I. S. Shivakumara

The effect of rotation on linear and non-linear double-diffusive convection in a sparsely packed, porous medium

The linear and non-linear stability of rotating double-diffusive convection in a sparsely packed porous medium is investigated considering a non-Darcy equation. In the case of linear theory both marginal and overstable motions are discussed. In the former case it is shown that the effect of Taylor number and porous parameter is to make the system more stable. In the latter case, however, it is shown that the bottom-heavy solute gradient and rotation destabilize the system under certain conditions. By drawing the stability boundaries in the Rayleigh number plane it is shown that the effect of rotation and porous parameter is to decrease the region of instabilities. Using the theory of self-adjoint operator the variation of critical eigenvalue with physical and boundary parameters is studied. In the case of non-linear theory, both steady and unsteady cases have been considered. In the unsteady case the transient behaviour concerning the variation of Nusselt number with time has been investigated, by solving numerically a seventh-order Lorenz model using the Runge-Kutta-Gill method. Interesting results are obtained by comparing these results with those of the steady case. Finally, the effect of porous parameter on streamfunction, isotherms, isohalines and zonal velocity is studied.

Effect of non-uniform basic temperature gradient on Rayleigh-Benard-Marangoni convection in ferrofluids

The effect of different basic temperature gradients on the onset of ferroconvection driven by combined surface tension and buoyancy forces is studied. The lower boundary is assumed to be rigid and either conducting or insulating to temperature perturbations while the upper boundary at which the surface tension acts is free insulating and non-deformable. The resulting eigenvalue problem is solved by the Galerkin technique for various basic temperature gradients. The results indicate that the stability of Rayleigh–Benard–Marangoni ferroconvection is significantly affected by basic temperature gradients and the mechanism for suppressing or augmenting the same is discussed in detail. It is shown that the results obtained under the limiting conditions compare well with the existing ones.

Effect of Boundary Conditions on the Onset of Thermomagnetic Convection in a Ferrofluid Saturated Porous Medium

The onset of thermomagnetic convection in a ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated for a variety of velocity and temperature boundary conditions. The Brinkman-Lapwood extended Darcy equation, with fluid viscosity different from effective viscosity, is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-ferromagnetic, while the upper boundary is considered to be either rigid-ferromagnetic or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective-radiative exchange at the upper boundary, which encompasses fixed temperature and heat flux as particular cases. The resulting eigenvalue problem is solved using the Galerkin technique and also by using regular perturbation technique when both boundaries are insulated to temperature perturbations. It is found that the increase in the Biot number and the viscosity ratio, and the decrease in the magnetic as well as in the Darcy number is to delay the onset of ferroconvection. Besides, the nonlinearity of fluid magnetization has no effect on the onset of convection in the case of fixed heat flux boundary conditions. Copyright © 2009 by ASME.

On the stability of double diffusive convection in a porous layer with throughflow

SummaryThe effect of throughflow on the stability of double diffusive convection in a porous layer is investigated for different types of hydrodynamic boundary conditions. The lower and upper boundaries are assumed to be insulating to temperature and concentration perturbations. The resulting eigenvalue problem is solved by the Galerkin technique. The curvature of the basic temperature as well as solute concentration gradients significantly affects the stability of the system. It is observed that, for a suitable choice of parametric values, Hopf bifurcation occurs always prior to direct bifurcation, and the throughflow alters the nature of bifurcation. In contrast to the single component system, it is found that throughflow is (a) destabilizing even if the lower and upper boundaries are of the same type, and (b) stabilizing as well as destabilizing, irrespective of its direction, when the boundaries are of different types.

Convective instability in superposed fluid and porous layers with vertical throughflow

A closed form solution to the convective instability in a composite system of fluid and porous layers with vertical throughflow is presented. The boundaries are considered to be rigid-permeable and insulating to temperature perturbations. Flow in the porous layer is governed by Darcy–Forchheimer equation and the Beavers–Joseph condition is applied at the interface between the fluid and the porous layer. In contrast to the single-layer system, it is found that destabilization due to throughflow arises, and the ratio of fluid layer thickness to porous layer thickness, ζ, too, plays a crucial role in deciding the stability of the system depending on the Prandtl number.

Boundary and inertia effects on convection in porous media with throughflow

SummaryThe effects of a nonuniform temperature gradient and inertia arising due to throughflow on the onset of convection in a porous layer for different types of boundaries are investigated. Closed form solutions are obtained for the boundaries which are insulating to temperature perturbations, and for the conducting boundaries solutions are obtained using Galerkin technique. It is found that when the two boundaries are of the same type, the effect of throughflow is to stabilize the system irrespective of its direction. However, when the lower and upper boundaries are of different types, a small amount of throughflow in one particular direction destabilizes the system depending upon the values of the Prandtl number and the porous parameter. The standard results available are obtained as limiting cases.

Non-darcian effects on the onset of convection in a porous layer with throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.

Non-Darcian effects on double diffusive convection in a sparsely packed porous medium

SummaryThe linear and non-linear stability of double diffusive convection in a sparsely packed porous layer is studied using the Brinkman model. In the case of linear theory conditions for both simple and Hopf bifurcations are obtained. It is found that Hopf bifurcation always occurs at a lower value of the Rayleigh number than one obtained for simple bifurcation and noted that an increase in the value of viscosity ratio is to delay the onset of convection. Non-linear theory is studied in terms of a simplified model, which is exact to second order in the amplitude of the motion, and also using modified perturbation theory with the help of self-adjoint operator technique. It is observed that steady solutions may be either subcritical or supercritical depending on the choice of physical parameters. Nusselt numbers are calculated for various values of physical parameters and representative streamlines, isotherms and isohalines are presented.

Effects of throughflow and internal heat generation on the onset of convection in a fluid layer

SummaryThe throughflow and internal heat generation effects on the onset of convection in an infinite horizontal fluid layer are investigated. The boundaries are considered to be rigid (however permeable) and perfectly conducting. The resulting eigenvalue problem is solved by using the Galerkin method, and the effects of various parameters in the stability results are analyzed. The results indicate that the stability of the system is significantly affected by both throughflow and internal heat generation in the fluid layer. The Prandtl number comes into play due to the presence of throughflow and it has a profound effect on the stability of the system. It is found that, in the presence of internal heating, throughflow in one direction supresses convection while throughflow in the other direction encourages it.

Double-diffusive convection with an imposed magnetic field

The linear and finite amplitude two-dimensional double-diffusive magnetoconvection (thermohaline convection in the presence of a magnetic field) has been studied analytically with free horizontal boundaries held at fixed temperature and concentration. It is shown that the magnetic field acts as a third diffusing component and its effect is to suppress convection. In the case of linear theory the conditions for direct and oscillatory modes are obtained and the stability boundaries for salt-finger and double-diffusive convection are predicted in the Rayleigh number plane. If τ2, the ratio of magnetic diffusivity to thermal diffusivity, is small and the solute Rayleigh number Rs and Chandrasekhar number Q are sufficiently large convection sets in as overstable oscillations and the onset of it is approximated by two straight lines in the Rayleigh number plane. It is found that the salt-finger and overstable modes may be simultaneously unstable over a wide range of conditions and the effect of the magnetic field is to suppress this region. In the case of nonlinear theory it is found that the finite amplitude magnetoconvection exists for subcritical values of the Rayleigh number R, for all Q and τ1 (which is the ratio of solute diffusivity to the thermal diffusivity) when τ2 = 0.1 and Rs = 104. It is found that the heat transport increases with an increase in R and decrease in τ2 but decreases with Q.