed in the light of observations made by the Radio and Plasma Wave Science (RPWS) instrument inbuilt in the Cassini spacecraft in the vicinity of Enceladus. The model consists of electrons, positive ions, and negative ions obeying nonthermal kappa distribution laws. In contrast, the constitutive massive charged dust grains are treated as inviscid fluid. A standard method of multiple scaling technique is systematically employed to yield a Korteweg-de Vries (KdV) equation on the first-order electrostatic potential fluctuations. A constructed numerical platform demonstrates that such fluctuations evolve as a solitary pulse-train or solitary wave chain of rarefactive type. The results are further compared with solitary structures detected by RPWS during Cassini's closest approach to Enceladus. It is interestingly found that the bipolar electrostatic field amplitude of the derived dust-acoustic solitary pulses (4−11 mV/m) theoretically investigated here falls well within the range of observationally obtained field value (≥10 mV/m). Finally, the non-trivial wave amplitude growth features with diverse plasma multi-parametric variations are spotlighted illustratively.

Numerical analysis of light-controlled drug delivery systems

In this paper, we solve a non-linear reaction–diffusion system with Dirichlet–Neumann mixed boundary conditions using a finite difference method (FDM) in space and the implicit midpoint method in time. This type of system appears, e.g., in the mathematical modeling of light-controlled drug delivery. One of the key results of this paper is the proof that the method has superconvergence second-order in space in a discrete H1-norm and optimal second-order convergence in time in a discrete L2-norm. Our result relies on the direct analysis of a suitable error equation, avoiding the classic construction of consistency plus stability implies convergence. One advantage of such an analysis technique is the establishment of the method’s non-linear stability in an elegant way. Numerical examples support the theoretical convergence result.

How Drivers Lose Control of the Car

<div>After a severe lane change, a wind gust, or another disturbance, the driver might be unable to recover the intended motion. Even though this fact is known by any driver, the scientific investigation and testing on this phenomenon is just at its very beginning, as a literature review, focusing on SAE Mobilus<sup>®</sup> database, reveals. We have used different mathematical models of car and driver for the basic description of car motion after a disturbance. Theoretical topics such as nonlinear dynamics, bifurcations, and global stability analysis had to be tackled. Since accurate mathematical models of drivers are still unavailable, a couple of driving simulators have been used to assess human driving action. Classic unstable motions such as Hopf bifurcations were found. Such bifurcations seem almost disregarded by automotive engineers, but they are very well-known by mathematicians. Other classic unstable motions that have been found are “unstable limit cycles.” The driving simulator results have been reproduced by experimental tests on track. We have assessed that the driver’s steering action can make the car motion unstable if a proper disturbance has acted. The delay of the driver’s steering action is the primary cause for the generation of limit cycles. Future automated vehicles should be conceived by focusing on the addressed phenomenon.</div>