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This paper is devoted to the investigation of the stability of plane-parallel flow in a vertical fluid layer with uniformly distributed heat sources in modulated gravity field. The layer boundaries are rigid and maintained at equal constant temperatures. Gravity is assumed to be vertical and consisting of both mean and sinusoidal modulation (‘jitter’). Specific feature of this problem is that in the absence of modulation, at zero Prandtl number, the decrements of normal-mode perturbations of the base state are complex-valued and hydrodynamic instability mode is caused by travelling perturbations (travelling vortices at the boundaries of counter flows). With the increase in Prandtl number the instability mode changes from hydrodynamic instability of the counter flows to growing thermal waves. In the presence of gravity modulation, the base flow is the superposition of the same stationary flow as in the absence of modulation and time-periodic flow. The linear stability of this base state is studied by the numerical solution of the linearized equations of small perturbations. Numerical data on temporal evolution of perturbations are used to determine the decrements of perturbations and instability boundaries at different values of the Prandtl number. The calculations confirm that all perturbations are quasi-periodic. Parameter ranges where modulation makes stabilizing or destabilizing effect are defined. Sharp stabilization of the base flow in low-frequency range is discovered and explained by transformation of the neutral curves with the decrease of frequency which incleds formation of a bottleneck, break into two instability regions (the isolated region of hydrodynamic instability at lower Grashof number values and bag-shaped region of thermal wave instability at higher Gr), decrease in the size of the hydrodynamic instability region and shift upward of the thermal wave instability region and vanishing the isolated region of hydrodynamic instability. [ABSTRACT FROM AUTHOR]

The present study investigates the gyrotactic microorganism flow in a rotating porous medium containing Newtonian fluid. Using gravity modulation, Darcy–Brinkman biothermal convection is examined. Linear theory describes the stationary convective mode which derives the expression for critical Rayleigh number. This indicates the onset of bioconvection. The system's marginal stability is demonstrated by graphical and tabular representation which has a good agreement with each other. The Ginzburg–Landau equation governs the Nusselt number, which is used to further explore heat transfer. The study provides an explanation and graphical representation of the effects of the following factors on heat transfer: cell eccentricity, modified Vadasz number and bioconvective Rayleigh–Darcy number, modulation frequency, and amplitude along with Taylor number. The mean Nusselt number has been plotted in the current study. The effect of rotating porous media and gravity modulation is explained in this work. Additionally, a comparison graph is plotted to examine the effects of gravity, both modulated and unmodulated, on the Nusselt number. This demonstrates how well gravity modulation on rotating porous media controls the system's heat transfer. A comparison between numerical and analytical results for unmodulated cases is also explained graphically. [ABSTRACT FROM AUTHOR]

The presented study is prepared on the triple‐diffusive convection (TDC) in the presence of an internal heat source to examine the rate of heat and mass transport. The weakly nonlinear instability analysis was used to show the effect of gravity modulation. Such phenomena have been found in space vehicle technology, wind turbines, geothermals, and many more. In the realm of geometry, our analysis encompasses the study of two parallel infinite horizontal plates with gravity in the downward direction. For TDC, we have considered two solutes. For the result validation, we have compared the critical Rayleigh number with the available research. The numerical outcomes are presented through graphs for different physical parameters. Based on the outcomes, we conclude that the Richardson, Prandtl, and Lewis numbers increase the mass and heat transport. The main finding of the present article is of the internal heat source that advances convection. [ABSTRACT FROM AUTHOR]

The current work establishes research into the non-Newtonian Casson fluid's linear and weakly non-linear stability analysis under the influences of internal heating and gravitational modulation. The study is confined to stationary convection, where marginal stability is determined by the critical Rayleigh number, which is derived from linear stability analysis. The marginal stability curves are plotted to observe the onset of convection due to different parameters that exist in the problem. Heat and mass transfer are measured in terms of the Nusselt number (Nu) and Sherwood number (Sh), respectively, in the non-linear stability analysis. These measurements are based on the Ginzburg-Landau (GL) equation. One of the main outcomes of this stability analysis is that the internal Rayleigh number and Casson parameter behave similarly for mass transport in a fluid whereas it operates oppositely for heat transfer in a fluid. [ABSTRACT FROM AUTHOR]

In this paper, the influence of gravitational modulation on weakly nonlinear biothermal convection in a porous rotating layer is investigated. We consider a layer of porous medium saturated with Newtonian fluid, containing gyrotactic microorganisms, and subject to gravitational modulation, rotation, and internal heating. To analyze linear stability, it is sufficient to represent disturbances in the form of normal modes, while nonlinear analysis includes a truncated Fourier series containing a harmonic of the nonlinear interaction. A six-dimensional nonlinear Lorentz-type model is constructed, exhibiting both reflection symmetry and dissipation. We determined heat and mass transfer using a weakly nonlinear theory based on the representation of a truncated Fourier series. Additionally, the behavior of nonstationary Nusselt and Sherwood numbers was investigated by numerically solving finite amplitude equations. Applying the expansion of regular perturbations in a small parameter to a six-dimensional model of Lorentz equations with periodic coefficients, we obtained the Ginzburg-Landau (GL) equation. This equation describes the evolution of the finite amplitude of the onset of convection. The amplitude of convection in the unmodulated case is determined analytically and serves as a standard for comparison. The study examines the effect of various parameters on the system, including the Vadasz number, modified Rayleigh-Darcy number, Taylor number, cell eccentricity, and modulation parameters such as amplitude and frequency. By varying these parameters, in different cases, we analyzed heat and mass transfer, quantitatively expressed by the Nusselt and Sherwood numbers. It has been established that the modulation amplitude has a significant effect on the enhancement of heat and mass transfer, while the modulation frequency has a decreasing effect.

The effect of gravity-field modulation is investigated in a nano liquid-confined Hele-Shaw cell. This study aims to finish the work described in (S. N. Rai, B. S. Bhadauria, K. Anish, and B. K. Singh, "Thermal instability in nanoliquid under four types of magnetic-field modulation within Hele-Shaw cell," Int. J. Heat Mass Transfer, vol. 145, no. 7, p. 072501, 2023) for oscillatory convection. The existence of the complex Ginzburg-Landau equation (CGLE) model is constrained by the requirement ω2 > 0. The magnetic fluxes in the Hele-shaw cell are governed by CGLE with g-jitter. The quantity of heat-mass transfer is examined in the presence of a g-jitter. In addition, the findings of our research on transport analysis indicate that oscillatory mode is preferable to stationary mode. It is also found that the gravity-driven Hele-Shaw layer has lower transport properties. Further, the transport analysis is compared to previous research and shown to have improved results. [ABSTRACT FROM AUTHOR]

In this paper, the influence of gravitational modulation on weakly nonlinear biothermal convection in a porous rotating layer is investigated. We consider a layer of porous medium saturated with Newtonian fluid, containing gyrotactic microorganisms, and subject to gravitational modulation, rotation, and internal heating. To analyze linear stability, it is sufficient to represent disturbances in the form of normal modes, while nonlinear analysis includes a truncated Fourier series containing a harmonic of the nonlinear interaction. A six-dimensional nonlinear Lorentz-type model is constructed, exhibiting both reflection symmetry and dissipation. We determined heat and mass transfer using a weakly nonlinear theory based on the representation of a truncated Fourier series. Additionally, the behavior of nonstationary Nusselt and Sherwood numbers was investigated by numerically solving finite amplitude equations. Applying the expansion of regular perturbations in a small parameter to a six-dimensional model of Lorentz equations with periodic coefficients, we obtained the Ginzburg-Landau (GL) equation. This equation describes the evolution of the finite amplitude of the onset of convection. The amplitude of convection in the unmodulated case is determined analytically and serves as a standard for comparison. The study examines the effect of various parameters on the system, including the Vadasz number, modified Rayleigh-Darcy number, Taylor number, cell eccentricity, and modulation parameters such as amplitude and frequency. By varying these parameters, in different cases, we analyzed heat and mass transfer, quantitatively expressed by the Nusselt and Sherwood numbers. It has been established that the modulation amplitude has a significant effect on the enhancement of heat and mass transfer, while the modulation frequency has a decreasing effect. [ABSTRACT FROM AUTHOR]

This paper investigates the impact of gravity modulation on weakly nonlinear magnetoconvection in a nanofluid layer that is nonuniformly rotating. The fundamental equations are obtained for the Cartesian approximation of the Couette flow using the Boussinesq approximation and gravitational modulation. The weakly nonlinear regime is analyzed using the method of perturbations with respect to the small supercritical parameter of the Rayleigh number, considering the effects of Brownian motion and thermophoresis in the nanofluid layer. Heat and mass transfer are evaluated in terms of finite amplitudes and calculated from the Nusselt numbers for the fluid and the volume concentration of nanoparticles. The findings demonstrate that gravitational modulation, nonuniform rotation, and differences in the volume concentration of nanoparticles at the layer boundaries can effectively control heat and mass transfer. Additionally, the negative rotation profile has a destabilizing effect. The study shows that the modulated system conveys more heat and mass than the unmodulated system.

The current paper considers the Boussinesq-Stokes suspensions and the temperature-dependent viscosity with the influence of gravity modulation to analyze a weak non-linear stability problem of Rayleigh Benard magnetoconvection. In the study of convective instability problems, the impact of time-periodic body force also known as gravity modulation or g-gitter is essential. In the problem of gravity modulation, the gravity field has two components: one is the constant part and another an externally imposed time periodic part, which can be produced by oscillating the fluid layer. The effect of varying frequency of gravitational oscillation on convection is examined. The truncated form of the Fourier series is used in the non-linear analysis. The effects of numerous factors on the onset of convection have been discussed in this paper. The thermal Nusselt number is computed and shown for slow time periods using non-linear theory. The impacts of gravity modulation frequency and amplitude have been investigated in order to study heat transport in the system, as well as other aspects that exist in the problem. [ABSTRACT FROM AUTHOR]

This paper investigates the effect of gravity modulation on Rayleigh–Bénard convection using the rigid isothermal boundary conditions. We calculate heat transfer results using the Nusselt and mean Nusselt numbers through the finite-amplitude of convection, which we got from the Ginzburg–Landau equation (GLE). The Ginzburg–Landau equation is derived analytically from the Fredholm solvability condition at third order. The finite amplitude equation (GLE) is a function of system parameters and solved numerically. The gravity modulation considered in terms of steady and sinusoidal parts. The sinusoidal part defines gravity modulation in terms of amplitude and frequency. Our study shows that gravity modulation controls the heat transfer results. The amplitude of modulation enhances heat transfer for low frequencies and diminishes for high frequencies. Further, we found that rigid isothermal boundary conditions are diminishing heat transfer than free and isothermal boundaries. Finally, we concluded that rigid isothermal boundary conditions and gravity modulation controls heat transfer results. [ABSTRACT FROM AUTHOR]

The present study performed three-dimensional numerical simulations of natural convection generated by nanofluids between plates under microgravity with gravity modulation. The Rayleigh number was fixed at 3 × 10 6 , and the dimensionless frequency of the gravity modulation was varied from F = 0 to 102 for the volume fraction ϕ = 0 –0.05. From an unsteady analysis, we clarified the time variation of three-dimensional thermal convection structures under gravity modulation and the frequency response of the heat transfer characteristics to the gravity modulation. When the frequency of gravity modulation and the volume fraction of nanoparticles are changed, two- or three-dimensional roll-shaped thermal convection occurs, and vortex pairs form between thermal plumes. For a low volume fraction of ϕ = 0.01 , the fluid follows the low-frequency gravity modulation of F = 1 well. Therefore, the thermal convection repeatedly develops and decays as the gravity acceleration increases and decreases. At high-frequency gravity modulation of F = 10 2 , since the followability of the fluid is poor, the convection remains during one cycle of gravity modulation. At the low frequency of F = 1 and low volume fraction of ϕ = 0.01 , the instantaneous Nusselt number is 1.0–1.1 times higher than that at ϕ = 0 over one period. For the time-averaged characteristics, the heat transfer coefficient is lower at low frequencies than at high frequencies regardless of the volume fraction. At the low volume fraction of ϕ = 0.01 , the average Nusselt numbers for the low frequencies of F = 1 and F = 5 are 1.0–1.01 times higher than that at ϕ = 0 , and the decrease in heat transfer coefficient owing to nanoparticles is suppressed. At ϕ = 0.05 for each frequency, the Nusselt number is much lower than that for ϕ = 0. Therefore, for low-frequency gravity modulations that make damping difficult, using nanofluids with a low volume fraction of ϕ = 0.01 can prevent the decrease in the heat transfer coefficient. [ABSTRACT FROM AUTHOR]

Effect of different boundaries on the gravity-modulated Rayleigh–Bénard convection has been investigated with an emphasis on rigid–free boundaries. Small-amplitude and large-amplitude modulations are studied using the linear stability analysis. The modified Venezian approach is used to study small-amplitude modulations using different modes of perturbations and the superposition principle. The existence of subharmonic motions for the case of large-amplitude modulations was explored using the Mathieu equation arising from the linear stability analysis. Floquet theory was used together with Hill’s infinite determinant method to compute the critical Rayleigh number for the case of large-amplitude modulations. Weakly non-linear analysis is performed leading to the cubic Stuart–Landau equation from the Lorenz system. Heat transport was quantified using the Nusselt number and the mean Nusselt numbers for different amplitudes and frequencies. It was found that gravity modulation has, in general, a stabilizing effect on the convection process in all three boundary types, and the heat transport was found to be an increasing function of amplitude. Another important outcome of the study is that the critical Rayleigh number for the onset of convection for rigid–free boundaries lies between those of the corresponding values of the free–free and rigid–rigid boundaries in the case of both harmonic and subharmonic motions which could be exploited in controlling convection. [ABSTRACT FROM AUTHOR]

This communication studies the importance of varying the radius D p of Copper nanoparticles for microgravity-modulated mixed convection in micropolar nanofluid flux under an inclined surface subject magnetic field and heat source. In the current era, extremely pervasive modernized technical implementations have drawn attention to free convection governed by g-jitter force connected with microgravity. Therefore, fixed inter-spacing of nanoparticles and effects of g-jitter on the inclined surface are taken into consideration. A mathematical formulation based on conservation principles was non-dimensionalized by enforcement of similarity transformation, yielding a related set of ODEs. The convective non-linearity and coupling, an FE discretization, was implemented and executed on the Matlab platform. The numerical process' credibility was ensured for its acceptable adoption with the defined outcomes. Then, the computational endeavor was continued to elucidate the impacts of various inputs of D p , the amplitude of modulation ϵ , material parameter β , mixed convection parameter λ , inclination angle γ , and magnetic parameter M. The enlarging size of nanoparticles accelerated the nanofluid flow due to the depreciation of viscosity and receded the fluid temperature by reducing the surface area for heat transportation. The modulated Nusselt number, couple stress, and skin friction coefficient are significantly affected by the variation of D p , M, β , λ , and ϵ. These results would benefit experts dealing with upper space transportation and materials' performance, such as the effectualness of chemical catalytic reactors and crystals. [ABSTRACT FROM AUTHOR]

The present paper deals with the theoretical study of a weakly non-linear Rayleigh–Bénard convection in temperature-dependent viscosity Newtonian liquids with temperature/gravity modulation. The effect of temperature-dependent variable viscosity and temperature/gravity modulation renders the problem analytically intractable. The imposed thermal boundary condition and gravity field consist of a steady part and a time-periodic oscillatory part as described by Venezian. The instability of convective rolls due to temperature/gravity modulation is expanded in terms of the power series of the amplitude of convection, which is quite small. The Ginzburg–Landau equation derived in this problem is a non-autonomous differential equation and is solved using the numerical technique. The authors have demonstrated that one can suppress or enhance heat transfer by tuning the parameters involved. The effect of different parameters on the heat transfer and streamlines is also studied. [ABSTRACT FROM AUTHOR]

The effect of three different types of time periodic modulations on the Rayleigh–Bénard magnetic system involving Newtonian nanoliquids is studied. Multiple-scale analysis (homogenization method) is used to arrive at the Ginzburg–Landau equation. The curiosity in the work is to know the individual effects of (1) rotation, (2) gravity and (3) temperature modulations on Rayleigh–Bénard magnetoconvection in weakly electrically conducting Newtonian nanoliquids. A significant effort in this research is devoted toward linear and nonlinear stability analyses as well as the homogenization method which leads to the Ginzburg–Landau evolution equation. Although several studies have concluded similar results for nanoliquids compared with those of pure base fluids, many fundamental issues like the choice of phenomenological models for the thermo-physical properties and "the" best type of nanoparticles are not well understood. This research focuses on several important issues involving mathematical and computational problems arising in heat transfer analysis in the presence of nanoliquids. Effects of various nanoliquid parameters, frequency and amplitude of modulation on heat transport are analyzed. This investigation focuses on five nanoliquids, with water as a carrier liquid and five nanoparticles, viz. copper, copper oxide, silver, alumina and titania. Enhanced heat transport was observed for rotation, gravity and temperature modulations. In the case of rotation modulation, it is found that increase in the amplitude of modulation results in a decrease in the critical Rayleigh number and thereby to an increase in the mean Nusselt number. The increase in the amplitude of the gravity modulation is shown to enhance the heat transport, whereas increase in frequency is to inhibit the heat transport. Two types of temperature modulations are considered, viz. in-phase (synchronous) and out-of-phase (asynchronous) temperature modulations with the assumption that the boundary temperatures vary sinusoidally with time. The amplitudes of modulation are considered to be very small. In the case of in-phase modulation, there is no significant difference between the heat transports in the presence and in the absence of temperature modulation. On this reason, out-of-phase temperature modulation is used to either enhance or diminish heat transport by suitably adjusting the frequency and phase difference of the modulated temperature. The effect of magnetic field, in all three cases of modulations, is to inhibit the onset of convection and thereby diminish the heat transport. [ABSTRACT FROM AUTHOR]

This paper sheds light on the impact of vertical oscillations (or gravity modulation) on triple-diffusive convection in a viscoelastic fluid using the Oldroyd-B model, in the presence of cross effects. Cross effects can significantly impact three-component convective systems, despite having small magnitudes. When the cross terms, indicating coupled molecular cross-diffusion of the mixture components, are included in the equations governing heat and species transport, then a deviation from the usual three-component convection process is observed. An analytical solution has been found using linear and nonlinear analysis. The conditions for the onset of convection have been obtained using the linear analysis, which is based on the perturbation technique and the Venezian method. In nonlinear analysis, the expressions for Nusselt and Sherwood numbers, which quantify the rate of heat and mass transport respectively, are obtained by deriving the Lorenz model. It has been found that the onset of convection and heat and mass transport can be controlled by choosing the appropriate values of the parameters. [ABSTRACT FROM AUTHOR]

This paper deals with the linear stability analysis and weakly non-linear analysis of Magneto-convection in a sparsely packed porous medium with constant vertical Magnetic field and gravity modulation. A linear stability analysis reported here and shows that the gravity modulation has significant effect on the stability limits of the system. The gravity modulation is know to have effect and is treated by a perturbation expansion in powers of the amplitude of modulation. The shift in the critical Rayleigh number is evaluated and depends on the prandtl number and frequency of modulation, using the Venezian method. It is also shown that the onset of convection can be advance or delay by the regulation of various parameters. Weakly nonlinear analysis is performed based on the method of power series, where the disturbance is expressed in terms of power series. A nonlinear Ginzburg-Landau equation to investigate the three different types of gravity modulation on heat transfer is derived as part of this work. Heat transfer have been shown to depend on Nusselt number, further the effect of different types of parameter on heat transport have been studied graphically. Nusselt number graph is also shown for different parameter and explain in detail. The effect of magnetic prandtl number and Chandrasekhar number are stabilize the system. The control of convection is a major issue in systems with fluids as a working media. This is all the more difficult if the fluid system is housed in a porous medium. The paper presents three mechanisms of controlling onset of convection and thereby the heat transfer in such fluid systems. In order the modulation effect is effective in its role, we have considered the system to be a fluid-saturated porous media.

This communication studies the importance of varying the radius Dp of Copper nanoparticles for microgravity-modulated mixed convection in micropolar nanofluid flux under an inclined surface subject magnetic field and heat source. In the current era, extremely pervasive modernized technical implementations have drawn attention to free convection governed by g-jitter force connected with microgravity. Therefore, fixed inter-spacing of nanoparticles and effects of g-jitter on the inclined surface are taken into consideration. A mathematical formulation based on conservation principles was non-dimensionalized by enforcement of similarity transformation, yielding a related set of ODEs. The convective non-linearity and coupling, an FE discretization, was implemented and executed on the Matlab platform. The numerical process’ credibility was ensured for its acceptable adoption with the defined outcomes. Then, the computational endeavor was continued to elucidate the impacts of various inputs of Dp, the amplitude of modulation ϵ, material parameter β, mixed convection parameter λ, inclination angle γ, and magnetic parameter M. The enlarging size of nanoparticles accelerated the nanofluid flow due to the depreciation of viscosity and receded the fluid temperature by reducing the surface area for heat transportation. The modulated Nusselt number, couple stress, and skin friction coefficient are significantly affected by the variation of Dp, M, β, λ, and ϵ. These results would benefit experts dealing with upper space transportation and materials’ performance, such as the effectualness of chemical catalytic reactors and crystals.

This communication explores the significance of suction/injection for gravity modulation mixed convection in micropolar fluid flow due to an inclined sheet in the presence of magnetic field and thermal radiation. In recent years, very extensive modern technological applications have rise the interest in mixed convection controlled by g-jitter forces associated with microgravity. The novelty of the present study is effects of g-jitter on dynamics of micropolar fluid adjustable inclination to the sheet taken into account. Mathematical formulation based on usual laws of conservation is non dimensionalized with emerging parameters through implementation of similarity transform to yield a corresponding set of partial differential equations. In the face of convective non linearity combined with coupling due to mixed convection and micropolarity, a finite element discretization is harnessed to be coded and run on Matlab platform. The credibility of numerical procedure is assured for its acceptable adoption with the established results. The growing strength of amplitude of modulation is showing proportional decrease and increase in the fluctuation of heat transfer and skin friction coefficients, and the fluctuation of reduced skin friction factor, couple stress, and Nusselt number improves with larger inputs of amplitude. These findings would be helpful to experts dealing with upper space heat transportation, performance of materials such as crystals and effectiveness of chemical catalytic reactors.

In this study, the analysis of nonlinear stability with viscoelastic ferromagnetic fluids as working media is performed by finite‐amplitude perturbations. The solution of the resulting nonautonomous system of the Lorenz model (generalized Khayat–Lorenz model of four modes) is obtained numerically to measure the amount of heat transport. The effects of elastic parameters, Prandtl number, modulation parameters, buoyancy magnetic parameter, and nonbuoyancy magnetic parameter on heat transport are studied. Heat transport is quantified through the average Nusselt number, which is determined by solving the scaled Lorenz model. As limiting cases of the study, the results of Newtonian, Maxwell, Rivlin–Ericksen fluids are determined. The results obtained have been presented graphically. [ABSTRACT FROM AUTHOR]

The present article aims at investigating the effect of gravity modulation on chaotic convection of a viscoelastic fluid in porous media. For this, the problem is reduced into Lorenz system (non-autonomous) by employing the truncated Galerkin expansion method. The system shows transitions from periodic to chaotic behavior on increasing the scaled Rayleigh number R. The amplitude of modulation advances the chaotic nature in the system while the frequency of modulation has a tendency to delay the chaotic behavior which is in good agreement with the results due to [1]. The behavior of the scaled relaxation and retardation parameter on the system is also studied. The phase portrait and time domain diagrams of the Lorenz system for suitable parameter values have been used to analyze the system.

In the present paper, a weakly nonlinear stability analysis is used to analyze the effect of time-periodic concentration/gravity modulation on mass transport. We have considered an infinite horizontal fluid layer with constant appliedmagnetic flux salted from above, subjected to an imposed time-periodic boundary concentration (ITBC) or gravity modulation (ITGM). In the case of ITBC, the concentration gradient between the plates of the fluid layer consists of a steady part and a time-dependent oscillatory part. The concentration of both walls is modulated. In the case of ITGM, the gravity fleld consists of two parts: a constant part and an externally imposed time periodic part, which can be realized by oscillating the fluid layer. We have expanded the infinitesimal disturbances in terms of power series of an amplitude of modulation, which is assumed to be small. Ginzburg-Landau equation is derived for dinding the rate of mass transfer. Effect of various parameters on the mass transport is also discussed. It is found that the mass transport can be controlled by suitably adjusting the frequency and amplitude of modulation.

In this research article, we investigated the weakly non-linear effect of gravity modulation for the temperature dependent viscous fluid in a horizontal porous layer in the presence of internal heat source. We use power series expansion in terms of the amplitude of gravity modulation, which is considered to be small for double-diffusive convection in porous media. We graphically show the effect of internal heat source, solute Rayleigh number, Lewis number, Vadász number, thermo-rheological parameter, the amplitude of gravity modulation, the frequency of modulation on the heat and mass transfer using Ginzburg-Landau equation. The effect of gravity modulation is found significant and is more effective for the low values of frequency of modulation.

This paper deals with the linear stability analysis and weakly non-linear analysis of Magneto-convection in a sparsely packed porous medium with constant vertical Magnetic field and gravity modulation. A linear stability analysis reported here and shows that the gravity modulation has significant effect on the stability limits of the system. The gravity modulation is know to have effect and is treated by a perturbation expansion in powers of the amplitude of modulation. The shift in the critical Rayleigh number is evaluated and depends on the prandtl number and frequency of modulation, using the Venezian method. It is also shown that the onset of convection can be advance or delay by the regulation of various parameters. Weakly nonlinear analysis is performed based on the method of power series, where the disturbance is expressed in terms of power series. A nonlinear Ginzburg-Landau equation to investigate the three different types of gravity modulation on heat transfer is derived as part of this work. Heat transfer have been shown to depend on Nusselt number, further the effect of different types of parameter on heat transport have been studied graphically. Nusselt number graph is also shown for different parameter and explain in detail. The effect of magnetic prandtl number and Chandrasekhar number are stabilize the system. The control of convection is a major issue in systems with fluids as a working media. This is all the more difficult if the fluid system is housed in a porous medium. The paper presents three mechanisms of controlling onset of convection and thereby the heat transfer in such fluid systems. In order the modulation effect is effective in its role, we have considered the system to be a fluid-saturated porous media. [ABSTRACT FROM AUTHOR]

The paper deals with the study of effect of gravity modulation on double-diffusive convection in a dielectric liquid for the cases of rigid-rigid and free-free boundaries. Using a modified Venezian approach, expressions for the Rayleigh number and its correction are determined. Fourier–Galerkin expansion is employed for a weakly nonlinear stability analysis and this results in a fifth-order Lorenz system that retains the structure of the classical one in the limiting case. A local nonlinear stability analysis using the method of multiscales leads to the time-periodic Ginzburg–Landau equation from the time-periodic generalized Lorenz system and the numerical solution of this simpler equation helps in quantifying unsteady heat and mass transports. Influence of various non-dimensional parameters (Lewis number, solutal Rayleigh number, electrical Rayleigh number and Prandtl number), amplitude and frequency of gravity modulation on onset of convection and heat and mass transports is discussed. The study reveals that the influence of gravity modulation is to stabilize the system and enhance heat and mass transports. The results from free-free boundaries are qualitatively similar to that of rigid-rigid boundaries. Further, it is shown that in the case of free-free boundaries the heat and mass transports are less compared to those of rigid-rigid boundaries. [ABSTRACT FROM AUTHOR]

In this paper, linear and non-linear analysis of Double-Diffusive convection in the presence of magnetic field and gravity modulation with heat and concentration source in a micropolar fluid is studied by assuming the strength of heat and concentration source same. The expression for Rayleigh number and correction Rayleigh number are obtained using regular perturbation method. The effects of parameters on heat and mass transport is investigated using non-linear analysis by deriving eighth order Lorenz equation. It is found that coupling parameter and Chandrasekhar number stabilizes the system. Whereas internal Rayleigh number and Darcy number destabilizes the system. [ABSTRACT FROM AUTHOR]

In this study, we consider Newtonian fluid with gyrotactic microorganisms flowing through a porous medium. The effects of gravity field and internal heating are investigated on Darcy–Brinkman bio-thermal convection. The threshold Rayleigh number expression in the form of stationary mode is derived to study the onset of bioconvection. Further, heat transfer is investigated using Nusselt number, which is governed by Ginzburg–Landau equation. The influence of internal Rayleigh number, Vadasz number, modified bioconvective Rayleigh–Darcy number, cell eccentricity, modulation frequency and modulation amplitude on heat transfer is explored by the research and depicted graphically. Also the effect of the above parameters on the threshold Rayleigh number against the wave number is studied graphically. These graphical study is called as marginal stability analysis. Also, a comparative graph is plotted to study modulated and unmodulated effect of gravity on Nusselt number. This highlights the effectiveness of the gravity modulation and internal heating effect in controlling heat transport within the system. • Theory of weakly nonlinear stability analysis is presented numerically. • Bioconvection analysis is investigated in a porous media in the presence of gravity modulation. • Ginzburg–Landau equation is employed to determine amplitude of bioconvection. • The effects of internal heating and gravitational modulation are discussed on bio-thermal convection. • Both biothermal convection and gravity modulations are contributed to stabilize the system also Transport analysis is presented within the bio-porous layer. [ABSTRACT FROM AUTHOR]

The effect of three different types of time-periodic vertical oscillations of the Rayleigh–Bénard system involving Newtonian nanoliquids is studied in the paper. Twenty nanoliquids are considered for investigation and it is found that enhanced heat transport is seen in all of them compared to what is observed in the corresponding carrier liquids without the nanoparticles. The increase in the amplitude of the vertical vibration is shown to augment convection in the nanoliquids whereas increase in frequency is to inhibit convection. On comparing the extent of heat transport in the presence of modulation with that in its absence, it was found that heat transport diminished in the former case. It was also found that the trigonometric sine type of gravity modulation transports heat intermediate to that of triangular and square types. The triangular type inhibits heat transport while square type enhances heat transport leading one to conclude that heat transport in a Rayleigh–Bénard system can be enhanced by using a combination of dilute concentration of nanoparticles and gravity modulation using a square wave. [ABSTRACT FROM AUTHOR]

Abstract We explore the modifications brought about by gravity modulation in self-aligned two-dimensional (2D) stationary rolls and higher-order convective instabilities with horizontal magnetic field. The Chandrasekhar number plays no role in determining critical values of the forcing parameters for convection to begin. The onset is in the form of 2D rolls with periodically varying intensity. The external magnetic field is solely responsible for aligning the rolls while the induced magnetic field comes into play after three-dimensional convection sets in. Gravity modulation destabilizes the system and results in chaotic flow much earlier above onset. Highlights • Gravity modulation affects thermal convection with horizontal magnetic field. • Chandrasekhar number does not affect the stability boundaries. • Oscillatory two dimensional convective rolls are found at onset. • The rolls align themselves along the magnetic field direction. • The system destabilizes and attains chaotic state soon above onset. [ABSTRACT FROM AUTHOR]

The influence of trigonometric sine, square and triangular wave-types of time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and in Newtonian nanoliquids is studied in the paper using the generalized Buongiorno two-phase model. The five-mode Lorenz model is derived under the assumptions of Boussinesq approximation, small-scale convective motion and some slip mechanisms. Using the method of multiscales, the Lorenz model is transformed to a Ginzburg–Landau equation the solution of which helps in quantifying the heat transport through the Nusselt number. Enhancement of heat transport in Newtonian liquids due to the presence of nanoparticles/nanotubes is clearly explained. The study reveals that all the three wave types of gravity modulation delay the onset of convection and thereby to a diminishment of heat transport. It is also found that in the case of trigonometric sine type of gravity modulation heat transport is intermediate to that of the cases of triangular and square types. The paper is the first such work that attempts to theoretically explain the effect of three different wave-types of gravity modulation on onset of convection and heat transport in the presence/absence of nanoparticles/nanotubes. Comparing the heat transport by the single-phase and by the generalized two-phase models, the conclusion is that the single-phase model under-predicts heat transport in nanoliquids irrespective of the type of gravity modulation being effected on the system. The results of the present study reiterate the findings of related experimental and numerical studies. [ABSTRACT FROM AUTHOR]

Thermovibrational flow can be seen as a variant of standard thermogravitational convection where steady gravity is replaced by a time-periodic acceleration. As in the parent phenomena, this type of thermal flow is extremely sensitive to the relative directions of the acceleration and the prevailing temperature gradient. Starting from the realization that the overwhelming majority of research has focused on circumstances where the directions of vibrations and of the imposed temperature difference are perpendicular, we concentrate on the companion case in which they are parallel. The increased complexity of this situation essentially stems from the properties that are inherited from the corresponding case with steady gravity, i.e., the standard Rayleigh–Bénard convection. The need to overcome a threshold to induce convection from an initial quiescent state, together with the opposite tendency of acceleration to damp fluid motion when its sign is reversed, causes a variety of possible solutions that can display synchronous, non-synchronous, time-periodic, and multi-frequency responses. Assuming a square cavity as a reference case and a fluid with Pr = 15, we tackle the problem in a numerical framework based on the solution of the governing time-dependent and non-linear equations considering different amplitudes and frequencies of the applied vibrations. The corresponding vibrational Rayleigh number spans the interval from Raω = 104 to Raω = 106. It is shown that a kaleidoscope of possible variants exist whose nature and variety calls for the simultaneous analysis of their temporal and spatial behavior, thermofluid-dynamic (TFD) distortions, and the Nusselt number, in synergy with existing theories on the effect of periodic accelerations on fluid systems.

In this paper, we investigate the combined effect of internal heating and time periodic gravity modulation in a viscoelastic fluid saturated porous medium by reducing the problem into a complex non-autonomous Ginzgburg-Landau equation. Weak nonlinear 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 is obtained in terms of the amplitude for oscillatory mode of convection. The influence of viscoelastic parameters on heat transfer has been discussed. Gravity modulation is found to have a destabilizing effect at low frequencies and a stabilizing effect at high frequencies. Finally, it is found that overstability advances the onset of convection, more with internal heating. The conditions for which the complex Ginzgburg-Landau equation undergoes Hopf bifurcation and the amplitude equation undergoes supercritical pitchfork bifurcation are studied.

The effect of vertical throughflow and internal heating effects on fluid saturated porous medium under gravity modulation is investigated. The amplitude of modulation is considered to be very small and the disturbances are expanded in terms of power series of amplitude of convection. A weakly nonlinear stability analysis is proposed to study stationary convection. The Nusselt number is obtained numerically to present the results of heat transfer while using Ginzburg–Landau equation. The vertical throughflow has dual effect either to destabilize or to stabilize the system for downward or upward directions. The effect of internal heat source (Ri>0) enhances or sink (Ri

This paper carried out a nonlinear thermal convection in a porous medium saturated with viscoelastic nanofluid under vibrations. The Darcy model has been used for the porous medium, while the nanofluid layer incorporates the effect of Brownian motion along with thermophoresis. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The non-uniform vertical vibrations of the system, which can be realized by oscillating the system vertically, is considered to vary sinusoidally with time. In order to find the heat and mass transports for unsteady state, a nonlinear analysis, using a minimal representation of the truncated Fourier series of two terms, has been performed. Effect of various parameters has been investigated on heat and mass transport and then presented graphically. It is found that gravity modulation can be used effectively to regulate either heat or mass transports in the system.

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.

The effect of vertical throughflow and time-periodic gravity field has been investigated on Darcy convection. The amplitude of gravity modulation is 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. As a consequence heat transport evaluated in terms of the Nusselt number, which is governed by the non-autonomous Ginzburg-Landau equation. Throughflow can stabilize or destabilize the system for stress free and isothermal boundary conditions. The amplitude and frequency of modulation, Prandtl Darcy number on heat transport have 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. Finally flow patterns are presented in terms of streamlines and isotherms.

A theoretical analysis of thermo-convective instability in an electrically conducting two component fluid layer is carried out when the gravity field vary with time in a sinusoidal manner. Newtonian liquid is considered between two horizontal surfaces, under a constant vertical magnetic field. The disturbance is expanded in terms of power series of amplitude of convection, which is assumed to be small. We use the linear matrix differential operator method to find the Ginzburg–Landau amplitude equation for the modulated problem. Use the solution of the Ginzburg–Landau equation in quantifying the amount of heat and mass transports in terms of Nusselt and Sherwood numbers. It is found that, the effect of magnetic field is to stabilize the system. Effect of various parameters on the heat and mass transport is also discussed. Further, it is found that the heat and mass transports can be controlled by suitably adjusting frequency and amplitude of gravity modulation.

In this research article, we investigated the weakly non-linear effect of gravity modulation for the temperature dependent viscous fluid in a horizontal porous layer in the presence of internal heat source. We use power series expansion in terms of the amplitude of gravity modulation, which is considered to be small for double-diffusive convection in porous media. We graphically show the effect of internal heat source, solute Rayleigh number, Lewis number, Vadász number, thermo-rheological parameter, the amplitude of gravity modulation, the frequency of modulation on the heat and mass transfer using Ginzburg-Landau equation. The effect of gravity modulation is found significant and is more effective for the low values of frequency of modulation. [ABSTRACT FROM AUTHOR]

On the basis of Darcy-Brinkman model, the chaotic convection in a couple stress fluid saturated porous media under gravity modulation is investigated using the nonlinear stability analyses. The transition from steady convection to chaos is analysed with the effect of Darcy-Brinkman couple stress parameter and the gravity modulation. The results show that the chaotic behavior is connected with the critical value of Rayleigh number which is dependent upon the oscillation frequency and the Darcy-Brinkman couple stress parameter. If the oscillation frequency Ω is not zero, the Rayleigh number value R of the chaotic behavior increases with the increase of the Darcy-Brinkman couple stress parameter. The Darcy-Brinkman couple stress parameter and the gravity modulation decrease the rate of heat transfer. [ABSTRACT FROM AUTHOR]

The article deals with a weakly nonlinear stability analysis in rotating fluid subjected to time-periodic concentration/gravity modulation. Navier–Stoke’s momentum equation with Coriolis term has been used to describe the flow. The system is considered rotating about z -axis with uniform angular velocity. We consider three types of imposed time-periodic boundary concentration (ITBC). The effect of time dependent sinusoidal gravitational acceleration i.e. imposed time-periodic gravity modulation (ITGM) is also studied in this problem. In the case of ITBC, the concentration gradient between the plates of the fluid layer consists of a steady part and a time-dependent periodic part. The concentration of both plates is modulated in this case. In the problem involving ITGM, the gravity field has two parts: a constant part and an externally imposed time-periodic part. The Ginzburg–Landau amplitude equation is obtained by using power series expansion in terms of the amplitude of modulation, which is assumed to be small. The individual effects of concentration and gravity modulation on mass transports have been investigated in terms of Sherwood number. Further, the effects of various parameters on mass transports have been analyzed and depicted graphically. It is found that the effect of increasing Taylor number is decreases the value of Sherwood number. Effect of Schmidt number and various parameters occurring in the system on mass transfer is also discussed. Further, it is found that the mass transport can be controlled by suitably adjusting the frequency and amplitude of the modulation. [ABSTRACT FROM AUTHOR]

In the present work the linear stability analysis of double diffusive convection in a binary fluid layer is performed. The major intention of this study is to investigate the influence of time-periodic vertical vibrations on the onset threshold. A regular perturbation method is used to compute the critical Rayleigh number and wave number. A closed form expression for the shift in the critical Rayleigh number is calculated as a function of frequency of modulation, the solute Rayleigh number, Lewis number, and Prandtl number. These parameters are found to have a significant influence on the onset criterion; therefore the effective control of convection is achieved by proper tuning of these parameters. Vertical vibrations are found to enhance the stability of a binary fluid layer heated and salted from below. The results of this study are useful in the areas of crystal growth in micro-gravity conditions and also in material processing industries where vertical vibrations are involved