Slip boundary condition effect on double‐diffusive convection in a porous medium: Brinkman Model

The model of double‐diffusive convection in a porous medium layer was analyzed using the Brinkman model and concentration based on an internal heat source. Linear instability analysis of the model was performed. Particularly, we analyzed the effect of slip boundary conditions on the instability of the system. We analyzed when the instability started and computed the critical Rayleigh number as a function of the slip coefficient.     

Cross flow on transient double‐diffusive natural convection in inclined porous trapezoidal enclosures

This paper investigates the cross‐diffusion effects subject to exponential variable boundary conditions on transient double‐diffusive natural convection flow in an enclosure. The flow domain is a two‐dimensional inclined trapezoidal cavity filled with a porous medium. The top wall is assumed to be insulated and permeable, while the enclosure's bottom wall is subject to exponential varying temperature and concentration. The prescribed temperature and concentration are different at the vertical walls. Conservation equations are used as the governing equations. The finite element Galerkin weighted residual method, in association with the Newton‐Raphson scheme is employed to solve the system of coupled nondimensional equations. The numerical tests are confirmed with existing literature and are found to be in excellent agreement. The simulations results for stream functions, isotherms, and isoconcentrations are discussed for the various flow parameters. A sensitivity analysis using the response surface method suggests that the average Nusselt and Sherwood numbers are more sensitive to the cross‐diffusion effects. It is further observed that the cross‐diffusion terms stabilize the sensitivity to the angle of inclination.     

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     

Mono‐diffusive Hadley‐Prats flow in a horizontal porous layer subject to variable gravity and internal heat generation

Analysis of internal heated and gravity effect on the onset of Hadley‐Prats flow in a horizontal porous layer with inclined temperature gradients is investigated using the linear and nonlinear instability analysis. The transformed eigenvalue problem is evaluated numerically to find the eigenvalue, which is treated as a vertical thermal Rayleigh number (R_z). It is evaluated by applying shooting and Runge‐Kutta method. Also, the critical R_z is investigated for different parameters governing the flow. A theoretical study is made to understand the influence of gravity field on the mechanism of mono‐diffusive instability of Hadley‐Prats convection in a fluid saturated horizontal porous layer. Nonlinear stability is evaluated by using energy functional. The comparison between linear and nonlinear instability results are presented and it is noted that linear theory of instability may not be useful to capture the complete picture of stability and instabilities may arise before one attains the linear stability threshold. This subcritical instability region is identified between the linear and energy thresholds in the parameter space of the problem considered.     

Unconditional nonlinear stability for double‐diffusive convection in a porous medium with temperature‐dependent viscosity and density

In this study, fluid flow in a porous medium is analyzed using a Forchheimer model. The problem of double‐diffusive convection is addressed in such a porous medium. We utilize a higher‐order approximation for viscosity‐temperature and density‐temperature, such that the perturbation equations contain more nonlinear terms. For unconditional stability, nonlinear stability has been achieved for all initial data by utilizing the or norms. It also shows that the theory of is not sufficient for such unconditional stability. Both linear instability and nonlinear energy stability thresholds are tested using three‐dimensional (3D) simlations. If the layer is salted above and salted below then stationary convection is dominant. Thus the critical value of the linear instability thresholds occurs at a real eigenvalue , and our results show that the linear theory produces the actual threshold. Moreover, it is known that with the increase of the salt Rayleigh number, R_c, the onset of convection is more likely to be via oscillatory convection as opposed to steady convection. The 3D simulation results show that as the value of R_c increases, the actual threshold moves towards the nonlinear stability threshold, and the behavior of the perturbation of the solutions becomes more oscillatory.     

Magnetohydrodynamic forced convection in bidisperse porous medium: Analysis of flow and temperature distributions in a parallel-plate channel

This study aims to explore magnetohydrodynamic forced convection in a parallel-plate channel filled with a bidisperse porous medium, while emphasizing the significance of viscous dissipation. The study utilizes the two-velocity two-temperature model to analyze the flow and temperature distributions in both the fluid phase and solid phase. Convective boundary conditions at the channel walls are considered, and momentum slip is incorporated into the analysis. By nondimensionalizing the governing equations and employing the Homotopy Analysis Method, the velocity and temperature profiles for both phases are determined. Notably, the findings of the study highlight a notable discrepancy in the temperature increase between the solid phase and the fluid phase. Furthermore, the study investigates the impact of various parameters, such as the Darcy number, Biot number, slip parameter, Hartmann number, and Brinkman number, on velocity, temperature, Nusselt number, and skin friction.     

Influence of Lewis number on double‐diffusive convection in a square cavity filled with binary gas

This paper presents double‐diffusive convection in a square cavity filled with binary gas, due to horizontal opposing temperature and concentration gradients. The effect of Lewis number was considered under the conditions of Prandtl number Pr = 1, buoyancy ratio N = 1, and thermal Rayleigh number Ra_T = 10⁴ and 10⁵. Numerical solutions are obtained by a Chebyshev collocation technique with high resolution. Depending on the Lewis number, three kinds of flow structures are identified: symmetric steady flow, asymmetric oscillatory flow, and symmetric oscillatory flow. Oscillatory flow occurs in the regime of thermal dominant flow and it leads to a periodic change between stable and unstable states in species stratification due to the thermo‐solutal interaction. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(1): 85–97, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10073     

Numerical research of double‐diffusive natural convection in elliptical cylinders: Effect of thermal Rayleigh number

This paper is the first part of a two‐part study, and it presented numerical research of double‐diffusive natural convection within an annulus area, situated in two horizontal confocal elliptic solids charged by a Newtonian fluid. The elliptical coordinates were used to transform the physical domain into a rectangular one. To resolve the governing equations and the boundary layer conditions, a calculator code based on the finite volume approach was developed. The details of the influences of thermal Rayleigh number on heat and mass transfer were investigated. Results obtained were compared with those existing in other reasearch works.     

Influence of Lewis number on double‐diffusive convection in a square cavity filled with a binary gas

This paper considers double‐diffusive convection in a square cavity filled with a binary gas, due to horizontal opposing temperature and concentration gradients. The effect of Lewis number was considered under the conditions of Prandtl number Pr = 1, buoyancy ratio N = 1, and thermal Rayleigh numbers R_aT = 10⁴ and 10⁵. Numerical solutions are obtained by a Chebyshev collocation technique with high resolution. Depending on the Lewis number, three kinds of flow structures are identified: symmetric steady flow, asymmetric oscillatory flow, and symmetric oscillatory flow. Oscillatory flow occurs in the regime of thermal dominant flow, and leads to a periodic change between stable and unstable states in species stratification due to the thermo‐solutal interaction. © 2000 Scripta Technica, Heat Trans Asian Res, 30(1): 63–75, 2001     

Numerical solution of the onset of Buoyancy‐driven nanofluid convective motion in an anisotropic porous medium layer with variable gravity and internal heating

The onset of buoyancy‐driven convective motion in a nanofluid saturated anisotropic porous medium layer is examined numerically in the occurrence of uniform internal heat source under the variable gravity field. Three kinds of gravity force variation functions: (a) G(z) = ?z (linear), (b) G(z) = ?z² (parabolic), and (c) G(z) = ?z³ (cubic) are considered. Wide‐range governing parameters impacts are inspected on the beginning of convective motion under the zero nanoparticle flux situation at the boundaries using the higher term Galerkin technique. It is established that the thermal anisotropy parameter η and the gravity variation parameter λ delay the arrival of convective motion, while the mechanical anisotropy parameter ξ, the internal heating parameter Hs, the nanoparticle Rayleigh‐Darcy number R_np, the modified diffusivity ratio NA_nf, and the modified nanofluid Lewis number Le_nf rapid the start of convective motion. The size of the convective cells reduces on raising the internal heating parameter Hs, while the gravity variation parameter λ, the mechanical anisotropy parameter ξ, the thermal anisotropy parameter η, the nanoparticle Rayleigh‐Darcy number R_np, the modified diffusivity ratio NA_nf, and the modified nanofluid Lewis number Le_nf amplify the dimension of the convective cells. It is also detected that the arrangement is more unstable for case (c), while it is more stable for case (a).     

Free convection in a square cavity filled with a bidisperse porous medium

The classical problem of steady Darcy free convection in a square cavity filled with a porous medium has been extended to the case of a bidisperse porous medium (BDPM) by following the recent model proposed by Nield and Kuznetsov [D.A. Nield, A.V. Kuznetsov, Natural convection about a vertical plate embedded in a bidisperse porous medium, Int. J. Heat Mass Transfer 51 (2008) 1658–1664] and Rees et al. [D.A.S. Rees, D.A. Nield, A.V. Kuznetsov, Vertical free convective boundary-layer flow in a bidisperse porous medium, ASME J. Heat Transfer 130 (2008) 1–9]. The transformed partial differential equations in terms of the dimensionless stream function and temperature are solved numerically using a finite-difference method for some values of the governing parameters when the Rayleigh number Ra is equal to 10² and 10³. Results are presented for the flow field with streamlines, temperature field by isotherms and heat transfer by local and mean Nusselt numbers are presented for both the f- and p-phases. It is found that the most important parameters that influence the fluid flow and heat transfer are the inter-phase heat transfer parameter H and the modified thermal conductivity ratio parameter γ.     

Thermosolutal convection in a fluid‐porous composite medium

In the present study analysis has been performed for thermosolutal convection in a fluid‐porous composite medium, consisting of a fluid‐saturated porous medium followed by an overlaying clean medium. The fluid‐porous composite medium is subjected to both a horizontal solutal and a thermal gradient. Top and bottom walls of the fluid‐porous composite medium are assumed to be impermeable and adiabatic. The Darcy‐Brinkman‐Forchheimer model is used to study the flow through the fluid‐porous composite medium. A single domain approach is taken into consideration for numerical simulation. The solution is done by control volume integration. A comprehensive analysis has been performed for various pertinent parameters to delineate their behavior. Results of the transport phenomenon have been provided in graphical and tabular form, for the complete understanding of the complex phenomenon. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21048     

Temporal analysis of dual phase‐lag double‐diffusive MHD flow within a porous microchannel with chemical reaction

A novel investigation is carried out to capture the transient effects of a dual phase‐lag (DPL) model for combined heat and mass transfer magnetohydrodynamic (MHD) flow within a porous microchannel in the presence of Dufour effects and homogenous first‐order chemical reaction. The governing equations for the fluid flow problem are solved using the Laplace transform method, which is a powerful technique for solving partial differential equations. Its inversion is done by using the INVLAP subroutine of MATLAB. The numerical values of fluid velocity, fluid temperature, and species concentration are demonstrated graphically and those of skin friction, heat transfer rate, and mass transfer rate are presented through tables. It is for the first time that the actual time gap between the DPL model, the Cattaneo‐Vernotte model, and the classical Fourier?s model has been deciphered and the results unique to the DPL model are presented. We observe a clear difference between the DPL and the other two models at a dimensionless time , which gradually diminishes as time progresses, and all models coincide together at , that is, where a steady state temperature is reached. An important contribution of this study lies in discovering the time‐bound effects of the phase‐lag parameters of the DPL model on fluid temperature, species concentration, and fluid velocity and support them by physical justification. A similar discussion is provided for all other flow parameters. The results conveyed through this study would undoubtedly help researchers to advance the design of mechanical systems in microdevices involving MHD flow in porous media.     

Prediction of flow and heat transfer through a microtube filled with bidisperse porous medium under local thermal nonequilibrium condition

In this paper, the thermal and hydrodynamic solutions of a microtube filled with bidisperse porous medium (BDPM) under the local thermal nonequilibrium (LTNE) condition are presented. Considering the LTNE condition, the energy equations have been numerically solved. The rarefaction effects are considered for Knudsen numbers ranging from 0 to 0.1; therefore, first‐order boundary condition is applied on the wall. The temperature distribution of each phase is examined with respect to the involved parameters in the BDPM system. For the first time, the Nusselt number ratio (NRDP) is introduced to study the influence of Darcy number on the Nusselt number more precisely. Also, the effect of different thermophysical parameters on the Nusselt number is studied. The advantage of BDPM system over monodisperse porous medium (MDPM) structure is examined through the heat transfer performance parameter. The findings exhibit a good agreement with the literature. Also, the LTNE condition produces more realistic results in comparison to local thermal equilibrium assumption. On the whole, although implementing the BDPM enhances the heat transfer rate compared with the MDPM, it does not improve the thermal hydrodynamic performance significantly.     

Thermo‐solutal convection in an inclined porous cavity with various aspect ratios under mixed thermal and species boundary conditions

The problem of laminar thermo‐solutal convective flow of a binary fluid mixture in an inclined rectangular cavity filled with a uniform porous medium is considered. Mixed heat and mass fluxes and uniform temperature and concentration conditions are applied on two opposing walls of the cavity while the other two walls are kept adiabatic and impermeable to mass transfer. The problem is put in terms of the stream function‐vorticity formulation. A numerical solution based on the finite‐difference methodology is obtained. Representative results illustrating the effects of the inclination angle of the cavity, buoyancy ratio, Darcy number, and the cavity aspect ratio on the contour maps of the streamline, temperature, and concentration as well as the profiles of velocity, temperature, and concentration at mid‐section of the cavity are reported. In addition, numerical results for the average Nusselt and Sherwood numbers as well as some useful correlations are presented for various parametric conditions and discussed. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20369     

An application of paired quasilinearization on double‐diffusive convection flow over a cone embedded in a porous medium in the presence of nanoparticles

Mixed convection flow of a nanofluid near a vertical cone embedded in a a porous medium with Soret and Dufour effects is exercised. The bearing of a porous medium is recounted by the Darcy model. The partial differential equations, modeling the concerned problem, is nondimensionalised by implementing compatible transformations, which results in a similar form. A new paired spectral quasilinearization method is adopted to get the accurate numerical solution. Convergence and accuracy of the solution is elaborated by analyzing the norm of residual and solution errors. Alteration of velocity, temperature, nanoparticle and solute concentration profiles due to flow controlling parameters, namely, Brownian motion, thermophoresis, Soret, Dufour, Lewis number, and buoyancy ratio is outlined by reproducing the obtained numerical solution in graphs and tables. Analysis reveals that the flow profiles are greatly influenced by the physical parameters under investigation.     

Natural convection heat transfer in an anisotropic porous cavity heated from the side: Part 1. Theory

Natural convection heat transfer and flow structure in an anisotropic porous medium in a square cavity saturated with a Boussinesq fluid have been studied analytically and numerically. Based on an asymptotic analysis, three distinctive regimes are found depending on the magnitude of the permeability ratio K. In the vicinity of K = 1 the average Nusselt number and fluid velocity are scaled with (KRa) ^1/2 when either K or the Rayleigh number Ra is varied. In the limit of K → 0 the heat transfer across the cavity approaches the conductive state, and the convecting velocity, which is primarily in the vertical direction, is scaled with KRa. At the other end of the spectrum, namely, K → ∞, the average Nusselt number and the convecting velocity are scaled with Ra and independent of K. The asymptotic results are verified with two‐dimensional numerical calculations. The ranges of K of the respective regimes are also determined based on the numerical results. © 2000 Scripta Technica, Heat Trans Asian Res, 29(5): 373–384, 2000     

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.     

Unsteady MHD chemically reactive double‐diffusive Casson fluid past a flat plate in porous medium with heat and mass transfer

In this paper, an attempt has been made to analyze the effects of various parameters, such as Soret and Dufour effects, chemical reaction, magnetic field, porosity on the fluid flow, and heat and mass transfer of an unsteady Casson fluid flow past a flat plate. Convective boundary conditions in heat and mass transfer and slip constant on velocity have been taken into account for analysis. The governing equations of the model have been solved numerically using the MATLAB program bvp4c. The impact of various parameters of the model on the velocity, temperature, and concentration profiles has been analyzed through different graphs. To get an insight into the physical quantities of engineering interest, viz, skin friction, Sherwood number, and Nusselt number, their numerical values have been computed for various parameters. The range of the parameters used in numerical computations are , , , , , , and . It has been noticed from the tabulated values that the skin friction gets enhanced with the increase in the thermal and solutal Grashof number, whereas its reverse effects have been observed with an increase in the Biot number. In limiting case, the present study is also compared with the available results in the literature.