Finite Amplitude Convection Research Articles

Nonlinear Analysis of Cross Rolls of Electrically Conducting Fluid under an Applied Magnetic Field with Rotation

The proposed planer layer dynamo physical model has real-world applications, especially in the Earth’s liquid core. Thus, in this paper, an attempt is made to understand the finite amplitude convection when there exists a coupling between the Lorentz force and the Coriolis force. In particular, the effect of a horizontally applied magnetic field is studied on the Rayleigh–Bénard convection (RBC) that contains the electrically conducting fluid and rotates about its vertical axis. Free–free boundary conditions are assumed on the geometry. Attention is focused on the nonlinear convective flow behavior during the occurrence of cross rolls which are perpendicular to the applied magnetic field and parallel to the rotation axis. The visualization of cross rolls is achieved using the Fourier analysis of perturbations up to the O(ε8). The relationship of the Nusselt number (Nu) with respect to the Rayleigh number (R), the Ekman number (E), and the Elsasser number (Λ) is investigated. It is observed that E generates a strong damping effect on the flow velocity and on the heat transfer at high rotation rates. Using the heatline concept, it is observed that the temperature within the central regime is enhanced as the Λ increases. The results show that either E decreases or Λ increases, then the heat transfer rate increases.

Time-Periodic Thermal Boundary Effects on Porous Media Saturated with Nanofluids: CGLE Model for Oscillatory Mode

Abstract The stability of nonlinear nanofluid convection is examined using the complex matrix differential operator theory. With the help of finite amplitude analysis, nonlinear convection in a porous medium is investigated that has been saturated with nanofluid and subjected to thermal modulation. The complex Ginzburg-Landau equation (CGLE) is used to determine the finite amplitude convection in order to evaluate heat and mass transfer. The small amplitude of convection is considered to determine heat and mass transfer through the porous medium. Thermal modulation of the system is predicted to change sinusoidally over time, as shown at the boundary. Three distinct modulations IPM, OPM, and LBMOhave been investigated and found that OPM and LBMO cases are used to regulate heat and mass transfer. Further, it is found that modulation frequency (ω f varying from 2 to 70) reduces heat and mass transfer while modulation amplitude (δ 1 varying from 0.1 to 0.5 ) enhances both.

Convective instability in multicomponent mixtures with Soret effect

We study the two-dimensional modes of gravity-dependent convective instability arising in a ternary mixture in a plane horizontal layer. The gravity force is oriented perpendicular to the layer heated from below. It is assumed that the system is in the state of mechanical equilibrium of a mixture considered as the base state of the system, which can become unstable under certain conditions. The mathematical model uses two-dimensional Navier-Stokes equations, transfer equations for describing mixture components with the Soret effect, and a heat transfer equation. Under the influence of a concentration gradient, the heat transfer symmetric to the Soret effect is neglected because it is usually small in water-based mixtures. The possible concentration-dependent diffusion and cross-diffusion of dissolved components are also neglected. We investigate a convective stability using linear approximation and under finite-amplitude convection regimes. The stability analysis of the base state after linearization of the governing equations with respect to the state of mechanical equilibrium predicts various convection modes. For each case, a neutral curve and the dependences of the disturbance growth decrements on both the Rayleigh number and the wave number are plotted. We show that in the case of the layer being heated from below, there are ranges of governing parameters where both long-wave and short-wave modes of oscillatory instability occur. The numerical analysis of a nonlinear coupling between these modes has revealed that such instabilities develop in the ternary mixtures whose components are arranged (due to the Soret effect) in different directions with respect to the temperature gradient. The effect of the Rayleigh number on the flow structure and heat mass transfer is demonstrated.

Thermocapillary-buoyancy convection of a power-law fluid layer heated from below

Buoyant Marangoni convection of a power-law fluid confined in a shallow cavity heated from below is investigated. The free upper surface of the system is assumed to be flat and underformable. A simplified analytical model, valid for an infinite layer, is derived on the basis of the parallel flow approximation. The criteria, for the onset of convection, are predicted for both shear-thinning and shear thickening fluids. For finite amplitude convection, the effects of the power law model on the fluid flow, temperature field and heat transfer are discussed. A good agreement is found between the simplified model and the numerical results obtained by solving the full governing equations, provided that the strength of convection is small enough for the flow pattern to remain parallel.

Detour induced by the piston effect in the oscillatory double-diffusive convection of a near-critical fluid

The nonlinear oscillatory double-diffusive convection in a thermodynamically near-critical binary fluid layer is investigated to explore the interactions between the piston effect and natural convection in the presence of subcritical bifurcation. The bifurcation diagram of the system is studied. Two subcritical bifurcation branches are depicted, which, together with the trivial branch of pure diffusion, are connected by two hysteresis loops. To understand the role of the piston effect, the Boussinesq counterpart of the near-critical system is considered and compared. Results show that the onset of convection is significantly altered by the piston effect. For the Boussinesq system, the lower boundary layer becomes unstable, brings on finite-amplitude perturbations, and leads to a statistically steady state. However, the near-critical system features a two-stage evolution. In the first stage, the lower boundary layer becomes unstable and then returns to stability. As soon as the temperature field relaxes into the second stage, a change of criterion occurs, and the fluid becomes unstable again. The residual convection motions amplify and finally result in finite-amplitude convection. By this means, the near-critical system becomes insensitive to the existence of the higher equilibrium state in hysteresis loops, and detours relative to the Boussinesq system are observed. This paper gives new insights into the piston effect and its interactions with natural convection from a dynamic system point of view. The conclusions can be extended to other situations where subcritical bifurcations exist.

Unsteady Finite Amplitude Convection of Water–Copper Nanoliquid in High-Porosity Enclosures

Unicellular Rayleigh–Bénard convection of water–copper nanoliquid confined in a high-porosity enclosure is studied analytically. The modified-Buongiorno–Brinkman two-phase model is used for nanoliquid description to include the effects of Brownian motion, thermophoresis, porous medium friction, and thermophysical properties. Free–free and rigid–rigid boundaries are considered for investigation of onset of convection and heat transport. Boundary effects on onset of convection are shown to be classical in nature. Stability boundaries in the R1*–R2 plane are drawn to specify the regions in which various instabilities appear. Specifically, subcritical instabilities' region of appearance is highlighted. Square, shallow, and tall porous enclosures are considered for study, and it is found that the maximum heat transport occurs in the case of a tall enclosure and minimum in the case of a shallow enclosure. The analysis also reveals that the addition of a dilute concentration of nanoparticles in a liquid-saturated porous enclosure advances onset and thereby enhances the heat transport irrespective of the type of boundaries. The presence of porous medium serves the purpose of heat storage in the system because of its low thermal conductivity.

Stability of triple diffusive convection in a viscoelastic fluid-saturated porous layer

The triple diffusive convection in an Oldroyd-B fluid-saturated porous layer is investigated by performing linear and weakly nonlinear stability analyses. The condition for the onset of stationary and oscillatory is derived analytically. Contrary to the observed phenomenon in Newtonian fluids, the presence of viscoelasticity of the fluid is to degenerate the quasiperiodic bifurcation from the steady quiescent state. Under certain conditions, it is found that disconnected closed convex oscillatory neutral curves occur, indicating the requirement of three critical values of the thermal Darcy-Rayleigh number to identify the linear instability criteria instead of the usual single value, which is a novel result enunciated from the present study for an Oldroyd-B fluid saturating a porous medium. The similarities and differences of linear instability characteristics of Oldroyd-B, Maxwell, and Newtonian fluids are also highlighted. The stability of oscillatory finite amplitude convection is discussed by deriving a cubic Landau equation, and the convective heat and mass transfer are analyzed for different values of physical parameters.

Convective hydromagnetic instabilities of a power-law liquid saturating a porous medium: Flux conditions

We investigate how an external imposed magnetic field affects thermal instability in a horizontal shallow porous cavity saturated by a non-Newtonian power-law liquid. The magnetic field is assumed to be constant and parallel to the gravity. A uniform heat flux is applied to the horizontal walls of the layer while the vertical walls are adiabatic. We use linear stability analysis to find expressions for the critical Rayleigh number as a function of the power-law index and the intensity of the magnetic field. We use nonlinear parallel flow theory to find some explicit solutions of the problem, and we use finite difference numerical simulations to solve the full nonlinear equations. We show how the presence of magnetic field alters the known hydrodynamical result of Newtonian flows and power-law flows and how it causes the presence of subcritical finite amplitude convection for both pseudoplastic and dilatant fluids. We also show that in the limit of very strong magnetic field, the dissipation of energy by Joule effect dominates the dissipation of energy by shear stress and gives to the liquid an inviscid character.

Pattern selection in ternary mushy layers

We consider finite-amplitude convection in a mushy layer during the primary solidification of a ternary alloy. Previous linear stability theories applied to ternary alloy primary-solidification models have identified an exceptional class of direct convective instability when all the individual stratifying agencies (one thermal and two solutal) were statically stabilizing. A reduced model, in which the effects of latent heat, solute rejection and background solidification are neglected, contains the essential interactions that admit qualitatively the same instability. We examine pattern selection for steady convection in this model. We find that roll, square or hexagonal convection patterns can be nonlinearly stable, depending on the relative importance of a number of physical effects, namely the solutal diffusion rates, the liquidus slopes and the background thermal and solutal density stratifications. The results for a special case are found to isolate a purely double-diffusive phase-change mechanism of pattern selection. Subcritical behaviour is identified inside the domain of individual static stability. A physical system is proposed that may be a promising one in which to experimentally identify these novel instabilities.

Finite amplitude convection and heat transfer in inclined porous layer using a thermal non-equilibrium model

Finite amplitude convection in a inclined porous layer heated from below is studied by using local thermal non-equilibrium (LTNE) as mathematical model which takes into account the heat transferred between the solid phase and the fluid phase. Consequently, in addition to Darcy-Rayleigh number Ra and the inclination angle ϕ, two further non dimensional numbers are introduced: the inter-phase heat transfer parameter H and the porosity modified conductivity ratio γ. In a recent paper (Barletta and Rees, 2015), the linear stability analysis of the basic monocellular flow indicated that the inclination angle promotes the appearance of longitudinal rolls as the preferred mode of convection. The current paper focuses on the nonlinear evolution of longitudinal rolls in a supercritical regime of convection. A weakly nonlinear analysis, using a derived amplitude equation, is adopted to determine the nonlinear effects of the parameters Ra,ϕ,H and γ. The results indicate that in inclined layers (i) the nonlinearity decelerates the mean flow; (ii) the heat transfer, determined by the evaluation of the Nusselt number (Nu) at the layer boundary, corresponds to the one obtained for horizontal layers by scaling Ra with cosϕ, i.e.Nu=Nu(Racosϕ,H,γ); (iii) in accordance with existing laboratory experiments, the slope of Nu is less than 2, where 2 is the value predicted by the local thermal equilibrium model, and the slope represents the derivative of Nu with respect to the distance of the critical parameter from the threshold value for the onset of instability; (iv) increasing values of both H and γ produce an enhancement of the heat transfer across the layer. Finally, the comparison between the LTNE theoretical predictions and existing experiments conducted with various combinations of solid matrix and fluids suggests a possible alternative way to determine the heat transfer coefficient H.

Effect of Carreau-Yasuda rheological parameters on subcritical Lapwood convection in horizontal porous cavity saturated by shear-thinning fluid

The present study is focused on Lapwood convection in isotropic porous media saturated with non-Newtonian shear thinning fluid. The non-Newtonian rheological behavior of the fluid is modeled using the general viscosity model of Carreau–Yasuda. The convection configuration consists of a shallow porous cavity with a finite aspect ratio and subject to a vertical constant heat flux, whereas the vertical walls are maintained impermeable and adiabatic. An approximate analytical solution is developed on the basis of the parallel flow assumption, and numerical solutions are obtained by solving the full governing equations. The Darcy model with the Boussinesq approximation and energy transport equations are solved numerically using a finite difference method. The results are obtained in terms of the Nusselt number and the flow fields as functions of the governing parameters. A good agreement is obtained between the analytical approximation and the numerical solution of the full governing equations. The effects of the rheological parameters of the Carreau-Yasuda fluid and Rayleigh number on the onset of subcritical convection thresholds are demonstrated. Regardless of the aspect ratio of the enclosure and thermal boundary condition type, the subcritical convective flows are seen to occur below the onset of stationary convection. Correlations are proposed to estimate the subcritical Rayleigh number for the onset of finite amplitude convection as a function of the fluid rheological parameters. Linear stability of the convective motion, predicted by the parallel flow approximation, is studied, and the onset of Hopf bifurcation, from steady convective flow to oscillatory behavior, is found to depend strongly on the rheological parameters. In general, Hopf bifurcation is triggered earlier as the fluid becomes more and more shear-thinning.

Penetration of steady fluid motions into an outer stable layer excited by MHD thermal convection in rotating spherical shells

Penetration of steady magneto-hydrodynamic (MHD) disturbances into an upper strongly stratified stable layer excited by MHD thermal convection in rotating spherical shells is investigated. The theoretical model proposed by Takehiro (2015) is reexamined in the case of steady fluid motion below the bottom boundary. Steady disturbances penetrate into a density stratified MHD fluid existing in the semi-infinite region in the vertical direction. The axis of rotation of the system is tilted with respect to the vertical. The basic magnetic field is uniform and may be tilted with respect to the vertical and the rotation axis. Linear dispersion relation shows that the penetration distance with zero frequency depends on the amplitude of Alfvén wave speed. When Alfvén wave speed is small, viscous diffusion becomes dominant and penetration distance is similar to the horizontal scale of the disturbance at the lower boundary. In contrast, when Alfvén wave speed becomes larger, disturbance can penetrate deeper, and penetration distance becomes proportional to the Alfvén wave speed and inversely proportional to the geometric average of viscous and magnetic diffusion coefficients and to the total horizontal wavenumber. The analytic expression of penetration distance is in good agreement with the extent of penetration of mean zonal flow induced by finite amplitude convection in a rotating spherical shell with an upper stably stratified layer embedded in an axially uniform basic magnetic field. The theory expects that the stable layer suggested in the upper part of the outer core of the earth could be penetrated completely by mean zonal flows excited by thermal/compositional convection developing below the stable layer.

Finite bandwidth, nonlinear convective flow in a mushy layer

Finite amplitude convection with a continuous finite bandwidth of modes in a horizontal mushy layer during the solidification of binary alloys is investigated. We analyze the nonlinear convection for values of the Rayleigh number close to its critical value by using multiple scales and perturbation techniques. Applying a combined temporal and spatial evolution approach, we determine a set of three coupled differential equations for the amplitude functions of the convective modes. A large number of new subcritical or supercritical stable solutions to these equations in the form of steady rolls and hexagons with different horizontal length scales are detected. We find, in particular, that depending on the parameter values and on the magnitude and direction of the wave number vectors for the amplitude functions, hexagons with down-flow or up-flow at the cells’ centers or rolls can be stable. Rolls or hexagons with longer horizontal wave length can be stable at higher amplitudes, and there are cases where hexagons are unstable for any value of the Rayleigh number, while rolls are stable only for the values of the Rayleigh number beyond some value. We also detected new stable and irregular flow patterns with two different horizontal scales in the form of superposition of either two sets of hexagons or two sets of inclined rolls.

Effect of Internal Heating on Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Medium

The effect of internal heat source on double diffusive convection in a couple stress fluid saturated horizontal anisotropic porous layer was studied analytically by performing linear and nonlinear stability analyses. For linear stability analysis, normal mode technique was used, whereas for nonlinear stability analysis a minimal representation of Fourier series upto two terms was used. Effects of anisotropic parameter, couple stress parameter, solute Rayleigh number, Vadasz number, Lewis number, internal heat source parameter on stationary, oscillatory and finite amplitude convection were obtained and shown graphically. Also the graphs for heat and mass transport, streamlines, isotherms, isohalines for steady and unsteady cases were drawn. It was found that internal heat source parameter had destabilizing effect on all modes of convection. Heat transport increased and mass transport decreased with the increase in internal heat source parameter.

Soret Effect on Darcy–Brinkman Convection in a Binary Viscoelastic Fluid‐Saturated Porous Layer

A linear and weakly nonlinear stability analyses is performed to study the onset of Darcy–Brinkman double diffusive convection in a binary viscoelastic fluid‐saturated porous layer in the presence of the Soret effect. The modified Darcy–Brinkman–Oldroyd model including the time derivative term is employed for the momentum equation. The expressions for stationary, oscillatory, and finite amplitude Rayleigh number are obtained as a function of the governing parameters. There is a competition between the processes of the Soret coefficient, viscoelasticity, thermal diffusion, and solute diffusion that causes the convection to set in through an oscillatory mode rather than a stationary mode. The effects of the Soret parameter, Darcy number, relaxation and retardation parameters, and Darcy–Prandtl number on the stationary, oscillatory, and finite amplitude convection is shown graphically. The weakly nonlinear theory is based on truncated representation of the Fourier series method and is used to find the Nusselt and Sherwood numbers. Further, the transient behavior of the Nusselt and Sherwood numbers is investigated by solving the nonlinear system of ordinary differential equations numerically using the Runge–Kutta method. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(4): 297–320, 2014; Published online 3 October 2013 in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21076

Onset of Darcy‐Brinkman Convection in a Binary Viscoelastic Fluid‐Saturated Porous Layer with Internal Heat Source

The onset of Darcy‐Brinkman convection in a binary viscoelastic fluid‐saturated sparsely packed porous layer with an internal heat source is studied using both linear and nonlinear stability analyses. The Oldroyd‐B model is employed to describe the rheological behavior of binary fluid. An extended form of the Darcy‐Oldroyd law incorporating Brinkman's correction and time derivative is used to describe the flow through a porous layer. The onset criterion for stationary, oscillatory, and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion, and viscoelasticity that causes the convection to set in through an oscillatory mode rather than a stationary mode. The effect of internal Rayleigh number, relaxation and retardation parameters, solute Rayleigh number, Darcy number, Darcy‐Prandtl number, and Lewis number on the stability of a system is investigated and is shown graphically. The nonlinear theory based on the truncated representation of the Fourier series method is used to find heat and mass transfer. The transient behavior of the Nusselt and Sherwood numbers is obtained using numerical methods. Some known results are recovered for the particular cases of the present study. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 42(8): 676–703, 2013; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21056

Onset of Double-Diffusive Reaction–Convection in an Anisotropic Rotating Porous Layer

The linear and non-linear stability of a rotating double-diffusive reaction–convection in a horizontal anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated considering a Darcy model that includes the Coriolis term. The effect of Taylor number, mechanical, and thermal anisotropy parameters, reaction rate, solute Rayleigh number, Lewis number, and normalized porosity on the stability of the system is investigated. We find that the Taylor number has a stabilizing effect, chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite-amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.

Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes

AbstractWe model buoyancy-driven convection with chimneys – channels of zero solid fraction – in a mushy layer formed during directional solidification of a binary alloy in two dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesized that the chimney spacing will adjust to optimize the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sublinear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carman–Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states.

Effect of a non-constant magnetic field on natural convection in a horizontal porous layer heated from the bottom

An analytical and numerical investigation is conducted to study the effect of an electromagnetic field on natural convection in a horizontal shallow porous cavity filled with an electrically conducting fluid. The magnetic field is assumed to be induced by two long wires, carrying current, parallel to the horizontal boundaries of the system. A uniform heat flux is applied to the horizontal walls of the layer while the vertical walls are adiabatic. The governing parameters of the problem under study are the thermal Rayleigh number, Ra, Hartmann number, Ha, position of the electrical wires, d, current intensity ratio, r , and aspect ratio of cavity, A. An analytical solution, valid for a shallow layer (A � 1), is derived on the basis of the parallel flow approximation. The critical Rayleigh number, Rac, for the onset of motion is derived in closed form in terms of the parameters of the problem. For finite-amplitude convection the heat and flow characteristics predicted by the analytical model are found to agree well with a numerical study of the full governing equations.

Onset of Convection in a Horizontal Porous Layer Saturated by a Power-Law Fluid

This paper investigates the onset of motion, and the subsequent finite-amplitude convection, in a shallow porous cavity filled with a non-Newtonian fluid. A power-law model is used to characterize the non-Newtonian fluid behavior of the saturating fluid. Constant fluxes of heat are imposed on the horizontal walls of the layer. The governing parameters of the problem under study are the Rayleigh number R, the power-law index n, and the aspect ratio of the cavity A. An analytical solution, valid for shallow enclosures (A ≫ 1), is derived on the basis of the parallel flow approximation. In the range of the governing parameters considered in this study, a good agreement is found between the analytical predictions and the numerical results obtained by solving the full governing equations. For dilatant fluids (n > 1), it is found that the onset of motion is linearly unstable, i.e., always occurs provided that the supercritical Rayleigh number RCsup≥0. For pseudoplastic fluids (n < 1), the supercritical Rayleigh number for the onset of motion is RCsup=∞. However, it is demonstrated, on the basis of the nonlinear parallel flow theory, that the onset of motion occurs above a subcritical Rayleigh number RCsub which depends upon the power-law index n. For finite-amplitude convection, the heat and flow characteristics predicted by the analytical model are found to agree well with a numerical study of the full governing equations.