A weakly nonlinear thermal instability is investigated under rotation speed modulation. Using the perturbation analysis, a nonlinear physical model is simplified to determine the convective amplitude for oscillatory mode. A non-autonomous complex Ginzburg-Landau equation for the finite amplitude of convection is derived based on a small perturbed parameter. The effect of rotation is found either to stabilize or destabilize the system.
The Nusselt number is obtained numerically to present the results of heat transfer. It is also found that modulation can be used alternately to control the heat transfer in the system.
Further, oscillatory mode enhances heat transfer rather than stationary mode. Ser A. Fluid Mech. A Math. Porous Media, vol. S and Kiran P. Heat Mass Transfer, vol. Heat Transfer, vol. Nonlinear Mech. Heat Mass Transf.2D High-resolution Rayleigh-Taylor Instability in UHD (4K, 2160p)
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Rayleigh-Benard Convection Cells
Sciendo degruyter. Sign In Create Profile. English Deutsch. Manjula 1P. Bhadauria 3. Access Metrics. Keywords: convection ; finite amplitude ; nonlinear theory ; rotation modulation ; Nusselt number. Export References. International Journal of Applied Mechanics and Engineering. Terms Privacy Latest News. Share Share.The onset of thermal convection of a Boussinesq fluid located in an unbounded layer heated from below and subject simultaneously to rotation and magnetic field, whose vectors act in different directions, is presented.
To the knowledge of the authors, the convective thermal instability analysis for this complex problem has not been previously reported.
In this paper, we use the Tau Chebyshev spectral method to calculate the value of the critical parameters wave number and Rayleigh number at the onset of convection as a function of i different kinds of boundaries, ii angle between the three vectors, and iii different values of the Taylor number rate of rotation and magnetic parameter strength of the magnetic force.
The linear stability analysis of this problem, in terms of normal modes, has been carried out and reported in several investigations. The amount of the stabilizing effect depends on the rate of rotation the Taylor number. Results for the problem in which the vectors and act in different directions, with an angle between them, were not explicitly reported by [ 14 ]; instead an analogy with the closely related problem of thermal convection subject to the action of a magnetic field was presented by [ 1 ].
The extent of the inhibition depends on the value of the nondimensional magnetic parameter. It has been found that when and are parallel, convection at marginal stability is characterized by a cellular pattern; hence, longitudinal and transverse rolls appear simultaneously. Results for the problem in which the vectors and act in different directions, with an angle between them, and when instability sets in as stationary convection have been previously obtained by considering two cases: i solution of the perturbation equations which are independent of and ii solutions of the perturbation equations that conduct to a more general patterns of motion.
In the former case, the onset of instability is characterized by rolls in the direction, whereas in the latter case, by using a variational procedure, [ 1 ] shows that as the parameterdefined as where is the wave number in the directionincreases, keeping the Chandrasekhar number defined bywhere,and are the magnetic permeability, the magnitude of the magnetic field, the distance between the two plates, the coefficient of electrical conductivity, the density of reference, and the kinematic viscosity, resp.
It has been shown that when is very slightly inclined to the direction of gravity, the extent to which transverse rolls are suppressed at marginal stability is also very slight.
According to [ 1 ], when longitudinal rolls and transverse rolls appear simultaneously, a cellular pattern of convection emerges. Therefore if and as well as and are not parallel, convection at marginal stability occurs as longitudinal rolls. On the other hand, when is impressed with an inclination angle to the direction of the vertical and lies in the -plane, the vertical component of the magnetic field is equal to zero.
The authors in [ 6 ] found that a magnetic field, impressed in a horizontal directiondid not inhibit convection, even though the magnetic field was five times stronger than that one needed to suppress convection when acting in the vertical direction ; hence, it was experimentally confirmed that the vertical component of the magnetic field is the critical parameter to inhibit the onset of convection as it is included in the Chandrasekhar number. When rotation and magnetic field act simultaneously and when both of them are parallel to the gravity vector, the stabilizing effects have conflicting tendencies; that is, the results reveal some very unexpected features showing the complex behaviour of the flow; thus, it has been found that the total effect is not always stabilizing with respect to both fields [ 1489 ].
The conflictual behaviour depends on the values of the parameters of the system: rotation rate Taylor number and magnetic parameter Chandrasekhar number. Reference [ 4 ] only shows explicit results on the dependence of the critical Rayleigh number on andfor the problem in which the conducting liquid is located between two free boundaries, and when the medium adjoining the fluid is nonconducting, because in this situation the system of equations can be solved explicitly.
To the knowledge of the authors, the solution of the problem, in which the three implicated vectors, and are not parallel, has not been previously reported. In the literature, very few investigations have been carried out aimed to study the cooperative work of perturbations in the flow to inhibit the thermal convection of an infinite layer of fluid; for example, [ 71011 ] have found that when the layer of fluid is simultaneously subject to rotation and to a salt concentration field, the stabilizing effects are cumulative; that is, rotation and salt concentration salted from below show a cooperative behaviour.We also show that the static conduction solution is linearly stable if and only if the Rayleigh number is less than or equal to a critical Rayleigh number.
Finally, we show that a measure of the thermal energy of the fluid decays exponentially which in turn implies that the L 2 norm of the perturbed temperature and velocity also decay exponentially.
This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Google Scholar. Boussinesq J. Gauthier-Villars, Paris Bridgman P. MacMillan, New York Chandrasekhar S. Clarendon Press, Oxford Hron J.
A1—20 Oberbeck A. Oberbeck, A. Pearson J. Fluid Mech. Rajagopal K. Nonlinear Anal. Real World Appl. Rajagopal, K. Nonlinear Analy. Rayleigh L. Rionero S. Pura Appl. Temam, R. Applied Mathematical Sciences, vol. Springer-Verlag, New York Download references. Duranti,Perugia, Italy. Lecce-Arnesano,Lecce, Italy.
Correspondence to K. Reprints and Permissions. Download citation. Received : 16 November Published : 19 February Issue Date : February Search SpringerLink Search.
Immediate online access to all issues from Subscription will auto renew annually.Our name has changed. Physical Sciences Laboratory. Observations Arctic Climate Shipboard Technology. This type of convection pattern occurs in a relatively shallow layer - this could mean a layer of fluid 1 millimeter thick in a petri dish, or the first 2 kilometers of the Earth's atmosphere.
Perfect Conditions, Perfect Pattern Almost! Under the right conditions, convection cells will take the shape of hexagons. Why don't we see hexagon-shaped clouds in the sky?
Take a look at the picture to the right, and notice the small glitch in the pattern. It was later discovered that there was a tiny dent in the copper plate under the fluid. This tells us that the pattern is very sensitive to the bottom surface.
Think about our earth - it's surface has millions of dents and bumps in the form of mountains, valleys, canyons, and more. All of these surface features affect the convection patterns in the atmosphere. Fluid in Motion This picture shows a time lapse view of Rayleigh-Benard cells. The picture was taken over ten seconds, so the aluminum flakes in the fluid look like long trails instead of small particles.
This helps to visulaize how the fluid is moving: up through the center of the cell, then spreading out and sinking at the edges of the cell. Images used with permission of The Parabolic Press.
Van Dyke, M.The classical problem of thermal-convection involving the classical Navier—Stokes fluid with a constant or temperature dependent viscosity, within the context of the Oberbeck—Boussinesq approximation, is one of the most intensely studied problems in fluid mechanics.
In this paper, we study thermal-convection in a fluid with a viscosity that depends on both the temperature and pressure, within the context of a generalization of the Oberbeck—Boussinesq approximation.
We show that the principle of exchange of stability holds and the Rayleigh numbers for the linear and non-linear stability coincide. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Duranti,Perugia, Italy. Lecce-Arnesano,Lecce, Italy. Correspondence to K. Reprints and Permissions. Rajagopal, K. Download citation. Received : 06 June Published : 04 March Issue Date : July Search SpringerLink Search.
Immediate online access to all issues from Subscription will auto renew annually. Taxes to be calculated in checkout. Duranti,Perugia, Italy G. Lecce-Arnesano,Lecce, Italy L. Vergori Authors K. Rajagopal View author publications. View author publications. Rights and permissions Reprints and Permissions. About this article Cite this article Rajagopal, K.Curator: A.
Eugene M. Such flows result from the development of the convective instabilityif the static vertical temperature gradient the gradient that would be present in a motionless fluid under the same conditions is large enough. A horizontal layer of convecting fluid is the most comprehensively studied example of nonlinear systems exhibiting self-organization pattern-forming systems. Convective motion enhances dramatically the heat transfer through the layer compared to the molecular heat conduction.
The moving fluid parcels, which are agents of heat exchange, normally have velocities and effective free paths much greater than the corresponding figures for molecules.
Therefore, the heat flux through the layer of convecting fluid may be several orders of magnitude higher than the heat flux due to molecular thermal conductivity. The role of non-uniform heating as the producer of most types of fluid motions in the Universe was first recognised in the mid-eighteenth century, nearly simultaneously by George Hadley and Mikhail Lomonosov.
Lord Rayleigh was the first to consider a linear problem of the onset of thermal convection in a horizontal layer, and a more comprehensive analysis of this problem was given by Pellew and Southwell Subsequent studies mainly dealt with nonlinear convection regimes and related pattern-formation processes.
The basic non-dimensional parameters controlling the regimes of convection are the Rayleigh and the Prandtl number. Assume that the equations of fluid motion, continuity, and heat transfer are linearised with respect to infinitesimal perturbations. The region above this curve corresponds to growing perturbations, while they all decay in the region below the curve. Nonlinear effects restrict their growth to a certain level.
The selection of this wavenumber is a very subtle issue, which has been addressed in numerous studies see, in particular, Chapter 6 in Getling If the temperatures at these boundaries are fixed, the critical Rayleigh numbers are:. Prior to Rayleigh's study, a necessary but insufficient condition of convective instability was found for compressible atmospheres in the context of stability of the solar atmosphere by K.
Schwarzschild to be. This criterion came to be known as the Schwarzschild criterion. It has a clear physical meaning.
Assume that the atmosphere is stratified adiabatically. If a fluid parcel that was initially in thermal and mechanical equilibrium with the ambient medium is displaced in a vertical direction from its initial position then, to a first approximation, its thermodynamic state experiences adiabatic changes and the parcel remains in equilibrium with the medium at any new height. If the temperature of the motionless medium varies with height more slowly and, accordingly, its density more rapidly than they would vary in the case of an adiabatic distribution, the parcel displaced upward proves to be heavier and the parcel displaced downward proves to be lighter than the medium.
In both cases, the parcel will tend to return to its initial position. If, conversely, the temperature of the medium varies with height more rapidly and the density more slowly than in the adiabatic case, the density difference between the parcel and medium will produce a buoyancy force, which can result in the development of the convective instability.
Dissipative factors, i. For this reason, the fact that the Schwarzschild criterion is satisfied is not sufficient for the development of convection. Since the driving force of convection in a compressible medium is directly related to the excess of the static temperature gradient in its absolute magnitude over the adiabatic gradient, precisely this excess appears, instead of the temperature gradient itself, in the equations of convection for a compressible medium.
Most frequently, the following idealised planforms are considered:. In experiments, nearly two-dimensional rolls Fig. If, however, such an up - down asymmetry is present e. As the Rayleigh number is increased, various flow instabilities can develop. The growth of numerous instability modes is possible, depending on the Rayleigh number, Prandtl number, horizontal wavenumber of the originally developed flow, and many other factors.
A comprehensive study of the instabilities of the convective flows was undertaken by F. Busse and his colleagues [see, in particular, Busse; a fairly complete survey of these studies carried out by was given by Getling ]. In particular, if the original flow forms a roll pattern, the rolls can acquire a wavelike form the zig-zag instabilitya secondary flow in the form of narrower rolls normal to the original ones can develop the cross-roll instabilitythe rolls can undergo a deformation with asymmetrically located left-side and right-side broadenings the skewed-varicose instabilitythe rolls can undulate the oscillatory instabilityetc.
There are also particular instability modes that can transform the original roll flow into a three-dimensional flow that resembles a cell pattern. At sufficiently high supercritical Rayleigh numbers, nonlinear effects can ultimately lead to the development of turbulence Busse ; Koschmieder ; for a summary of some more recent results see Getling GetlingScholarpedia, 7 7 Jump to: navigationsearch.Suspended fibres significantly alter fluid rheology, as exhibited in for example solutions of DNA, RNA and synthetic biological nanofibres.
It is of interest to determine how this altered rheology affects flow stability. A transversely-isotropic fluid treats these suspensions as a continuum with an evolving preferred direction, through a modified stress tensor incorporating four viscosity-like parameters.
We consider the linear stability of a stationary, passive, transversely-isotropic fluid contained between two parallel boundaries, with the lower boundary at a higher temperature than the upper.
To determine the marginal stability curves the Chebyshev collocation method is applied, and we consider a range of initially uniform preferred directions, from horizontal to vertical, and three orders of magnitude in the viscosity-like anisotropic parameters. Determining the critical wave and Rayleigh numbers we find that transversely-isotropic effects delay the onset of instability; this effect is felt most strongly through the incorporation of the anisotropic shear viscosity, although the anisotropic extensional viscosity also contributes.
Our analysis confirms the importance of anisotropic rheology in the setting of convection. We consider the linear stability of a transversely-isotropic fluid contained between two infinitely-long horizontal boundaries of different temperatures as shown in Figure 1to a small arbitrary perturbation. Three different combinations of boundary types will be considered, 1 both boundaries are rigid, 2 both are free, and 3 the bottom boundary is rigid and the top is free. One application of our theory is to fibre-laden fluids, however it holds for any fluid which may be described as transversely-isotropic.
The stress tensor depends on the fibre orientation and linearly on the rate of strain; it takes the simplest form that satisfies the required invariances. We work with the Boussinesq approximation that the flow is incompressible with non-constant density entering only through a buoyancy term.
Given we consider infinitesimal motion of a liquid the Boussinesq approximation is equally valid as for a Newtonian fluid. In his original study Rayleigh was able to find a closed-form solution in the case of both upper and lower boundaries being free, i.
To determine the conditions where instability occurs for other combinations of boundary types, numerical techniques are required Drazin, We briefly discuss the equations and derive the steady state of the transversely-isotropic model section 2and then undertake a linear stability analysis, leading to an eigenvalue problem which is solved numerically sections 3 - 4. The effect of variations in viscosity-like parameters and the steady state preferred direction on the marginal stability curves is considered section 5then we conclude with a discussion of the results in section 6.
In formulating our governing equations we make use of the Boussinesq approximation Chandrasekhar,treating the density as constant in all terms except bouyancy.
Stability analysis of Rayleigh–Bénard convection in a porous medium
Mass conservation and momentum balance leads to the generalised Navier-Stokes equations. In the present study, we assume there is no active behaviour, i. We will consider two types of bounding surfaces; for both types of surface we assume perfect conduction of heat and that the normal component of velocity is zero, i.
The distinction between the types of bounding surfaces is then made through the final two boundary conditions. If the surface is rigid we impose no-slip boundary conditions, if the surface is free we impose zero-tangential stress, i. Results will be presented from three groups of boundary conditions: both surfaces are rigid, both surfaces are free, and the bottom surface is rigid and the top surface is free. The model is non-dimensionalised by scaling the independent and dependent variables via:.
The incompressibility condition 1 and the kinematic equation 4 remain unchanged by this scaling. The momentum balance 2 becomes. The Rayleigh number R is a dimensionless parameter relating the stabilising effects of molecular diffusion of momentum to the destabilising effects of buoyancy Drazin, ; Koschmieder, ; Sutton,and the Prandtl number P relates the diffusion of momentum to diffusion of thermal energy Chandrasekhar, Non-dimensionalising the stress tensor 5 yields.
The constitutive equation 7 for variable density is non-dimensionalised to give. Finally, the boundary conditions 8 and 9in dimensionless form, are.