Effect of rotation on the onset of thermal convection in a sparsely packed porous layer using a thermal non-equilibrium model

Linear and nonlinear stability of a rotating fluid-saturated sparsely packed porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium. The extended Darcy–Brinkman model that includes the time derivative and Coriolis terms is employed as a momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for both stationary and oscillatory convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of rotation and thermal diffusion that causes the convection to set in through oscillatory mode rather than stationary. The rotation inhibits the onset of convection in both stationary and oscillatory mode. The Darcy number stabilizes the system towards the oscillatory mode, while it has dual effect on stationary convection. Besides, the effect of porosity modified conductivity ratio, Darcy–Prandtl number and the ratio of diffusivities on the stability of the system is investigated. The nonlinear theory is based on the truncated representation of Fourier series method. The effect of thermal non-equilibrium on heat transfer is brought out. The transient behavior of the Nusselt number is investigated by using the Runge–Kutta method. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

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     

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     

The onset of convection in a binary fluid saturated anisotropic porous layer

The double diffusive convection in a horizontal fluid saturated anisotropic porous layer, which is heated and salted from below, is studied analytically. The generalized Darcy model is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory. The effect of anisotropy parameters, solute Rayleigh number, Lewis number and Prandtl number on the stationary, oscillatory and finite amplitude convection is shown graphically. The transient behavior of the Nusselt and Sherwood numbers is studied, by solving numerically a fifth order Lorenz type system using Runge–Kutta method. Some of the convection systems previously reported in the literature are shown to be special cases of the system presented in this study.     

Effect of anisotropic permeability on double‐diffusive bidisperse porous medium

The model of thermosolutal convection in a fluid‐saturated bidisperse porous medium of Darcy type is studied in this paper. The permeability is allowed to be horizontally isotropic for both the macro‐ and microphases. The linear instability and nonlinear stability are analyzed by taking the Soret effect into account. Furthermore, the effect of anisotropy parameter, Soret coefficient, and other physical parameters on the stability of the system are investigated. It is shown that the linear instability boundaries and the energy stability boundaries do not coincide when the layer is heated and salted from below, where a region of potential subcritical instability occurs. The results reveal that the horizontal to vertical permeability ratio plays a crucial role in the stability of the system. It is also observed that for large values of the salt Rayleigh number, the onset of thermal convection is more likely to be via oscillatory convection rather than stationary convection. Furthermore, the onset of stationary convection is significantly influenced by the presence of the Soret coefficient.     

On the onset of thermal convection in rotating nanofluid layer saturating a Darcy–Brinkman porous medium

In this paper, the effect of rotation on the onset of thermal convection in a horizontal layer of nanofluid saturated by a Darcy–Brinkman porous medium is considered. A linear stability analysis based upon normal mode is used to find solution of the fluid layer confined between two free boundaries. The onset criterion for stationary and oscillatory convection is derived analytically and graphically. The effects of the concentration Rayleigh number, Taylor number, Lewis number, Darcy number and modified diffusivity ratio on the stability of the system are investigated. The sufficient conditions for the non-existence of overstability are also derived.     

Nonlinear convection in an elasticoviscous fluid‐saturated anisotropic porous layer using a local thermal nonequilibrium model

By adopting a perturbation method and a local thermal nonequilibrium model, nonlinear thermal convection in an anisotropic porous layer saturated by an elasticoviscous fluid is investigated. An elasticoviscous fluid is modeled by a modified Darcy‐Oldroyd‐B model, and the fluid and solid phase temperatures are represented using a two‐field model for the heat transport equation. Anisotropy in permeability and fluid and solid thermal conductivities are considered. A cubic Landau equation is derived separately to study the stability of bifurcating solution of both stationary and oscillatory convection, and the results of linear instability theory are delineated. The boundary between stationary and oscillatory convection is demarcated by identifying codimension‐two points in the viscoelastic parameters plane. It is found that the subcritical instability is not possible, and the linear instability analysis itself completely captures the behavior of the onset of convection. Heat transfer is obtained in terms of Nusselt number, and the effect of governing parameters on the same is discussed. The results of the Maxwell fluid are obtained as a particular case from the present study.     

g‐Jitter–Induced Electrothermoconvection in a Dielectric Fluid Saturated Porous Layer

This paper presents linear and nonlinear stability analyses of thermal convection in a dielectric fluid saturated sparsely packed porous layer subject to the combined effect of time‐periodic gravity modulation (GM) and an AC electric field. In the domain of linear theory, the critical stability parameters are computed by the regular perturbation method in the form of a perturbation series in powers of frequency of modulation. The local nonlinear theory based on the truncated Fourier series method gives information on convection amplitudes and heat transfer. The principle of the exchange of stabilities is found to be valid and subcritical instability is ruled out. Based on the governing linear autonomous system, several qualitative results on stability are discussed. The sensitive dependence of the solution of a Lorenz system of electrothermal convection subject to the choice of initial conditions points to the possibility of chaos. Low‐frequency g‐jitter is found to have a significant stabilizing influence, which is in turn diminished by an imposed AC electric field. The role of sparseness of the porous layer, viscosity ratio, and normalized porosity on the stability criterion and on heat transport is determined.     

Low Prandtl number chaotic convection in porous media with uniform internal heat generation

Using the theory of dynamical systems, this study investigated the effects of a uniform internal heat generation on chaotic behaviour in thermal convection in a fluid-saturated porous layer subject to gravity and heated from below for low Prandtl number. A low-dimensional, Lorenz-like model was obtained using Galerkin truncated approximation. The fourth-order Runge–Kutta method was employed to solve the nonlinear system. We found that there is an inverse proportional relation between the level of internal heat G and the scaled Rayleigh number R, and consequently the porous media gravity-related Rayleigh number Ra.     

Double diffusive LTNE porous convection with Cattaneo effects in the solid

The impact of Cattaneo heat flux law in the solid on the onset of double‐diffusive Darcy porous convection with local thermal nonequilibrium temperatures is investigated. The Fourier law of heat transfer is invoked for the fluid, whereas the Cattaneo heat flux law used to transfer heat in solid skeleton alters the temperature equation from parabolic to hyperbolic. The results are obtained for porous skeletons of aluminum and copper oxides. Both Cattaneo and solute concentration effects reinforce in controlling the onset of oscillatory convection and some novel consequences are observed. Compared with the results perceived in the absence of solute concentration, a manifestation of oscillatory convection with scaled‐interphase heat transfer coefficient as well as solid thermal relaxation time parameter initiates earlier in its presence. The effect of increasing interphase heat transfer coefficient and the Lewis number is to delay and hasten the onset of stationary and oscillatory convection. Besides, the increase in the value of solid thermal relaxation time parameter advances the oscillatory onset. Although the increase in the solute Darcy–Rayleigh number is to delay the stationary onset, it shows a twofold behavior on the onset of oscillatory convection. Before the onset of oscillatory convection, the size of the convection cell gets narrower and after which it becomes much wider. The existing results are retrieved as limiting cases from the current study.     

Influence of anisotropy on the Jeffrey fluid convection in a horizontal rotary porous layer

Stability analysis of thermal convection for a Jeffrey fluid with rotation in an anisotropic porous medium is examined utilizing a modified Jeffrey–Darcy model. The linear stability theory is applied to examine how the Jeffrey parameter, rotation parameter, and anisotropic parameters affect the convective motion. It is observed that the rotation and the anisotropic in the thermal diffusivity act to delay the start of Jeffery fluid convection, while the Jeffery parameter and the anisotropic in the permeability show a dual effect in the presence of rotation. The extent of the convection cell diminishes with rotation and Jeffery parameters, while it augments with the thermal anisotropy parameter. Also, some previous outcomes are regained as special cases of the current analysis.     

Effect of rotation and internal heat generation on Darcy–Brinkman convection in Oldroyd-B liquid

Convection in an Oldroyd-B liquid saturated highly permeable porous medium is studied via both linear and nonlinear theories. Estimating a convection threshold is the objective of linear-stability analysis whereas convection amplitudes and heat transfer are elucidated by performing nonlinear-stability analysis. The eigenvalue problem is solved by the Galerkin method of weighted residuals. The oscillatory mode becomes dominant over the stationary mode. This is because of the race among diffusivity, viscoelasticity, internal-heat generation, and rotation. The increasing permeability, internal heat generation coefficient, and stress-relaxation parameter are liable to subcritical motions while the rotation, viscosities ratio, heat capacities ratio, and strain retardation parameter are responsible for the system attaining a supercritical state. The Runge–Kutta–Gill method presents the mechanism to evaluate the amount of heat transfer. The increasing Rayleigh number, internal Rayleigh number, Darcy number, Deborah number, Prandtl number, and the heat capacities ratio enhance the heat transfer. This offers a convenient mechanism for regulating convection. The results obtained in the present paper are expected to play a decisive role in some of the real-life applications such as oil-reservoir modeling, crude oil extraction, crystal growth, medicine industries, geothermal-energy utilization, and so on.     

Double‐Diffusive Convective Transport in a Nanofluid‐Saturated Porous Layer with Cross Diffusion and Variation of Viscosity and Conductivity

The onset of double‐diffusive nanofluid convection in a fluid‐saturated horizontal porous layer is studied with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. The Darcy model has been used for the porous medium, while the nanofluid incorporates the effects of Brownian motion along with thermophoresis. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation in the vertical direction of conductivity and viscosity. In addition, the thermal energy equation includes regular diffusion and cross diffusion terms. The linear stability analysis is based on the normal mode technique, while for nonlinear analysis, minimal representation of the truncated Fourier series representation involving only two terms has been used. It is found that for the stationary mode the Soret parameter, Dufour parameter, viscosity ratio, and conductivity ratio have a stabilizing effect, while the solutal Rayleigh number destabilizes the system. For the oscillatory mode, the Soret parameter, Dufour parameter, and viscosity ratio have a stabilizing effect while the solutal Rayleigh number and conductivity ratio destabilize the system. For steady finite amplitude motions, the heat and mass transport decreases with an increase in the values of the Dufour parameter and solutal Rayleigh number. The Soret parameter enhances the solute concentration Nusselt number while it retards the thermal Nusselt number and concentration Nusselt number. The viscosity ratio and conductivity ratio enhances the heat and mass transports. We also study the effect of time on transient Nusselt numbers which is found to be oscillatory when time is small. However, when time becomes very large, all three transient Nusselt values approach a steady value. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 628–652, 2014; Published online 11 November 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21102     

On oscillatory rotating convection of a couple-stress fluid with helical force

Linear and weakly nonlinear stability analyses of thermosolutal convection in a couple-stress fluid with effects of helical force and rotation are performed. The governing nondimensional equations are solved using the normal modes. We have shown the effect of the helical force parameter, solutal Rayleigh number, Couple stress parameter, Lewis number, Taylor number, and Prandtl number on stationary and oscillatory convection regions and presented graphically. Solutal Rayleigh number, Couple stress parameter, Lewis number, and Taylor number have a stabilizing effect on the system whereas the helical force parameter has a destabilizing effect on the system. To study heat transport by convection we have derived the Ginzburg–Landau equation.     

Further results on stability boundaries and a weak nonlinear analysis for double diffusive convection with rotation

We review some results for linear stability when a rotating doubly diffusive layer is studied. Some additional stability boundaries for ‘salt-finger’ and diffusive convection are predicted. Using the theory of self adjoint operators, the variation of the critical eigenvalue with physical parameters is examined at steady conditions. A weak non-linear analysis that uses a truncated Fourier series representation provides concentration and temperature profiles, and shows that heat and mass transport increase with Rayleigh number but decrease with Prandtl number diffusivity ratio and Taylor number.     

Effect of horizontal pressure gradient on Rayleigh–Bénard convection of a Newtonian nanoliquid in a high porosity medium using a local thermal non-equilibrium model

A study of linear and weakly nonlinear stability analyses of Darcy–Brinkman convection in a water–alumina, nanoliquid-saturated porous layer for stress-free isothermal boundaries, when the solid and nanoliquid phases are in local thermal nonequilibrium, is conducted. The critical eigenvalue is found using the Galerkin approach. The effect of the pressure gradient, thermal conductivity ratio, interphase heat transfer coefficient, inverse Darcy number, and Brinkman number on the heat transport and onset of convection is examined and represented graphically. The critical values of wavenumber and nanoliquid Rayleigh number are found for different problem parameter values. The effect of increasing the porosity-modified ratio of thermal conductivity advances the onset of convection and increases the amount of heat transport, whereas the remaining parameters have the opposite impact on the onset of convection and amount of heat transport. The classical results of the local thermal equilibrium case and Darcy–Bénard convection in the presence of pressure gradient are obtained as a limiting case of the present problem.     

The effects of combined horizontal and vertical heterogeneity and anisotropy on the onset of convection in a porous medium

The effects of both horizontal and vertical hydrodynamic and thermal heterogeneity together with anisotropy of both permeability and thermal conductivity, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. For a square enclosure, horizontal heterogeneity is invariably destabilizing, but vertical heterogeneity can be either stabilizing or destabilizing. For an enclosure whose aspect ratio is optimized to give the minimum value of the critical Rayleigh number, both horizontal and vertical heterogeneity are destabilizing, by an amount dependent on the ratio of the conductivity and permeability anisotropy measures.     

Effect of thermal modulation on the onset of convection in a rotating fluid layer

The stability of a rotating horizontal fluid layer heated from below is examined when, the walls of the layer are subjected to time-periodic temperature modulation. The linear stability analysis is used to study the effect of infinitesimal disturbances. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and wavenumber. The shift in critical Rayleigh number is calculated as a function of frequency of modulation, Taylor number and Prandtl number. It is established that the instability can be enhanced by the rotation at low frequency symmetric modulation and with moderate to high frequency lower wall temperature modulation, whereas the stability can be enhanced by the rotation in case of asymmetric modulation. The effect of Taylor number and Prandtl number on the stability of the system is also discussed. We found that by proper tuning of modulation frequency, Taylor number and Prandtl number it is possible to advance or delay the onset of convection.     

Onset of surface tension driven convection in a fluid layer overlying a layer of an anisotropic porous medium

The paper deals with the criterion for the onset of surface tension-driven convection in the presence of temperature gradients in a two-layer system comprising a fluid saturated anisotropic porous layer over which lies a layer of fluid. The lower rigid surface is assumed to be insulated to temperature perturbations, while at the upper non-deformable free surface a general thermal condition is invoked. Both the Beavers-Joseph and the Jones conditions have been used at the interface to know their preference and prominence in the study of the problem. The resulting eigenvalue problem is solved exactly and also by regular perturbation technique when both the boundaries are insulating to temperature perturbations. It is found that the depth of the relative layers, mechanical and thermal anisotropy parameters have a profound effect on the stability of the system. Decreasing the mechanical anisotropy parameter and increasing the thermal anisotropy parameter leads to stabilization of the system. Besides, the possibility of control of Marangoni convection by suitable choice of physical parameters is discussed in detail.     

Thermal instability of viscoelastic fluids in porous media

A theoretical analysis of thermal instability driven by buoyancy forces is conducted in an initially quiescent, horizontal porous layer saturated by viscoelastic fluids. Modified Darcy’s law is used to explain characteristics of fluid motion. The linear stability theory is employed to find the critical condition of the onset of convective motion. The results of the linear stability analysis show that the overstability is a preferred mode for a certain parameter range. Based on the results of linear stability analysis, a nonlinear stability analysis is conducted. The onset of convection has the form of a supercritical and stable bifurcation independent of the values of the elastic parameters. The Landau equations and the Nusselt number variations are derived for steady and oscillatory modes.