Stability analysis of double-diffusive convection in superposed fluid and porous layers using a one-equation model

Linear stability analysis has been carried out to predict the onset of double-diffusive convection in superposed fluid and porous layers using a one-equation model. The eigenvalue problem is solved numerically by a finite difference scheme. Results have been obtained for the thermal convection and salt-finger cases. Comparing with the results obtained for the same problems by Chen and Chen [F. Chen, C.F. Chen, J. Heat Transfer 110 (1988) 403–409] using a two-equation model, we find that these two methods give the same general characteristics of the marginal stability curves, however, there are differences in the critical conditions and the flow streamlines at onset. Carefully conducted experiments are needed to determine which model gives the more realistic results.     

Stability of natural convection in superposed fluid and porous layers: Equivalence of the one- and two-domain approaches

Stability analyses of thermal and/or solutal natural convection in a configuration composed by a fluid layer overlying a homogeneous porous medium have been performed using different modeling approaches, especially for the treatment of the interfacial region. Comparisons between the one-domain approach and the two-domain formulation have shown important discrepancies of the marginal stability curves. This note shows that, according to Kataoka [I. Kataoka, Local instant formulation of two-phase flow, Int. J. Multiphase Flow 12(5) (1986) 745–758.], the differentiation of the macroscopic properties of the porous layer at the interface (porosity, permeability, thermal effective diffusivity) must be considered in the meaning of distributions. In that case, the one- and the two-domain approaches are shown to be equivalent and very good agreement is indeed found when comparing the results obtained with both approaches.     

Critical Rayleigh number of natural convection in high porosity anisotropic horizontal porous layers

The critical Rayleigh number Ra_C at the onset of natural convection was studied by linear stability analysis for high porosity anisotropic horizontal porous layers of uniformly arraying vertical thin circular wires stretched across a hot and a cold surface. Navier–Stokes equations including flow resistance by the wires were solved since Darcy’s law cannot be applied due to high porosity. Ratio of permeability in the horizontal direction to the vertical direction is constant to be 0.5 but ratio of effective thermal diffusivity ξ is changed dependent on wire materials. The critical Rayleigh number Ra_C for the case of ξ = 1 obtained by the analysis agreed well with the experiment.     

Darcy–Benard–Marangoni convection in porous media

The onset of coupled Darcy–Benard–Marangoni convection in a liquid saturated porous layer of high permeability of practical importance is investigated by employing the Brinkman–Forchheimer– Lapwood-extended Darcy flow model with fluid viscosity different from effective viscosity. The lower boundary is taken to be rigid and insulating to temperature perturbations, while the upper surface is open to atmosphere and subject to a general thermal condition. The critical eigenvalues are obtained numerically, in general, using Galerkin method. However, closed form solution is also obtained using regular perturbation technique for insulated boundaries. Besides, the eigenvalue problem is solved exactly for pure Darcy–Marangoni convection. The numerical and analytical results are found to be in excellent agreement with each other. It is observed that the effect of buoyancy is destabilizing, while an increase in the permeability parameter is to delay the onset of convection. The Biot number and the ratio of effective viscosity to fluid viscosity are found to increase the critical conditions. Some known results are recovered as special cases.     

Coupled thermocapillary convection on Marangoni convection in liquid layers with curved free surface

This paper numerically studied the coupled Marangoni convection and thermocapillary convection in a finite liquid layer (Pr = 11.6) in the microgravity conditions. The multi-cellular flow structure and the marginal instability boundary of the coupled convection are predicted. Oscillatory coupled convection is also reported in concave liquid layers of volume ratio between 0.80 and 0.85.     

Control of oscillatory Marangoni convection in a rotating fluid layer

The effect of feedback control on the onset of steady and oscillatory surface-tension-driven (Marangoni) convection in a rotating horizontal fluid layer with a flat free upper surface and heated from below is considered theoretically using linear stability theory. The role of the controller gain parameter on the Ta–Pr parameter space, dividing stability domains into which either steady or oscillatory convection is preferred, is determined.     

Stabilities of combined radiation and Rayleigh–Bénard–Marangoni convection in an open vertical cylinder

This paper studies the combined radiation and natural convection in an open vertical cylinder. The cylinder is heated through its sidewall at a constant heat flux and cooled at the top surface via radiation. The high order finite difference method is used to simulate the fluid flow and heat transfer inside the cylinder and the internal radiation is solved using discontinuous finite element method. The two numerical methods are coupled through an iterative process. Based on the numerical simulations, a linear stability analysis is carried out to investigate how the internal radiation changes the stability of the flow.     

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.     

The onset of convection in porous layers with multiple horizontal partitions

In this paper we investigate the onset of convection in a horizontally partitioned porous layer which is heated from below. Identical sublayers are separated by thin impermeable barriers. A linear stability analysis is performed, and dispersion relations are obtained directly and explicitly for two- and three-layer configurations. A systematic numerical procedure is devised to compute the dispersion relation for an arbitrary number of sublayers, but from this it is possible to guess the correct analytical form of the dispersion relation for general cases.Neutral stability curves are found to organise themselves into natural groups of N members when there are N sublayers. When the disturbance wavenumber, k, is large, each member of any group lies within an O(k^?1) distance of all other members, but within an O(1) distance of other groups. When the number of sublayers is large, the system tends towards one with a critical Darcy–Rayleigh number of 12 and a critical wavenumber of zero; this is the well-known property of a single porous layer with constant heat flux boundary conditions. An asymptotic analysis is performed in order explore these two apparently disparate configurations. Finally, another asymptotic analysis is used to determine the critical Rayleigh number and its associated wavenumber when the number of sublayers is large.     

Thermosolutal Marangoni convection of Bingham non-Newtonian fluids within inclined lid-driven enclosures full of porous media

Multiconvection modes together with the entropy generation due to Marangoni effects, movement of the side walls, and double-diffusive convection within lid-driven enclosures filled by a porous medium are examined. The current flow domain is an inclined non-Darcy porous geometry having a top free surface where linear expressions in terms of the temperature and concentration are presented for the surface tension. The dynamic viscosity of the suspension has an exponential profile and the convective boundary conditions are taken into account. The finite volume method in which values of the pressure are computed using the SIMPLE algorithm is applied to solve the governing system. The non-Newtonian Bingham fluids are applied for different governing parameters, such as Grashof number, Reynolds number, Marangoni number, Darcy number, inclination angle of the cavity, Biot number, Bingham number, and geometrical parameters. It is observed that a rise in the Bingham number from 0 to 0.5 causes a decrease in the average Bejan number up to 4.77%. Also, the averaged and mixing temperatures are augmented as the inclination angle is altered, regardless values of the Biot number.     

Onset of Marangoni convection in variable-viscosity fluid layer subject to uniform heat flux from below

The onset of stationary Marangoni instabilities in a horizontal fluid layer with free surface deformation and heated from below with a uniform heat flux is considered theoretically using linear stability theory. The explicit solution is obtained and the influences of temperature-dependent viscosity, surface deformability, gravity waves and heat transfer mechanism at the free surface on the stability thresholds are investigated. Small stabilizing effect is observed in fluids with a small viscosity variation while large viscosity variation strongly destabilizes the fluid layer. The stability thresholds are critically dependent on viscosity variation, surface deformation and heat transfer mechanism.     

Unsteady MHD free convection of a micropolar fluid between two parallel porous vertical walls with convection from the ambient

The present work is concerned with the unsteady free convection flow of an incompressible electrically conducting micropolar fluid, bounded by two parallel infinite porous vertical plates submitted to an external magnetic field and the thermal boundary condition of forced convection. The governing equations are solved using a numerical technique based on the electrical analogy, where only previous spatial discretization is necessary to obtain a stable and convergent solution with very low computational times. To solve the system of algebraic equations with time as continuous function, an electric circuit simulator is used. This method permits the direct visualization of the local and/or integrated transport variables (temperatures and velocities) at any point or section of the medium. Numerical results for temperature, velocity and microrotation are illustrated graphically.     

Non-Darcy natural convection of a non-Newtonian fluid in a porous cavity

A numerical study of non-Darcy natural convection in a porous enclosure saturated with a power-law fluid is presented. Hydrodynamic and heat transfer results are reported for the configuration in which the enclosure is heated from a side-wall while the horizontal walls are insulated. The flow in the porous medium is modeled using the modified Brinkman–Forchheimer-extended Darcy model for power-law fluids, which accounts for both inertia and boundary effects. The results indicate that when the power law index is decreased, the circulation within the enclosure increases leading to a higher Nusselt number and these effects are enhanced as the Darcy number is increased. Consequently as the power law index decreases, the onset of the transitions from Darcy regime to Darcy–Forchheimer–Brinkman regime to asymptotic convection (boundary layer) regime shift to higher corresponding values of the Darcy number. An increase in Rayleigh number produces similar effects as a decrease in power law index.     

Experimental study of natural convection in fluid-superposed porous layers heated locally from below

An experimental study of natural convection in fluid-superposed porous layers heated locally from below is reported. Measurements are made in a rectangular chamber with 3 mm DIA glass beads as the porous layer and distilled water as the saturating fluid. The effects of the heater-to-cavity length ratio and the porous layer-to-cavity height ratio on the overall heat transfer coefficients are reported. Average heat transfer coefficients over the heated surface increase with a decrease in porous layer-to-cavity height ratio, but no clear effect of heater-to-cavity length ratio is seen. Temperature profiles in the domain reveal a plume like flow with a single pair of circulating cells and evidence of convective motion inside the porous layer.     

Stability of Natural Convection of Power—law Fluid and non—Darcy Flow in Forous Media

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On effect of non-uniform basic temperature gradient on Bénard–Marangoni convection in micropolar fluid

Linear stability analysis is performed to study the effect of non-uniform basic temperature gradients on the onset of Bénard–Marangoni convection in a micropolar fluid. The influence of various parameters on the onset of convection has been analysed. The possibility of delaying the onset of convection by the application of a cubic basic state temperature profile is demonstrated.