Non-autonomous Ginzburg Landau Equation Research Articles

Influence of Two-Frequency Rotational Modulation on the Dynamics of the Rayleigh–Bénard Convection in Water-Based Nanoliquids with Either AA7072 or AA7075 Nanoparticles

The effect of time-periodic two-frequency rotation modulation on Rayleigh–Bénard convection in water with either AA7072 or AA7075 nanoparticles is investigated. The single-phase description of the Khanafer–Vafai–Lightstone model is used for modeling the nanoliquids. An asymptotic expansion procedure is adopted in the case of the linear stability to obtain the correction (due to modulation) to the Rayleigh number at marginal stability of unmodulated convection. A nonlinear regime of convection is considered with a nonautonomous generalized Lorenz model as the governing equation. The method of multiscales is then employed to obtain the coupled nonautonomous Ginzburg–Landau equations with cubic nonlinearity from the Lorenz model. These equations are presented in the phase-amplitude form and the amplitude is used to quantify the heat transport. The modulation amplitude is considered to be small (of order less than unity) and moderate frequencies of modulation are considered. We found that there is a threshold frequency beyond which the system behavior reverses. At frequencies below the threshold, the mean Nusselt number increases with an increase in the amplitude of modulation while an opposite influence is seen for values above the threshold. Such a behavior is a consequence of what is analogously seen in the case of the critical Rayleigh number. The influence of two-frequency modulation is more pronounced on the results of the linear and nonlinear regimes compared to that of the single-frequency one. The heat transport is enhanced due to the presence of dilute concentration of suspended nanoparticles (either AA7072 or AA7075 nanoalloys) in water. The influence of nanoparticles is to modify the threshold values generating chaos but it does not qualitatively alter the dynamical behavior of the system. The plots of Lyapunov exponents reveal that there is no possibility of hyper-chaos in the generalized Lorenz model when there is a rotational modulation.

Effect of gravity modulation on weakly nonlinear bio-thermal convection in a porous medium layer

Investigating thermal convection within porous media permeated by fluids and micro-organisms stands as a significant inquiry with broad relevance across geophysical and engineering domains. Studying convection within porous media can aid in controlling temperature and nutrient distribution for cell growth and tissue regeneration, as well as the efficiency of biofuel fermentation and production processes. Hence, the primary objective of this study is to investigate the influence of time-periodic gravitational forces on Darcy–Brinkman bio-thermal convection within a porous medium layer. This medium is saturated with a Newtonian fluid that encompasses gyrotactic micro-organisms. The gravity modulation amplitude is assumed to be very small. A weak nonlinear stability analysis is performed to analyze the stationary mode of bioconvection. The heat transport, measured by the Nusselt number, is governed by a non-autonomous Ginzburg–Landau equation. The research explores the influence of several parameters on heat transport, including the Vadaszs number, the modified bioconvective Rayleigh–Darcy number, cell eccentricity, modulation frequency, and modulation amplitude. The results are presented graphically, illustrating the impact of these parameters on heat transfer. The findings reveal that both the Vadaszs number and the modulation amplitude have a positive effect on heat transfer, enhancing the process. On the other hand, an increase in the modified bioconvection Rayleigh–Darcy number and cell eccentricity leads to a decrease in heat transfer. Furthermore, a comparison between the modulated and unmodulated systems indicates that the modulated systems have a more significant influence on the stability problem compared to the unmodulated systems. This highlights the effectiveness of external modulation in controlling heat transport within the system.

Onset of stability and heat transfer by Rayleigh-Bénard magnetoconvection with variable viscosity effect using Ginzburg-Landau model: Effect of thermal/gravity modulation

In modulating the commencement of Rayleigh-Bénard convection, thermal/gravitational modulation refers to its promising uses in various technical and scientific disciplines, including crystal formation, space laboratory research, and atmospheric convection. The aim of the study is to demonstrate the effect of thermal/gravitational modulation on the onset of stability and heat transfer of the Rayleigh-Bénard magnetoconvection with variable viscosity. The disturbance produced by the thermal/gravitational modulation is expanded in terms of the power series of the amplitude of the convection, and a non-autonomous Ginzburg-Landau equation is derived. The numerical solution of the Ginzburg-Landau equation is obtained using the classical Runge-Kutta method, which helps to determine the necessary amplitudes for the quantification of heat transfer. The examination of the data demonstrates that the commencement of convection and heat transfer can be advanced or suppressed by appropriately adjusting the problem’s parameters. The issue also shows how different parameters affect the Nusselt number, which measures the heat transfer coefficient.

Regularity of random attractors for non-autonomous stochastic discrete complex Ginzburg-Landau equations

In this paper, we consider the asymptotic behaviour of non-autonomous stochastic discrete complex Ginzburg-Landau equations with additive noise in weighted space . The existences of tempered random attractors for this equation in spaces and are proved respectively by tail estimates, which implies that the obtained -random attractor is compact and attracting in the topology of space. The main difficulty here is the lack of compactness on infinite lattices. To deal with this, we introduce a common embedding space of and and derive some tail-estimates of solutions.

Weakly Nonlinear Magnetic Convection in a Nonuniformly Rotating Electrically Conductive Medium Under the Action of Modulation of External Fields

In this paper we studied the weakly nonlinear stage of stationary convective instability in a nonuniformly rotating layer of an electrically conductive fluid in an axial uniform magnetic field under the influence of: a) temperature modulation of the layer boundaries; b) gravitational modulation; c) modulation of the magnetic field; d) modulation of the angular velocity of rotation. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number nonlinear non-autonomous Ginzburg-Landau equations for the above types of modulation were obtaned. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various types of modulation of external fields and for different profiles of the angular velocity of the rotation of electrically conductive fluid.

A Local Nonlinear Stability Analysis of Modulated Double Diffusive Stationary Convection in a Couple Stress Liquid

The non-autonomous Ginzburg-Landau equation with time-periodic coefficients is derived for two modulated double-diffusive stationary convection involving couple stress liquid. The heat and mass transports are quantified in terms of Nusselt and Sherwood numbers, which are obtained as functions of the slow time scale. Effects of Prandtl number, Lewis number, solute Rayleigh number and couple stress parameter have been discuused in detail.

Random attractors for stochastic discrete complex non-autonomous Ginzburg-Landau equations with multiplicative noise

In this paper, we study the asymptotic behavior of stochastic discrete complex non-autonomous Ginzburg-Landau equations with multiplicative noise. We prove the existence and uniqueness of the random attractor.

Nonlinear throughflow effects on thermally modulated porous medium

Abstract Effect of vertical throughflow on Darcy convection has been investigated subject to time-periodic temperature modulation of the boundaries. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small, and the disturbances are expanded in terms of power series of amplitude of convection. A weak nonlinear stability analysis has been performed for the stationary mode of convection, and heat transport in terms of the Nusselt number, which is governed by the non-autonomous Ginzburg–Landau equation, is calculated. The effect of vertical throughflow is found to be either to destabilize or stabilize the system for downward or upward throughflows in the case of impermeable boundary conditions. The effect of amplitude and frequency of modulation, Prandtl–Darcy number on heat transport has been analyzed and depicted graphically. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system.

Chaotic and oscillatory magneto-convection in a binary viscoelastic fluid under g-jitter

A weak onlinear analysis of double diffusive convection in an electrically conducting viscoelastic fluid layer heated from below, has been performed for the range of viscoelastic parameters where oscillatory mode exists. A uniform magnetic field has been applied vertically. Employing complex non-autonomous Ginzburg–Landau equation, the effects of various parameters on double diffusive convection is investigated. Weak nonlinear analysis reveals that the values of viscoelastic parameters have significant effect on the instability. The present study is to investigate the effect of time periodic gravity field on heat and mass transfer in the presence of external magnetic field, where controlling convection external to the system is important. It is found that the variation of Nusselt, Sherwood numbers with respect to the slow time becomes rapid upon increasing the solutal Rayleigh number Rs, Prandtl number Pr, time relaxation parameter λ and amplitude of modulation δ2 or decreasing on Chandrasekhar number Q, diffusivity ratio Γ, time retardation parameter ε and frequency of modulation Ω. It is found that the applied magnetic field has a stabilizing effect, hence reducing heat and mass transport in the system. Further, modulated gravity field can be used either to delay or enhance the heat and mass transfer in the system. Chaotic convection under gravity modulation has also been investigated using Lorenz model.

Heat and mass transfer for oscillatory convection in a binary viscoelastic fluid layer subjected to temperature modulation at the boundaries

Weak nonlinear hydrodynamic thermal instability analysis has been performed for double diffusive oscillatory mode of convection in a horizontal layer of viscoelastic fluid, heated from below. Employing complex non-autonomous Ginzburg–Landau equation, effects of various viscoelastic parameters on thermal instability have been investigated. The weak nonlinear analysis reveals that values of viscoelastic parameters have significant effect on the instability. The present study is to investigate the effect of time-periodic temperature modulation on heat and mass transfer, where controlling convection external to the system is important. Sinusoidal profile is taken to modulate the temperature of the boundaries. It is found that the variation of Nusselt number and Sherwood number with respect to the slow time scale becomes rapid as either increasing Rs,Pr,λ,δ or decreasing Γ,ε,Ω. Further, it is found that in-phase temperature modulation has negligible effect, while out of phase temperature modulation and only lower plate temperature modulation have oscillatory effects on heat and mass transport.

Weak nonlinear analysis of magneto–convection under magnetic field modulation

An analytic study of heat transport in an electrically conducting fluid layer is performed under a non-uniform time-dependent magnetic field. The applied vertical magnetic field consists of two parts: a constant part and a time-dependent periodic part, which varies sinusoidally with time. A weakly nonlinear theory has been considered to investigate heat transfer in the fluid layer. The heat transfer coefficient is obtained by deriving the non-autonomous Ginzburg–Landau equation for an amplitude of convection. This amplitude of convection is derived by using NDSolve Mathematica 8, and the results are verified using Runge–Kutta–Fehlberg method. The Nusselt number is obtained in terms of various system parameters and the effect of each parameter on heat transport is reported in detail. The effect of magnetic Prandtl number , amplitude of modulation δ is to enhance the heat transfer. The Chandrasekhar number Q, modulation frequency ω is to stabilize the system. Further, it is found that magnetic modulation can be used effectively in either enhancing the heat transfer or diminishing it.

Weak Nonlinear Double-Diffusive Magnetoconvection in a Newtonian Liquid under Temperature Modulation

The present paper deals with a weak nonlinear theory of double-diffusive magnetoconvection in an electrically conducting Newtonian liquid, confined between two horizontal surfaces, under a constant vertical magnetic field, and subjected to imposed time-periodic thermal boundaries. The temperature of both walls is varied time periodic in this case. The disturbances are expanded in terms of power series of amplitude of convection, which is assumed to be small. Using nonautonomous Ginzburg-Landau equation, the Nusselt and Sherwood numbers obtained analytically and studied heat and mass transport in the system. Effect of various parameters on the heat and mass transport is discussed extensively. It is found that the effect of magnetic field is to stabilize the system. Further, it is also notified that the heat and mass transport can be controlled by suitably adjusting the external parameters of the system.

Weak Nonlinear Oscillatory Convection in a Viscoelastic Fluid-Saturated Porous Medium Under Gravity Modulation

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal porous medium saturated with viscoelastic fluid. Modified Darcy’s law is used to explain characteristics of fluid motion. The time periodic gravity field has been considered, and its effect on the system has been investigated. A weak nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex non-autonomous Ginzburg–Landau equation, is calculated. The influence of relaxation and retardation times of viscoelastic fluid on heat transfer has been discussed. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Finally, it is found that overstability advances the onset of convection, and hence increases heat transfer.

Weakly nonlinear oscillatory convection in a viscoelastic fluid saturating porous medium under temperature modulation

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal porous medium saturated with viscoelastic fluid. Modified Darcy’s law is used to explain characteristics of fluid motion. The time-periodic temperature on the boundaries has been considered and its effect on the system has been investigated. A weak nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex non-autonomous Ginzburg–Landau equation, is calculated. The influence of relaxation and retardation times of viscoelastic fluid on heat transfer has been discussed. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Finally, it is found that supercritical flow advances the onset of convection hence increases heat transfer.

Weak non-linear oscillatory convection in a viscoelastic fluid layer under gravity modulation

Using a complex non-autonomous Ginzburg–Landau equation, we have investigated oscillatory Rayleigh–Bénard convection under Gravity modulation, considering a viscoelastic fluid layer. The influence of (stress) relaxation and (strain) retardation times of viscoelastic fluid on heat transfer has been discussed. The study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Modulation has a destabilization effect at low frequencies and a stabilization effect at high frequencies, which increases with increasing the amplitude of modulation. Finally, it is found that overstability advances the onset of convection, hence increases heat transfer.

Pullback Attractor for Nonautonomous Ginzburg-Landau Equation with Additive Noise

Long time behavior of stochastic Ginzburg-Landau equations with nonautonomous deterministic external forces, dispersion coefficients, and nonautonomous perturbations is studied. The domain is taken as a bounded intervalIinR. By making use of Sobolev embeddings and Gialiardo-Nirenberg inequality we obtain the existence and upper semicontinuity of the pullback attractor inL2(I)for the equation. The upper semicontinuity shows the stability of attractors under perturbations.

Heat Transport in an Anisotropic Porous Medium Saturated with Variable Viscosity Liquid Under Temperature Modulation

Thermal instability in a horizontal porous medium saturated with temperature-dependant viscous fluid has been considered, and the effect of time-periodic temperature modulation has been investigated. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small and the disturbances are expanded in terms of power series of amplitude of convection. A weak non-linear stability analysis has been performed for the stationary mode of convection, and heat transport in terms of the Nusselt number, which is governed by the non-autonomous Ginzburg–Landau equation, is calculated. The effects of thermo-rheological parameter, amplitude and frequency of modulation, thermo-mechanical anisotropies, and Vadasz number on heat transport have been analyzed and depicted graphically. It is found that an increment in the value of thermo-rheological parameter results in the enhancement of heat transport in the system. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system.

Effect of internal-heating on weakly non-linear stability analysis of Rayleigh–Bénard convection under g-jitter

In this paper, we study the combined effect of internal-heating and time-periodic gravity modulation on thermal instability in a viscous fluid layer, heated from below. The time-periodic gravity modulation, considered in this problem can be realized by vertically oscillating the fluid layer. A weak non-linear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number has been obtained in terms of the amplitude of convection which is governed by the non-autonomous Ginzburg–Landau equation derived for the stationary mode of convection. Effects of various parameters such as internal Rayleigh number, Prandtl number, and amplitude and frequency of gravity modulation have been analysed on heat transport. It is found that the response of the convective system to the internal Rayleigh number is destabilizing. Further, it is found that the heat transport can be controlled by suitably adjusting the external parameters of the system.

Effects of Time-Periodic Thermal Boundary Conditions and Internal Heating on Heat Transport in a Porous Medium

The effects of time-periodic boundary temperatures and internal heating on Nusselt number in the Benard–Darcy convective problem has been considered. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small. By performing a weakly non-linear stability analysis, the Nusselt number is obtained in terms of the amplitude of convection, which is governed by the non-autonomous Ginzburg–Landau equation, derived for the stationary mode of convection. The effects of internal Rayleigh number, amplitude and frequency of modulation, thermo-mechanical anisotropies, and Vadasz number on heat transport have been analyzed and depicted graphically. Increasing values of internal Rayleigh number results in the enhancement of heat transport in the system. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system.

Synchronous and asynchronous boundary temperature modulations of Bénard–Darcy convection

A theoretical analysis of thermo-convective instability in a densely packed porous medium is carried out when the boundary temperatures vary with time in a sinusoidal manner. By performing a weakly non-linear stability analysis, the Nusselt number is obtained as a function of amplitude of convection which is governed by a non-autonomous Ginzburg–Landau equation derived for the stationary mode of convection. The paper succeeds in unifying the modulated Bénard–Darcy, Bénard–Rayleigh, Bénard–Brinkman and Bénard–Chandrasekhar convection problems and hence precludes the study of these individual problems in isolation. A new result that shows that asynchronous temperature modulation may be effectively used to either enhance or reduce heat transport by suitably adjusting the frequency and phase-difference of the modulated temperature is presented.