Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium

The nonlinear stability of the conduction-diffusion solution of a fluid mixture heated and salted from below (and of a homogeneous fluid heated from below) and saturating a porous medium is studied with the Lyapunov direct method. Both Darcy and Brinkman models have been used. The porous medium is bounded by two horizontal parallel planes and is rotating about a vertical axis. Necessary and sufficient conditions of unconditional stability are proved (i.e., the critical linear and nonlinear stability Rayleigh numbers coincide). Our results generalize those given by Straughan [21] for a homogeneous fluid in the Darcy regime. In the case of a mixture two stabilizing effects act: that of the rotation and of the concentration of the solute. Received March 05, 2002 / Published online June 4, 2002 RID="a" ID="a" e-mail: lombardo@dmi.unict.it RID="b" ID="b" e-mail: mulone@dmi.unict.it Communicated by Brian Straugham, Durham     

Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a porous medium

Stability analysis of double-diffusive convection for viscoelastic fluid with Soret effect in a porous medium is investigated using a modified-Maxwell-Darcy model. We use the linear stability analysis to investigate how the Soret parameter and the relaxation time of viscoelastic fluid effect the onset of convection and the selection of an unstable wavenumber. It is found that the Soret effect is to destabilize the system for oscillatory convection. The relaxation time also enhances the instability of the system. The effects of Soret coefficient and relaxation time on the heat transfer rate in a porous medium are studied using the nonlinear stability analysis, the variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Some previous results can be reduced as the special cases of the present paper.     

Rayleigh–Benard convection subject to time dependent wall temperature/gravity in a fluid-saturated anisotropic porous medium

 The effect of time-periodic temperature/gravity modulation at the onset of convection in a Boussinesq fluid-saturated anisotropic porous medium is investigated by making a linear stability analysis. Brinkman flow model with effective viscosity larger than the viscosity of the fluid is considered to give a more general theoretical result. The perturbation method is applied for computing the critical Rayleigh and wave numbers for small amplitude temperature/gravity modulation. The shift in the critical Rayleigh number is calculated as a function of frequency of the modulation, viscosity ratio, anisotropy parameter and porous parameter. We have shown that it is possible to advance or delay the onset of convection by time-periodic modulation of the wall temperature and to advance convection by gravity modulation. It is also shown that the small anisotropy parameter has a strong influence on the stability of the system. The effect of viscosity ratio, anisotropy parameter, the porous parameter and the Prandtl number is discussed. Received on 28 July 2000 / Published online: 29 November 2001     

Study of nonlinear convection in a sparsely packed porous medium using spectral analysis

Nonlinear study cellular convection in a sparsely packed fluid saturated porous medium is investigated, considering the Brinkman model, using the technique of spectral analysis. It is established how the Brinkman model with free-free boundaries generalizes the study of convection in a porous medium in the sense that it yields the results tending to those of viscous and Darcy flows respectively for very small and very large values of the permeability parameter σ². It also provides results for the transition zone (i.e. 10²<σ²<10³). The cross-interaction of the linear modes caused by non-linear effects are considered through the modal Rayleigh number R_γ. The possibility of the existence of steady solution with two self-excited modes in certain regions is predicted. The similarities of present analysis with and advantages over that of the power integral technique are brought out. A detailed discussion of the heat transport, with the effect of permeability thereon, is made. The theoretical values of the Nusselt number are found to be in good agreement with experimental results.     

Effect of gravity modulation on linear,weakly-nonlinear and local-nonlinear stability analyses of stationary double-diffusive convection in a dielectric liquid

Meccanica - 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...     

Non-Darcy double-diffusive natural convection in axisymmetric fluid saturated porous cavities

Double-diffusive natural convection in a fluid saturated porous medium has been investigated using the finite element method. A generalised porous medium model is used to study both Darcy and non-Darcy flow regimes in an axisymmetric cavity. Results indicate that the Darcy number should be a separate parameter to understand flow characteristics in non-Darcy regime. The influence of porosity on heat and mass transfer is significant and the transport rates may differ by 25% or more, at higher Darcy and Rayleigh numbers. When compared with the Darcy and other specialised models of Brinkman and Forchheimer, the present generalised model predicts the least heat and mass transfer rates. It is also observed that an increase in radius ratio leads to higher Nusselt and Sherwood numbers along the inner wall.     

Onset of double-diffusive convection in a saturated porous layer with time-periodic surface heating

The stability of a fluid saturated, horizontal porous layer in the presence of a solute concentration gradient and time-periodic thermal gradient is examined. The modulated gradient is the result of a sinusoidal upper surface temperature which models the effect of variable solar radiation heating of the layer. Darcy's law and the Boussinesq approximation are employed, and we assume an equation of state linear in temperature and concentration. A linear stability analysis is carried out to obtain predictions for the onset of convection and critical wavenumbers for the system. The critical conditions are obtained via the Galerkin method and Floquet theory. The effects of variable concentration gradient, temperature modulation amplitude and frequency are examined, and compared with the results obtained analytically from the corresponding unmodulated problem. It is shown that instabilities can occur as convective motions which are synchronous or subharmonic with the surface heating, or can be identified via complex conjugate Floquet exponents. The neutral stability curves at the transitions between instabilities are found to be bimodal when the temperature is time-periodic, and are characterized by jumps in the critical wavenumbers. Received February 5, 1998     

Horizontal thermal convection in a porous medium

Linearised instability and nonlinear stability bounds for thermal convection in a fluid-filled porous finite box are derived. A nonuniform temperature field in the steady state is generated by maintaining the vertical walls at different temperatures. The linearised instability threshold is shown to be well above the global stability boundary, which is strongly suggested by the lack of symmetry in the perturbed system. The numerical results are evaluated utilising a newly developed Legendre-polynomial-based spectral method.     

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.     

Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium

The stability analysis of the quiescent state in a Maxwell fluid-saturated densely packed porous medium subject to vertical concentration and temperature gradients is presented. A single phase model with local thermal equilibrium between the porous matrix and the Maxwell fluid is assumed. The critical Darcy–Rayleigh numbers and the corresponding wave numbers for the onset of stationary and oscillatory convection are determined. A Lorenz like system is obtained for weakly nonlinear stability analysis.     

Double diffusive convection in a porous medium with modulated temperature on the boundaries

The effect of temperature modulation on the onset of double diffusive convection in a sparsely packed porous medium is studied by making linear stability analysis, and using Brinkman-Forchheimer extended Darcy model. The temperature field between the walls of the porous layer consists of a steady part and a time dependent periodic part that oscillates with time. Only infinitesimal disturbances are considered. The effect of permeability and thermal modulation on the onset of double diffusive convection has been studied using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. The effects of other parameters are also discussed on the stability of the system. Some results as the particular cases of the present study have also been obtained. Also the results corresponding to the Brinkman model and Darcy model have been compared.     

Measurements of thermally-induced convection in a model porous medium

The velocity field generated by thermal convection in a model porous medium is experimentally determined by means of both PIV and LDA techniques. Details of matching refraction index under non isothermal conditions are given. Fields are measured in the empty parallelepipedic cell and in a model medium made of parallel circular bundles. Results are in good agreement. Moreover, by an averaging technique, we are able to measure seeping velocity profiles.     

Magnetic effect on instability and nonlinear stability of double-diffusive convection in a reacting fluid

We study the problem of double-diffusive convection in a reacting fluid with a concentration and magnetic field effect–based internal heat source. A linear instability analysis and nonlinear stability analysis are performed, and using the finite element method of p order, we get the numerical results of each case. The numerical results are presented for fixed–fixed and free–free boundary conditions.     

An approximate solution for free convection flows in a thermally stratified porous medium

A simple and sufficiently accurate method for finding the rate of heat transfer for buoyancy-induced flows over bodies of arbitrary shape embedded in a thermally stratified porous medium is given.     

Onset of convection in a couple-stress fluid-saturated porous medium: effects of non-uniform temperature gradients

Onset of convection in a layer of couple-stress fluid-saturated porous medium is investigated for different types of basic temperature gradients. The boundaries are considered to be adiabatically insulated to temperature perturbations. The eigenvalue equations of the perturbed state obtained from the normal mode analysis are solved analytically using a regular perturbation technique with wave number as a perturbation parameter and also numerically using the Galerkin technique. The critical stability parameters obtained from these two techniques are in excellent agreement and an increase in the value of couple-stress parameter is found to delay the onset of convection. The results also indicate that the piecewise linear temperature profile hastens the onset of convection when compared to linear, parabolic, and inverted parabolic temperature profiles. In addition, the influence of thermal depth on the critical conditions is assessed in the case of piecewise linear temperature profiles, and it is observed that the critical thermal depth decreases marginally with an increase in the couple-stress parameter.     

Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium

Convection induced by the selective absorption of radiation is investigated, for the case of an internal heat source that is modelled quadratically with respect to concentration. The growth rate for the linearised system is shown to be real, and a linear instability analysis is performed. To establish conditional and unconditional nonlinear stability results, both the Darcy and Forchheimer models are employed to describe fluid flow. Due to the presence of significant regions of potential subcritical instabilities, the results indicate that linear theory may only be accurate enough to predict the onset of convective motion when the model for the internal heat source is predominantly linear.Received: 6 May 2003, Accepted: 9 August 2003, Published online: 12 December 2003     

STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

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Combined convection along a non-isothermal wedge in a porous medium

A boundary layer analysis has been presented for the combined convection along a vertical non-isothermal wedge embedded in a fluid-saturated porous medium. The transformed conservation laws are solved numerically for the case of variable surface temperature. Results are presented for the details of the velocity and temperature fields as well as the Nusselt number. The wedge angle geometry parameter m ranged from 0 to 1.