Onset Of Oscillatory Convection Research Articles

Weakly nonlinear thermohaline rotating convection in a sparsely packed porous medium

This paper discusses the analytical study of linear and weakly nonlinear stability of thermohaline convection in a sparsely packed porous due to rotation. In the linear stability analysis, we used the normal mode technique to find the critical Rayleigh number and it is a function of q,Ta,Λ,R2, and L. Marginal stability curves were traced corresponding to thermal Rayleigh number and studied bifurcation points. In the weakly nonlinear analysis, a nonlinear two-dimensional Landau Ginzburg equation was derived at the onset of stationary convection, studied the secondary instabilities and heat transport due to convection. One dimensional coupled Landau Ginzburg equations are derived at the onset of oscillatory convection and analyzed the stability regions of steady-state, standing waves, and travelling waves.

Double diffusive LTNE porous convection with Cattaneo effects in the solid

AbstractThe impact of Cattaneo heat flux law in the solid on the onset of double‐diffusive Darcy porous convection with local thermal nonequilibrium temperatures is investigated. The Fourier law of heat transfer is invoked for the fluid, whereas the Cattaneo heat flux law used to transfer heat in solid skeleton alters the temperature equation from parabolic to hyperbolic. The results are obtained for porous skeletons of aluminum and copper oxides. Both Cattaneo and solute concentration effects reinforce in controlling the onset of oscillatory convection and some novel consequences are observed. Compared with the results perceived in the absence of solute concentration, a manifestation of oscillatory convection with scaled‐interphase heat transfer coefficient as well as solid thermal relaxation time parameter initiates earlier in its presence. The effect of increasing interphase heat transfer coefficient and the Lewis number is to delay and hasten the onset of stationary and oscillatory convection. Besides, the increase in the value of solid thermal relaxation time parameter advances the oscillatory onset. Although the increase in the solute Darcy–Rayleigh number is to delay the stationary onset, it shows a twofold behavior on the onset of oscillatory convection. Before the onset of oscillatory convection, the size of the convection cell gets narrower and after which it becomes much wider. The existing results are retrieved as limiting cases from the current study.

On the Oscillatory Convective Instability in Nanofluid Layer with Temperature Dependent Viscosity

In the present paper, the problem of thermal convection in nanofluid layer heated from below with temperature dependent viscosity is studied mathematically for general boundary conditions, when the instability sets in as oscillatory convection. The expressions for Rayleigh numbers for exponential temperature dependent viscosity variation are derived using Galerkin method from the eigenvalue problem for each combination of rigid (the surfaces with non-slip condition) and dynamically free (the surfaces with stress free condition) boundaries. The effects of viscosity variation parameter, Lewis number, concentration Rayleigh number, modified diffusivity ratio and Prandtl number on the onset of oscillatory convection for each case of boundary conditions are investigated numerically. It is observed that the exponential variation of temperature dependent viscosity has stabilizing effect on the onset of oscillatory convection in nanofluid layer for each case of boundary conditions. However, the Lewis number and the concentration Rayleigh number have destabilizing effect on the onset of oscillatory convection.

Effect of side wall conditions and aspect ratio on convective pattern formation in Rayleigh-Benard convection

Effect of side wall boundary conditions and aspect ratio in very small boxes on patterns formed during Raleigh-Benard (RB) convection is studied using an indigenously developed flow solver based on the finite difference method. While keeping the bottom (hot) and top (cold) walls isothermal, the side wall thermal conditions are kept either constant heat flux or isothermal type. For constant heat flux side wall condition, the considered value of heat flux lies between 0 and maximum corresponding to conductive heat flux between top and bottom walls. For isothermal side wall condition, the chosen value of temperature is between the top and bottom wall temperatures. The computational domain is a cube or a square prism with aspect ratio in longitudinal direction (ARx) varied between 1.2 and 2 in steps of 0.2. For fixed value of ARx = 1.6, aspect ratio in transverse direction (ARz) has been varied for adiabatic side wall conditions. Effect of side wall condition on orientation of roll axis and the number of rolls has been investigated with observations well supported by stream traces and temperature contours. Moreover, for selected parameters, based on nature of pattern formation for steady flow, onset of oscillatory convection is explained.

English

English

Porous ferroconvection with local thermal nonequilibrium temperatures and with Cattaneo effects in the solid

The influence of local thermal nonequilibrium with Cattaneo effects in the solid on the onset of thermal-convective instability in a horizontal layer of Darcy porous medium saturated by a ferrofluid in the presence of a uniform vertical magnetic field is investigated. The presence of the Cattaneo effect is to instill instability via oscillatory motion as well which is not reminiscent of the observed instability phenomenon in its absence. Increase in the value of the solid thermal relaxation time parameter τ is found to advance the onset of oscillatory ferroconvection. The onset of stationary ferroconvection is delayed, while the onset of oscillatory convection is accelerated with an increase in the value of the interphase heat transfer coefficientHt. The threshold value of Ht, at which the transition from stationary to oscillatory convection takes place, decreases with increasing τ noticeably and marginally with increasing magnetic parameter M3, while it increases with increasing ratio of conductivities α and magnetic number M1. The critical wave number for stationary convection is found to be higher than for oscillatory convection.

Effects of a temperature-dependent viscosity on thermal convection in binary mixtures.

We investigate the effect of a temperature-dependent viscosity on the onset of thermal convection in a horizontal layer of a binary fluid mixture that is heated from below. For an exponential temperature dependence of the viscosity, we find, in binary mixtures as a function of a positive separation ratio ψ and beyond a certain viscosity contrast, a discontinuous transition between two stationary convection modes having different wavelengths. In the range of negative values of the separation ratio ψ, a (continuous or discontinuous) transition from an oscillatory to a stationary onset of convection occurs beyond a certain viscosity contrast, and for large values of the viscosity ratio, the oscillatory onset of convection is suppressed.

Oscillatory instability and fluid patterns in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation

We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number (Pr = 0.1) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box (Lx × Ly × 1) and a wide range of Taylor numbers (750 ⩽ Ta ⩽ 5000) for the purpose. The horizontal aspect ratio η = Ly/Lx of the box was varied from 0.5 to 10. The primary instability appeared in the form of two-dimensional standing waves for shorter boxes (0.5 ⩽ η < 1 and 1 < η < 2). The flow patterns observed in boxes with η = 1 and η = 2 were different from those with η < 1 and 1 < η < 2. We observed a competition between two sets of mutually perpendicular rolls at the primary instability in a square cell (η = 1) for Ta < 2700, but observed a set of parallel rolls in the form of standing waves for Ta ⩾ 2700. The three-dimensional convection was temporally quasiperiodic for 750 ⩽ Ta ⩽ 1114, chaotic for 1115 ⩽ Ta < 1125, and quasiperiodic once again for 1125 ⩽ Ta < 2700. The three-dimensional quasiperiodic flow bifurcated into a two-dimensional periodic flow for Ta ⩾ 2700. The convection at the primary instability were three-dimensional and quasiperiodic in time for all values of Ta (⩽3000) in a box with η = 2. Küppers-Lortz type of convection was observed at the primary instability in longer boxes (4 ⩽ η ⩽ 10). The convective structures consisted of the appearance and disappearance of straight rolls, rhombic patterns, and wavy rolls inclined at an angle \documentclass[12pt]{minimal}\begin{document}$\phi = \frac{\pi }{2} - \arctan {(\eta ^{-1})}$\end{document}ϕ=π2−arctan(η−1) with the straight rolls. We also present a low-dimensional model constructed for a square box (η = 1).

Influence of Chemical Reaction on Stability of Thermo-Solutal Convection of Couple-Stress Fluid in a Horizontal Porous Layer

We consider the onset of thermo-solutal convection in a couple-stress fluid-saturated anisotropic porous medium, where the chemical equilibrium on the bounding surfaces and the solubility of the dissolved components depend on temperature. The entire study has been spilt into two parts: (i) linear stability analysis (ii) weakly non-linear stability analysis. Stationary case of linear stability analysis is discussed for two modes of bounding surfaces (a) realistic bounding surfaces i.e. Rigid-Rigid and Rigid-Free (R/R and R/F), (b) non-realistic bounding surfaces i.e. Free-Free (F/F). Howsoever, investigation of oscillatory state and weakly non-linear stability are restricted to F/F case. Galerkin method is used to solve the eigenvalue problem for R/R and R/F cases, whereas, exact solutions are obtained for F/F case.A comparative study among flow stability for above different cases is made as function of ratio of viscosities ( i.e., couple-stress viscosity to fluid viscosity which is defined as couple-stress parameter, \((C)\)) and effective chemical reaction (i.e. chemical reaction parameter, \((\chi )\)). It has been found that increasing viscosity of the couple-stress fluid, in terms of increasing \(C\), increases flow stability in all three cases, but among all cases its stabilization effect for R/R is maximum. However, in the absence of couple-stress parameter the maximum stability of flow is observed for F/F. Apart from this, the chemical reaction stabilizes the flow for all the three cases. Furthermore, stability analysis for F/F case indicates that couple-stress parameter stabilizes the system in all modes (stationary, oscillatory and finite amplitude) of convection.Damkohler number \((\chi )\) is found to delay the stationary convection, however, it speeds up the onset of oscillatory convection. The non-linear theory based on truncated representation of Fourier series method predicts the occurrence of sub-critical instability in the form of finite amplitude motion. The effect of \(C\) and \(\chi \) on heat and mass transfer is also examined.

Nonlinear rotating convection in a sparsely packed porous medium

We investigate linear and weakly nonlinear properties of rotating convection in a sparsely packed Porous medium. We obtain the values of Takens–Bogdanov bifurcation points and co-dimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to rotating convection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We derive a nonlinear two-dimensional Landau–Ginzburg equation with real coefficients by using Newell–Whitehead method [16]. We investigate the effect of parameter values on the stability mode and show the occurrence of secondary instabilities viz., Eckhaus and Zigzag Instabilities. We study Nusselt number contribution at the onset of stationary convection. We derive two nonlinear one-dimensional coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discuss the stability regions of standing and travelling waves.

Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer

Linear and nonlinear stability analyses were performed on a fluid layer with a concentration-based internal heat source. Clear bimodal behaviour in the neutral curve (with stationary and oscillatory modes) is observed in the region of the onset of oscillatory convection, which is a previously unobserved phenomenon in radiation-induced convection. The numerical results for the linear instability analysis suggest a critical value γc of γ, a measure for the strength of the internal heat source, for which oscillatory convection is inhibited when γ > γc. Linear instability analyses on the effect of varying the ratio of the salt concentrations at the upper and lower boundaries conclude that the ratio has a significant effect on the stability boundary. A nonlinear analysis using an energy approach confirms that the linear theory describes the stability boundary most accurately when γ is such that the linear theory predicts the onset of mostly stationary convection. Nevertheless, the agreement between the linear and nonlinear stability thresholds deteriorates for larger values of the solute Rayleigh number for any value of γ.

Non-linear convection due to compositional and thermal buoyancy in Earth's outer core

The linear stability of convection due to compositional and thermal buoyancy in Earth's outer core has been investigated. We have obtained the values of Takens–Bogdanov bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters. We have derived a non-linear two-dimensional Landau–Ginzburg equation with real coefficients near the onset of stationary convection at a supercritical pitchfork bifurcation and two non-linear one-dimensional coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation. We have studied Nusselt number contribution from a Landau–Ginzburg equation at the onset of stationary convection. We have discussed the stability regions of standing and travelling waves. We have also discussed the occurrence of secondary instabilities such as Eckhaus, zigzag and Benjamin–Feir instabilities. We have also derived the non-linear amplitude equation near the Takens–Bogdanov bifurcation point.

Surface tension driven convection in viscoelastic liquids with thermorheological effect

Oscillatory onset of convection is studied numerically for Rivlin–Ericksen, Maxwell and Jeffreys liquids by considering free–free and rigid–free isothermal/adiabatic boundaries. The effect of variable viscosity parameter is shown to destabilize the system. The problem reveals the stabilizing nature of strain retardation parameter and the destabilizing nature of stress relaxation parameter, on the onset of convection. The Maxwell liquids are found to be more unstable than the one subscribing to the Jeffreys description whereas the Rivlin–Ericksen liquids are comparatively more stable. Rigid-free adiabatic boundary combination is found to give rise to a most stable system, whereas the free isothermal free adiabatic combination gives rise to a most unstable system.

Nonlinear Magnetoconvection in a Sparsely Packed Porous Medium

Linear and weakly nonlinear properties of magnetoconvection in a sparsely packed porous medium are investigated. We have obtained the values of Takens-Bogdanov bifurcation points and codimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to magnetoconvection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We have derived a nonlinear two-dimensional Ginzburg-Landau equation with real coefficients by using Newell-Whitehead (1969) method. The effect of the parameter values on the stability mode is investigated and shown the occurrence of secondary instabilities namely, Eckhaus and Zigzag instabilities. We have studied Nessult number contribution at the onset of stationary convection. We have also derived two nonlinear one-dimensional coupled Ginzburg-Landau-type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discussed the stability regions of standing and travelling waves.

Interactions between steady and oscillatory convection in mushy layers

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

Effect of interfacial heat transfer on the onset of oscillatory convection in liquid bridge

In present study, effect of interfacial heat transfer with ambient gas on the onset of oscillatory convection in a liquid bridge of large Prandtl number on the ground is systematically investigated by the method of linear stability analyses. With both the constant and linear ambient air temperature distributions, the numerical results show that the interfacial heat transfer modifies the free-surface temperature distribution directly and then induces a steeper temperature gradient on the middle part of the free surface, which may destabilize the convection. On the other hand, the interfacial heat transfer restrains the temperature disturbances on the free surface, which may stabilize the convection. The two coupling effects result in a complex dependence of the stability property on the Biot number. Effects of melt free-surface deformation on the critical conditions of the oscillatory convection were also investigated. Moreover, to better understand the mechanism of the instabilities, rates of kinetic energy change and “thermal” energy change of the critical disturbances were investigated

Onset of oscillatory convection in two liquid layers with phase change

We perform linear stability calculations for horizontal fluid bilayers that can undergo a phase transformation in the presence of a vertical temperature gradient. We reconsider the oscillatory instability calculated by Huang and Joseph [J. Fluid Mech. 242, 235 (1992)] for the water-steam system cooled from below at temperatures near 100 °C, where there is a large difference in the densities of the two fluids. We find that buoyancy and surface tension gradients are unimportant for this instability. Numerical solutions demonstrate that the properties of the vapor and liquid systems at these temperatures are sufficiently different that an approximate treatment is possible in which the equations for the vapor phase can be eliminated from the overall governing equations. Further analytical approximations suggested by the numerical solution are also presented, and the results are in good agreement with the numerical solution for the full set of governing equations. A simple model of the oscillatory instability is developed which gives insight into its origins.

Oscillatory Marangoni convection in variable-viscosity fluid layer: The effect of thermal feedback control

The effect of a feedback control strategy on the onset of oscillatory convection in an infinite horizontal layer of fluid with temperature-dependent viscosity is investigated theoretically using linear stability analysis. It is shown that small viscosity variations stabilizes the fluid layer. Large controller gains, large viscosity variations, and high surface tension, however, promote the onset of overstability leading to oscillatory motions.

Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below

Based on a modified Darcy–Brinkman–Oldroyd model, linear and nonlinear thermal stability analyses of a horizontal layer of an Oldroyd-B fluid in a porous medium heated from below were performed. By using the linear stability theory, the critical Rayleigh number, wave number, and frequency for stationary and oscillatory convections were determined. The effects of the viscoelastic parameters and the porous parameter on the critical Rayleigh number for oscillatory convection were analyzed. Based on the results of the linear stability analysis, a nonlinear stability analysis was also conducted. It is shown that the onset of stationary convection has the form of a supercritical and stable bifurcation independent of the viscoelastic parameters. However, the onset of oscillatory convection has the forms of supercritical or subcritical bifurcations. The nature of the oscillatory mode depends strongly on the viscoelastic parameters. The variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Although the critical Rayleigh number for stationary convection is independent of the viscoelastic parameters, the Nusselt number depends on the viscoelastic parameters of the fluids, which is different from that for the modified Darcy–Oldroyd model.

Directional effect of a magnetic field on oscillatory low-Prandtl-number convection

The directional effect of a magnetic field on the onset of oscillatory convection is studied numerically in a confined three-dimensional cavity of relative dimensions 4:2:1 (length:width:height) filled with mercury and subject to a horizontal temperature gradient. The magnetic field suppresses the oscillations most effectively when it is applied in the vertical direction, and is the least efficient when applied in the longitudinal direction (parallel to the temperature gradient). In all cases, however, exponential growths of the critical Grashof number, Grc (Gr, ratio of buoyancy to viscous dissipation forces) with the Hartmann number (Ha, ratio of magnetic to viscous dissipation forces) are obtained. Insight into the damping mechanism is gained from the fluctuating kinetic energy budget associated with the time-periodic disturbances at threshold. The kinetic energy produced by the vertical shear of the longitudinal basic flow dominates the oscillatory transition, and when a magnetic field is applied, it increases in order to balance the stabilizing magnetic energy. Moreover, subtle changes in the spatial distribution of this shear energy are at the origin of the exponential growth of Grc. The destabilizing effect of the velocity fluctuations strongly decreases when Ha is increased (due to the decay of the velocity fluctuations in the bulk accompanied by the appearance of steep gradients localized in the Hartmann layers), so that an increase of the shear of the basic flow at Grc is required in order to sustain the instability. This yields an increase in Grc, which is reinforced by the fact that the shear of the basic flow naturally decreases at constant Gr with the increase of Ha, particularly when the magnetic field is applied in the vertical direction. For transverse and longitudinal fields, the decay of the velocity fluctuations is combined with an increase of the shear energy term due to a sustained growth in stabilizing magnetic energy with Ha.