Cross‐diffusion and viscoelastic effects on multidiffusive porous convection

The onset of triply cross‐diffusive convection in a viscoelastic fluid‐saturated porous layer is investigated as the study is found very relevant for describing natural phenomena (contaminant transport, underground water flow, improved oil recovery, polymer processing). A modified Darcy‐Oldroyd‐B model is used to describe the viscoelastic fluid flow in a porous medium with full cross‐diffusion terms in the diffusivity matrix. A normal mode analysis yields an exact dispersion equation of fifth degree and accordingly the criterion for the onset of stationary and oscillatory convection is obtained. The numerical computations are carried out for diffusivity elements experimentally determined for lysozyme‐sodium chloride‐bovine serum albumin (BSA)‐water system. Instability is found to occur via oscillatory mode for a certain choice of governing parameters. The relaxation and retardation viscoelastic parameters portray opposing contributions on the oscillatory onset and an increase in the relaxation parameter is to increase the range of retardation parameter up to which the oscillatory convection is preferred. The cross‐diffusion is to either delay/hasten the onset of instability based on the magnitude of the stratifying agents. Even minute variations in the cross‐diffusion elements indict complete change in the linear instability criteria. The topology of neutral curves disclosed the occurrence of disconnected closed convex oscillatory neutral curve revealing the requirement of three critical solute Darcy‐Rayleigh numbers to state fully the instability criteria instead of the usual single value; a novel result ensured from the study. Moreover, the nature of instability for Oldroyd‐B, Maxwell and Newtonian fluids turns out to be dissimilar for the same governing parameters.     

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.     

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.     

Couple stress effects on the stability of three-component convection-diffusion in a porous layer

In this study, the stability (linear and weak nonlinear) of triple-diffusive convection in a couple stress fluid-saturated porous layer has been studied. A normal mode analysis yields an exact dispersion equation of fourth degree, and the criterion for the onset of stationary and oscillatory convection is obtained accordingly. The numerical computations are carried out for diffusivity elements experimentally determined for an aqueous NaCl-KCl-Sucrose system. Contrary to the double-diffusive couple stress fluid system, it is found that (i) oscillatory convection occurs even if the diffusivity ratios are greater than unity and (ii) a disconnected heart-shaped oscillatory neutral curve exists for certain choices of physical parameters, demonstrating the requirement of three critical values of Darcy-Rayleigh number to specify the linear instability criteria. The impact of a couple stress parameter on some of these unusual behaviors is emphasized. The cubic Landau equations are derived by performing a weak nonlinear stability analysis and the stability of stationary and oscillatory bifurcating solutions is discussed. The heat and mass transfer for stationary and oscillatory convection modes is presented, and the influence of the couple stress parameter on the same is analyzed.     

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.     

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.     

Double diffusive convection of a micropolar fluid saturated in a sparsely packed porous medium

This study examines the double diffusive convection of a sparsely packed micropolar fluid‐saturated porous medium by using a linear stability analysis. The Darcy–Brinkman–Forchheimer model is employed for the porous fluid layer. The stability criterion is sought analytically with the simple free‐free, iso‐thermal, and iso‐solutal boundary conditions. The dependence of stationary or oscillatory convection on the porous parameter, Lewis number, solutal Rayleigh number, and parameters involved in micropolar fluids is drawn and discussed. The results show that the critical wave number is found to be insensitive to the variation of governing parameters except for the porous parameter. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21052     

The effect of rotation on the onset of convection in a horizontal anisotropic porous layer

The effect of rotation and anisotropy on the onset of convection in a horizontal porous layer is investigated using a linear theory and a weak nonlinear theory. The linear theory is based on the usual normal mode technique and the nonlinear theory on the truncated Fourier series analysis. Darcy model extended to include time derivative and Coriolis terms with anisotropic permeability is used to describe the flow through porous media. A modified energy equation including the thermal anisotropy is used. The effect of rotation, mechanical and thermal anisotropy parameters and the Prandtl number on the stationary and overstable convection is discussed. It is found that the effect of mechanical anisotropy is to allow the onset of oscillatory convection instead of stationary. It is also found that the existence of overstable motions in case of rotating porous medium is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The steady finite amplitude analysis is performed using truncated Fourier series to find the Nusselt number. The effect of various parameters on heat transfer is investigated.     

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     

Thermal convection with the Cattaneo–Christov model

We consider the problem of thermal convection in a horizontal layer of incompressible Newtonian fluid with gravity acting downward. The constitutive equation for the heat flux is taken to be one of Cattaneo type. Since we are considering a fluid one has to be careful with the choice of objective derivative for the rate of change of the heat flux. Here we employ a recent model due to Professor C. Christov. The thermal relaxation effect is found to be significant if the Cattaneo number is sufficiently large, and the convection mechanism switches from stationary convection to oscillatory convection with narrower cells.     

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.     

Modified stability analysis of double-diffusive convection in viscoelastic fluid layer saturating porous media

The present analysis mathematically investigates the thermohaline convection problem in viscoelastic fluid layer saturating porous media by utilizing the modified Boussinesq approximation. By performing linear stability analysis, the Darcy–Rayleigh numbers for stationary and oscillatory modes of convection are derived. The effects of different parameters describing the problem are studied numerically. In nonlinear stability analysis, the heat and mass transfer rates in the form of Nusselt and Sherwood numbers, respectively, are obtained for oscillatory convection using the derived Ginzburg–Landau equation. From the results, it is observed that overstability is the preferred mode of instability in linear stability. It is found that in linear double-diffusive convection problems, the stress relaxation imparts a destabilizing effect whereas the strain retardation time, the coefficient of specific heat variation due to temperature, and the concentration gradient have a stabilizing effect on the system's stability. The numerical values of heat and mass transfer rates varied with the coefficient of specific heat showing that the heat transport decreases while the mass transport increases. Also, the stress relaxation time, the concentration gradient, and the gravity modulation's amplitude increase while the strain retardation time decreases the heat and mass transfer rates. The wavelength of oscillations remains unaltered with the variation of specific heat variation due to temperature. The modulation frequency does not affect the heat/mass transfer rate; though, the wavelength of oscillations decreases with increasing frequency.     

Effects of quadratic drag and throughflow on double diffusive convection in a porous layer

The linear stability theory is used to investigate analytically the effects of quadratic drag and vertical throughflow on double diffusive convection in a horizontal porous layer using the Forchheimer-extended Darcy equation. The boundaries of the porous layer are considered to be either impermeable or porous, but perfect conductors of heat and solute concentration. Conditions for the occurrence of stationary and oscillatory convection are obtained using the Rayleigh-Ritz method. Stability boundaries are drawn in the Rayleigh numbers plane and the throughflow is found to influence the mode of instability. It is found that, irrespective of the nature of boundaries, a small amount of throughflow in either of its direction destabilizes the system; a result which is in contrast to the single component system.     

Nonlinear thermohaline convection in rotating fluids

Linear and weakly nonlinear properties of thermohaline convection in rotating fluids are investigated. Linear stability analysis is studied by plotting graphs for different values of physical parameters relevant to the Earth’s outer core and oceans. We have derived a nonlinear two-dimensional Landau–Ginzburg equation with real coefficients near the onset of stationary convection at the supercritical pitchfork bifurcation and shown the occurrence of Eckhaus and zigzag instabilities. We have studied heat transfer by using Nusselt number which is obtained from Landau–Ginzburg equation at the onset of stationary convection for the steady case. A coupled two-dimensional Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection are derived and the stability regions of travelling and standing waves discussed.     

Thermal Interaction between Two Forced‐Convection Boundary Layers across a Conductive Solid Wall

This paper presents an analytical model to the problem of thermal interaction between two forced convection layers of parallel flow on opposite wall sides. The problem is formulated in dimensionless terms to generalize the solution. The two convection layers are analyzed separately by employing the integral technique. The two analyses are then coupled by applying the solid–fluid interfacial conditions. The study indicates that the thermal interaction process is governed mainly by two dimensionless parameters relating the heat transfer effectiveness of two interactive convection modes and wall conduction. The effects of governing parameters on the flow and heat transfer characteristics of two coupled convection layers are documented. Results regarding mean conjugate Nusselt number are obtained for wide ranges of governing parameters.     

Onset of oscillatory double-diffusive buoyancy instability in an inclined rectangular cavity

Double-diffusive buoyancy convection in an inclined rectangular closed cavity with imposed temperatures and concentrations along two opposite sidewalls is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude (buoyancy ratio R_ρ = ?1). In this case a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Gr_c. For both infinite and finite layers, linear stability analysis shows that, when the cavity inclination α with respect to gravity decreases from 0° to ?90°, Gr_c for the onset of stationary instability increases exponentially while that for the onset of oscillatory instability decreases exponentially. Below a critical α_c, the first onset of instability is oscillatory, rather than stationary. For the infinite layer, the influences of α on the critical wave number and frequency of the oscillatory mode are shown and the corresponding flow structure of the eigenfunction consists of counterrotating vortices travelling from one end to the other. For a bounded layer, the neutral stability curves of the first two oscillatory modes, centrosymmetric and anticentrosymmetric, cross each other successively at a series of double Hopf bifurcation points as the aspect ratio increases. These two curves are not smooth, but each contains several abrupt changes, after every one of which a pair of counterrotating vortices is added to the flow field and thus the parity of the mode remains unchanged. The neutral curves showing the influences of Pr and Le are also obtained. The present work expands the work of Bergeon et al. (1999) [8] in which the same physical problem was studied and yet no oscillatory onset of instability was considered.     

Thermal instability of a power-law fluid-saturated porous layer with an internal heat source and vertical throughflow

The thermal instability of internally heated convection in a porous medium saturated by a power-law (PL) fluid is studied. The governing dimensionless equations are solved using the normal mode approach, which leads to an eigenvalue problem for the linear stability theory. The neutral curves are obtained for different prescribed values of the other physical parameters. The effect of the PL index, internal heat source parameter, and Peclet number on the onset of instability was analyzed. Furthermore, an analytical solution is obtained for the regime of small wave numbers by using asymptotic analysis.     

The onset of Lapwood-Brinkman convection using a thermal non-equilibrium model

The stability of a horizontal fluid saturated sparsely packed porous layer heated from below and cooled form above when the solid and fluid phases are not in local thermal equilibrium is examined analytically. The Lapwood-Brinkman model is used for the momentum equation and a two-field model is used for energy equation each representing the solid and fluid phases separately. Although the inertia term is included in the general formulation, it does not affect the stability condition since the basic state is motionless. The linear stability theory is employed to obtain the condition for the onset of convection. The effect of thermal non-equilibrium on the onset of convection is discussed. It is shown that the results of Darcy model for the non-equilibrium case can be recovered in the limit as Darcy number Da → 0. Asymptotic analysis for both small and large values of the inter phase heat transfer coefficient H is also presented. An excellent agreement is found between the exact solutions and asymptotic solutions when H is very small.     

Heat and mass transfer for oscillatory convection in a binary viscoelastic fluid layer subjected to temperature modulation at the boundaries

Weak nonlinear hydrodynamic thermal instability analysis has been performed for double diffusive oscillatory mode of convection in a horizontal layer of viscoelastic fluid, heated from below. Employing complex non-autonomous Ginzburg–Landau equation, effects of various viscoelastic parameters on thermal instability have been investigated. The weak nonlinear analysis reveals that values of viscoelastic parameters have significant effect on the instability. The present study is to investigate the effect of time-periodic temperature modulation on heat and mass transfer, where controlling convection external to the system is important. Sinusoidal profile is taken to modulate the temperature of the boundaries. It is found that the variation of Nusselt number and Sherwood number with respect to the slow time scale becomes rapid as either increasing Rs, Pr, λ, δ or decreasing Γ, ε, Ω. Further, it is found that in-phase temperature modulation has negligible effect, while out of phase temperature modulation and only lower plate temperature modulation have oscillatory effects on heat and mass transport.