Nonlinear Stability Analysis Research Articles

Pollution overturning instability in an incompressible fluid with a Maxwell-Cattaneo-Mariano model for the pollutant field

We develop a model for a pollutant dissolved in or dispersed in an incompressible Navier–Stokes fluid when the diffusion theory for the pollutant obeys a second order in time system of equations rather than the first order in time system obtained from Fourier’s law. A detailed analysis is performed for a layer of fluid where a pollutant is such that the top of the layer will be heavier in concentration. A detailed expression for the critical pollutant Rayleigh number is found indicating precise conditions under which a convective overturning motion will arise. The investigation is performed by a linear instability analysis, but additionally we provide a completely nonlinear energy stability analysis. Diffusion of flux is also added to a Cattaneo-like equation and this leads to surprising results.

Control of chaos in Darcy–Bénard axisymmetric convection in a cylindrical enclosure using a uniform vertical cross-flow

The linear and weakly nonlinear stability analyses of Darcy–Bénard convection of a Newtonian fluid experiencing a uniform vertical cross-flow is investigated in the paper for various aspect ratios. Making use of the Maclaurin series representation, an expression for axial eigenfunctions is obtained with the radial eigenfunction being a Bessel function of first kind. These eigenfunctions are influenced by the Peclet number, Pe, the non-dimensional number that signifies the rate of vertical cross-flow. The modified-Vadasz-Lorenz model obtained in this paper has newly defined non-dimensional parameters that capture the influence of vertical cross-flow. From the linear stability analysis, it is found that the effect of introducing vertical cross-flow is to stabilize the system. Using a weakly nonlinear stability analysis, the closed-form expression of the Hopf-Rayleigh number as a function of Pe is obtained. Furthermore, the behavior of the modified-Vadasz–Lorenz model is analyzed using the largest Lyapunov exponent and the bifurcation diagram. This gives information about the intensity of chaos and occurrence of the periodic motion. We observe that the influence of vertical cross-flow is to increase the value of the Hopf–Rayleigh number and thereby to delay the onset of chaos. Furthermore, the appearance of the first periodic point is preponed due to the vertical cross-flow. As the rate of vertical cross-flow increases, the intensity of chaos decreases, thereby indicating that the effect of introducing vertical cross-flow is to suppress chaos.

An analytical investigation on nonlinear vibrations and stability of Timoshenko pipes conveying two-phase flow

The primary objective of this study is to conduct a comprehensive investigation on the nonlinear free vibration and stability analysis of the multilayer graphene reinforced pipe conveying liquid–gas two–phase flow based on the Timoshenko beam model. The Halpin–Tsai micromechanics model is employed which provides a means to evaluate and analyze the mechanical behavior and performance of the multilayer FG reinforced pipe. According to Von–Kármán strain relations and applying the Hamilton's principle, the governing equations of motion are derived. The Galerkin technique is operated to convert the partial differential equation (PDE) to the ordinary differential equation (ODE). The homotopy analysis method is employed to solve the fourth order nonlinear differential equation. Closed–form expressions for the nth nonlinear frequency and time history as well as the critical flow velocity based on the Timoshenko theory are achieved. The results demonstrate that neglecting the effects of rotary inertia and shear deformation may cause some errors by the increment of the initial amplitude. Furthermore, the critical flow velocity, depends not only on the mechanical and geometrical properties of the pipe but also on the initial mode shape excitation. The results indicate that by the increment of flow velocity, liquid density and slenderness ratio, the nonlinear frequency decreases, on the other hand, by the increment of the initial amplitude, the nonlinear frequency increases.

Improved Car-Following Model for Connected Vehicles on Curved Multi-Lane Road

Under the development of intelligent network technology, drivers can obtain the surrounding traffic situation in real time, which is conducive to improving the stability of traffic flow. Therefore, this paper proposes a new curve-car-following model considering multi-vehicle information of adjacent lanes in connected environment, and conducts linear and nonlinear stability analyses of the model to demonstrate the effectiveness of the proposed model and its ability to improve the stability of traffic system; in addition, numerical simulation experiments of traffic flow convoys are designed to analyze the effects of different parameters in the proposed model on the stability of the traffic flow and test the proposed model’s ability to maintain the following behavior in a convoy. Furthermore, numerical simulation experiments are designed to analyze the effects of different parameters in the proposed model on the stability of traffic flow, and to test the ability of the proposed model to maintain the following behavior in the convoy. The model can provide theoretical guidance to alleviate traffic congestion and improve safety, and extend the application of the following model in curved multi-lane road scenarios.

Numerical study of linear and nonlinear stability in double-diffusive Hadley-Prats flow through a horizontal porous layer with Soret and internal heat source effects

Numerical analysis is constructed to study the onset of the double-diffusion convection mechanism through an infinite parallel permeable channel with heat generation, mass flow and Soret impacts. Homogenous isotropic Darcy’s flow model is deployed to elucidate the porous features. The roll instabilities pertaining to longitudinal and transverse cases are examined through a linear and nonlinear stability analysis. The unit-less eigenvalue problem is constructed through linear and nonlinear stability assumptions and which is solved numerically using a fourth order Runge-Kutta scheme. The critical values of wave and thermal Rayleigh parameters are evaluated. Extended graphical and tabular visualization is presented to describe the onset of the convection mechanism. The results of this semi-numerical investigation may be useful in environmental, geothermal and chemical engineering processes. In addition, the critical R z value is also determined for a range of thermo-physical numbers. Significant modifications in the flow patterns are computed with mass flow parameter and vertical thermal and solutal Rayleigh number. With an increment in Soret (thermo-diffusive) parameter Sr the regime become more unstable. Based on the nonlinear stability analysis, an elevation in solutal Rayleigh number also induces earlier instability. The collective influence of the Lewis number and horizontal concentration parameter is observed to render the system more stable.

On the nonlinear wave structures and stability analysis for the new generalized stochastic fractional potential-KdV model in dispersive medium

On the nonlinear wave structures and stability analysis for the new generalized stochastic fractional potential-KdV model in dispersive medium

Unsteady double-diffusive Brinkman–Bénard convection in cylindrical enclosure saturated with hybrid bi-viscous Bingham nanoliquid

This paper aims to present an analytical as well as a comparative study to investigate the effect of the Brinkman porous medium on the onset of regular and chaotic motion in cylindrical enclosures of different heights. The hybrid bi-viscous Bingham nanoliquid is considered as the working fluid. Modified Brinkman–Buongiorno and bi-viscous Bingham fluid models are incorporated to obtain the flow governing dynamics. The thermophysical properties of the hybrid nanoliquid are calculated using phenomenological laws and the mixture theory. The study is carried out for the axisymmetric mode, and the Bessel functions are taken as the eigenfunctions of the problem. Double Fourier–Bessel series expansions are used for weakly non-linear stability analysis. The limiting cases of the study are obtained, and the results on the onset of convection, heat, and mass transport are discussed graphically. The behavior of the dynamical system is analyzed using the maximum Lyapunov exponent plot, the bifurcation diagram, and phase plots. Outcomes suggest that convection sets in earlier in the water-based hybrid nanoliquid than in the bi-viscous Bingham hybrid nanoliquid. The use of Single-walled carbon nanotubes enhances the heat transfer rate by approximately 17%. Further, it is concluded that a tall cylindrical enclosure is the most favorable geometry for achieving a higher heat transfer rate among the others.

Weakly nonlinear stability analysis of non-isothermal parallel flow in a vertical porous annulus

The study examines the stability of parallel airflow within a tall vertical coaxial circular annulus filled with a highly permeable porous medium under stable stratification. The linear stability theory predicts that the least unstable mode can be either axisymmetric or non-axisymmetric, depending on the values of the governing parameters. The curvature parameter, defined as the gap between coaxial circular cylinders, contributes to the stabilization of the flow. Depending on various controlling parameter values, the stably stratified non-isothermal annular parallel flow (SNAPF) can manifest three instability modes: thermal-shear, thermal-buoyant, and mixed or interactive. The instability type can change abruptly with Reynolds number, curvature parameter, and media permeability. A finite-amplitude instability analysis is examined using weakly nonlinear stability theory. The analysis demonstrates both subcritical and supercritical bifurcations near the instability boundary. The range of Reynolds numbers for subcritical bifurcation expands when switching from axisymmetric to non-axisymmetric disturbance. Bifurcation transitions between supercritical and subcritical occur due to changes in the least stable azimuthal mode. The study also investigates the equilibrium/threshold amplitude within the supercritical/subcritical instability regime, and it verifies the existence of subcritical/supercritical bifurcation through nonlinear kinetic energy balance. Furthermore, the finite-amplitude analysis anticipates that the lower critical Rayleigh number increases as the Reynolds number, curvature parameter, or media permeability decreases. An increase in the Reynolds number shifts the point of inflection in the velocity profile toward the inner cylinder, which subsequently leads to a reduction in the size of the core vortex region. In the subcritical instability regime, the radius of the primary vortex cell decreased significantly as the Rayleigh number increased. When the curvature is altered, a structural transition occurs in the isotherms from multicellular to unicellular.

Global nonlinear stability of bidispersive porous convection with throughflow and depth-dependent viscosity

The linear instability and the nonlinear stability analyses have been performed to examine the combined impact of a uniform vertical throughflow and a depth-dependent viscosity on bidispersive porous convection using the Darcy theory with a single temperature field. The validity of the principle of exchange of stability is proved. The eigenvalue problems resulting from both linear instability and nonlinear stability analyses with variable coefficients are numerically solved using the Chebyshev pseudo-spectral method. The equivalence of linear instability and nonlinear stability boundaries is established in the absence of throughflow, while in its presence, the subcritical instability is shown to be evident. The stability of the system is independent of the direction of throughflow in the case of constant viscosity, whereas upflow is found to be more stabilizing than downflow when the viscosity is varying with depth. While the viscosity parameter offers a destabilizing influence on the onset of convection in the absence of throughflow, it imparts both stabilizing and destabilizing effects on the same in its presence. The influence of the ratio of permeabilities and the interphase momentum transfer parameter is to make the system more stable. The findings of a mono-disperse porous medium are presented as a specific case within the broader context of this investigation.

Thermal convection with a Cattaneo heat flux model

The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.

EFFECTS OF ROTATIONAL MODULATION ON CONVECTION IN ETHYLENE GLYCOL-BASED HYBRID NANOFLUIDS WITH INTERNAL HEATING

The rotational modulation effects on Rayleigh-Bénard convection in ethylene glycol-based hybrid nanofluids with internal heating are investigated. Due to their improved thermophysical properties as compared to base fluid, nanofluids are frequently used in numerous heat transfer applications. Hybrid nanofluids with suitable nanoparticle combinations can have better thermophysical characteristics than mono nanofluids. As a result, this study investigates the impact of hybridizing the base fluid on system stability and heat transfer. A single-phase model is employed to perform a linear and weakly nonlinear stability analysis of the nanofluid. The nonautonomous Ginzburg-Landau equation is derived and solved, and the solution is used to obtain the Nusselt number expression. Based on the linear analysis, the critical Rayleigh number attained in the case of hybrid nanofluids is less than the value found in the case of mono nanofluids. Therefore, the convection onset is faster in a hybrid nanofluid than in a mono nanofluid. The study further shows that hybrid nanofluid ethylene glycol-alumina-copper increases the heat transportation rate as compared to the mono nanofluid ethylene glycol-alumina, presenting evidence that hybrid nanofluid facilitates heat transfer better than the mono nanofluid. Increasing the volume of hybrid nanoparticles qualitatively improved heat transfer by up to 5.96%. Further, the effects of important fluid parameters on heat transfer are presented. Among other results, we found that increasing the modulation's amplitude improves heat transmission in the hybrid nanofluid.

Thermal instability of Tri-hybrid Casson nanofluid with thermal radiation saturated porous medium in different enclosures

In this paper, authors have studied the thermal instability of Tri-hybrid Casson nanofluid with thermal radiation saturated porous medium within four different types of enclosures using both linear and nonlinear ways. Linear stability analysis is done using the normal mode approach, whereas nonlinear stability analysis is done using the truncated Fourier series method. The results demonstrate that the porosity, thermal radiation, and porous parameter stabilize the system. On the other hand, the Casson parameter advances the initiation of convective motion in the system. To understand the behaviour of mass/heat transfer in the system two different approaches viz Mathematica NDSolve and Runge–Kutta–Fehlberg method (RKF-45) are used, and it is found that the rate of heat/mass transfer is exactly the same in both cases. The study examines three distinct enclosures and compares their effects on the onset of convection and heat/mass transport. In comparison of different enclosures the tall enclosures have a quicker onset of convection other than square and shallow enclosures. The tall enclosure accelerates mass/heat transfer more quickly than the square and shallow enclosure. The results are illustrated graphically and in tabular form.

Modeling vegetation patterning on sloped terrains: The role of toxic compounds

Vegetation patterning processes taking place on sloped terrains are here investigated through a class of three-compartments 1D reaction–advection–diffusion models enclosing the effects associated with the presence of toxic compounds. Focus is given to the transition from a uniformly-vegetated area to the desert state which occurs through the formation of an intermediate state characterized by non-stationary vegetation stripes. To this aim, linear stability analysis is addressed to characterize the mechanism of wave instability, responsible for the emergence of oscillatory Turing patterns. In detail, the analysis provides the critical value of the main control parameter as well as the wavelength and the migration speed at the onset of instability. Moreover, multiple-scale weakly nonlinear stability analysis is performed to describe the time evolution of the pattern amplitude close to the bifurcation threshold. Theoretical investigations are complemented by numerical simulations carried out for an extension of the Klausmeier kinetics that explicitly takes into account the interaction between autotoxicity and vegetation biomass. Numerical results provide several insights on how the interplay among mean annual rainfall, plant mortality and plant’s sensitivity to toxicity gives rise to different ecological scenarios.

Nonlinear Vibrations and Stability Analysis of GPL Reinforced Pipes Conveying Fluid Excited by Arbitrary Initial Conditions: An Optimized Analytical Solution

In this study, nonlinear vibrations and stability of multilayer functionally graded (FG) pipes conveying fluid laying on nonlinear elastic foundation and reinforced by graphene nanoplatelets (GPLs) are investigated with an emphasis on optimized analytical method. Various types of GPL distribution patterns are considered and to estimate the mechanical characteristics of the reinforced pipe the Halpin–Tsai micromechanics theory is applied. The nonlinear differential equation of motion is obtained by using Von–Kármán strain relations in conjunction with the Euler–Bernoulli beam theory and applying Hamilton’s principle. The Galerkin technique has been used for discretizing the partial differential equation. The optimized homotopy analysis method (HAM), as a strong analytical method, is utilized to solve the nonlinear differential equation by considering different initial excitations. The convergence-control parameter is taken into account to guarantee the convergence of the analytical solution. In this study, an exact solution based on HAM for the critical flow velocity and [Formula: see text]th nonlinear frequency is presented. Additionally, series solutions for the nonlinear time responses of the transverse and longitudinal vibrations of fluid-conveying pipe based on optimized HAM are obtained for the first time. In the numerical results, the effects of arbitrary initial conditions and different physical characteristics on the nonlinear frequencies and time responses are extensively examined. It is shown that the nonlinear frequencies and critical flow velocity of the pipe depend not only on the initial excitation amplitude, but also on the entire initial excitation function applied on the pipe.

Effect of Three Types of Gravity Modulation on Rivlin Ericksen Nanofluid Saturated in Porous Medium with Internal Heating

In this paper, authors have studied the Rivlin-Ericksen elastico-viscous nanofluid thermal instability with internal heating under three types of gravity modulation in a horizontal layer of porous medium using both linear and nonlinear ways. Linear stability analysis is done using normal mode approach, whereas nonlinear stability analysis is done using truncated Fourier series method. The results demonstrate that the internal heating, modified diffusivity ratio, and concentration Rayleigh number advance the initiation of convective motion in the system. On the other hand, Darcy number stabilize the system. To understand the behaviour of mass/heat transfer in the system two different approaches viz Mathematica NDSolve and Runge-Kutta-Fehlberg method (RKF-45) are used, and it is found that the rate of heat/mass transfer is exactly the same in both cases. In comparison to other gravity modulation profiles, day-night gravity modulation profile accelerates mass/heat transfer more quickly than the other two types of gravity modulation profiles. Steady state graph has demonstrated, convection occurs earlier when internal heating and amplitude values are increased. Moreover, it is also observed that heat/mass transport is very much affected by the variation of kinematic viscoelasticity parameter.

Effect of Three Types of Magnetic-Field Modulation and Different Shapes of Nanoparticles on the Instability of Nanofluid (Al2O3 + H2O) Saturated Porous Medium with Internal Heating

In this article the authors investigate the effect of three different types of magnetic-field modulation namely triangular-wave form, sine-wave form, and square-wave form on the instability of nanofluid (Al2O3 + H2O) saturated porous medium with internal heating. Linear stability analysis is done analytically using the normal mode approach, whereas nonlinear stability analysis is done by using the truncated Fourier series method. We obtained that the internal heating parameter destabilizes the system, whereas Darcy number and magnetic Chandrasekhar number stabilizes the system. The impact of three different shapes (Sphere/Bricks/Cylinder) of nanoparticles over instability is also analyzed, and it is found that the system is most stable for sphericalshaped nanoparticles while least stable for cylindrical-shaped nanoparticles. Mass/heat transfer in the system is analyzed by two different approaches, Mathematica NDSolve and Runge-Kutta-Fehlberg method (RKF-45), and it is found that the rate of mass/heat transfer is exactly the same in both cases. The internal heating and magnetic Chandrasekhar number play a crucial role in the heat/mass transfer in the system. Moreover, it is also noticed that heat and mass transfer start earliest in the case of triangle-waveforms as compared to other types of magnetic-field modulations.

Effect of variable viscosity, porous walls and mixed thermal boundary condition on the onset of Rayleigh-Bénard convective instability

In this paper, we have studied the criteria for the onset of instability in the Rayleigh-Bénard convection problem, by considering a fluid layer confined between two infinitely extended rough boundaries, which are subjected to mixed-type thermal boundary conditions, physically representing the imperfectly conducting boundaries. The rough boundaries are assumed to be shallow porous layers with different pore size, properties, and permeabilities. Mathematically, it is given by the Saffman type boundary condition. Therefore, we have two generalized forms of boundary conditions, one for the flow field and the other for the thermal field. The two limiting cases of the mixed-type thermal boundary condition correspond to a perfectly conducting boundary and to a perfectly insulating boundary, whereas the two limiting cases of the hydrodynamic boundary condition correspond to a no-slip condition and to a free-surface condition, depending upon the limiting values of the parameters involved in these two generalized boundary conditions. The linear and nonlinear energy stability analyses are performed, and the existence of the region of subcritical instability is checked. Through the principle of exchange of stability analysis, it is found that the instability occurs only in the stationary mode. The adiabatic boundary condition is found to be more restrictive than the isothermal boundary condition. Under given conditions, the instability is observed to be occurring in the infinite wavelength mode for the case of adiabatic boundaries. In the present work, the effect of temperature and pressure dependent viscosity on the stability of the system has been shown and found to be of destabilizing nature. However, the roughness of the boundaries is found to be of stabilizing nature.

Nonlinear property and dynamic stability analysis of a novel bio-inspired vibration isolation–absorption structure

Nonlinear property and dynamic stability analysis of a novel bio-inspired vibration isolation–absorption structure

Retrieval of diverse soliton, lump solutions to a dynamical system of the nonlinear ($$4+1$$) Fokas equation and stability analysis

Retrieval of diverse soliton, lump solutions to a dynamical system of the nonlinear ($$4+1$$) Fokas equation and stability analysis

Interaction and breakdown induced by multiple optimal disturbances in hypersonic boundary layer

The present paper finds that the coexistence of multiple primary instability waves may cause a non-trivial nonlinear interaction and breakdown process, which has not been reported before. In the considered Mach 6 flat-plate boundary layer, a global resolvent analysis reports three optimal disturbances (local maxima): a high-frequency planar wave, a low-frequency oblique wave and a stationary streak. For the dominant planar and oblique waves, a parabolised stability equation analysis identifies the initial non-modal transient growth and downstream modal growth. Initiated by these two optimal disturbances jointly, the complete linear and nonlinear instability processes until breakdown to turbulence are shown with direct numerical simulation. Owing to the transient growth, the oblique wave may be more significant than the planar wave in the breakdown. The oblique wave and scales of nonlinear interactions are pronounced in the outer layer, whose significance may not be comprehensively characterised by the wall pressure measurement. Fourier modes characterising the oblique-wave oblique breakdown, the planar-wave fundamental resonance, the planar-wave subharmonic resonance and the combination resonance related to a detuned mode are observed successively. The detuned mode seems to dominate the near-wall dynamics in the late nonlinear stage, characterised by $\varLambda$ -like structures. Meanwhile, the existence of this detuned mode is independent of the initial amplitude ratio and the absolute amplitude of the oblique and planar waves. Weakly nonlinear stability analyses demonstrate that the detuned mode is mainly a consequence of the secondary instability under the combination of planar and oblique primary waves. Wave vector plots reveal the resonant state of multiple triads. Energy budget and amplitude-correction analyses provide a clear physical image of energy transfer.