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.     

Mixed convection inside a fluid‐porous composite cavity with centrally rotating cylinder

The mixed convection in a fluid‐porous composite medium lying inside a square cavity with a centrally rotating cylinder has been investigated in the present work. The bottom half of the cavity is filled with a porous material and the top half is filled with a clear fluid. The bottom wall of the cavity is at a higher temperature, and the top wall is at a lower temperature. The vertical walls are thermally insulated. The convection inside the cavity sets through the combined mechanisms of the thermal buoyancy force and the shearing action of the centrally rotating cylinder. The relative importance of each driving mechanism over the other is featured through the Richardson number. The Darcy–Brinkman–Forchheimer equation is used for the flow modeling in the porous medium, and a single‐domain approach is adopted for the numerical solution in the fluid‐porous composite medium. The simulation is carried out with ANSYS Fluent software, and a parametric analysis involving the Rayleigh number (), Richardson number (), and the Darcy number () is conducted showing their effects on the flow and heat transfer. The phenomena are quite interesting at higher Darcy number and Rayleigh number. The distributions of isotherms, streamlines, and vector plots are plotted, along with the local Nusselt numbers for different parameters, to explore the underlying physics of the phenomenon. The system is found stable at lower Darcy number, and the heat transfer is minimum around Ri = 10. From the numerical study, an empirical correlation for the average Nusselt number is developed as a function of the other dimensionless numbers.     

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     

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.     

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.     

Linear stability of convection in a vertical channel filled with nanofluid saturated porous medium

The convection in a vertical channel filled with a porous medium saturated by a nanofluid is studied numerically. The effects of Brownian motion and thermophoresis are incorporated in the model used for nanofluid. Also, the flow within the porous region is governed by Brinkman's equation. The generalized eigenvalue problem for the perturbed state is obtained from a normal mode analysis and solved using the Chebyshev spectral collocation method. The Rayleigh number is expressed as an implicit function of the wavenumber with other parameters. The critical wavenumber and the critical Rayleigh number are calculated for different parameters. The preferred modes under critical conditions are detected.     

Double diffusive unsteady free convection on two-dimensional and axisymmetric bodies in a porous medium

The unsteady laminar free convection flow of an incompressible electrically conducting fluid over two-dimensional and axisymmetric bodies embedded in a highly porous medium with an applied magnetic field has been studied. The unsteadiness in the flow field is caused by the variation of the wall temperature and concentration with time. The coupled nonlinear partial differential equations with three independent variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. It is observed that the skin friction, heat transfer and mass transfer increase with the permeability parameter but decrease with the magnetic parameter. The results are strongly dependent on the variation of wall temperature and concentration with time. The skin friction and heat transfer increase or decrease as the buoyancy forces from species diffusion assist or oppose the thermal buoyancy force. However, the mass transfer is found to be higher for small values of the ratio of the buoyancy parameters than for large values.     

Double diffusive natural convection along an inclined wavy surface in a porous medium

This paper studies the double diffusive natural convection near an inclined wavy surface in a fluid saturated porous medium with constant wall temperature and concentration. A coordinate transformation is employed to transform the complex wavy surface to a smooth surface, and the obtained boundary layer equations are solved by the cubic spline collocation method. Effects of angle of inclination, Lewis number, buoyancy ratio, and wavy geometry on the heat and mass transfer characteristics are studied. Results show that a decrease in the angle of inclination leads to a greater fluctuation of the local Nusselt and Sherwood numbers. Moreover, increasing the angle of inclination tends to increase the total heat and mass transfer rates.     

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.     

Dissolution-driven convection of a power-law fluid in a porous medium in the presence of chemical reaction

The flow through porous medium accounts for numerous applications in various fields namely, agriculture, geothermal sciences, and engineering. Furthermore, dissolution-driven convection in porous media has grabbed great attention in recent years due to its practical applications in long-term geological storage of carbon dioxide, in the production of mineral deposits, and other industrial applications. In this regard, the current numerical analysis focuses on addressing the thermal instability of dissolution-driven convective phenomena of a power-law fluid through a porous horizontal domain with a first-order chemical reaction. For linear stability analysis, the method of normal modes has been employed to solve governing dimensionless equations which give rise to an eigenvalue problem. The bvp4c routine in MATLAB R2020a has been used to solve the raised problem for the onset of convection. The impact of Damköhler number, Péclet number, and power-law index on the onset of convection has been investigated. The role of these critical parameters is found to be highly significant in stabilizing the system. An increase in the power-law index causes stabilization or destabilization in the system, depending on the Péclet number. An enhancement in the magnitude of Damköhler number makes the system stable for all values of the Péclet number. Also, Damköhler and critical Rayleigh number are inter-related, that is, an increment in Damköhler number results in the enhancement of critical Rayleigh numbers, which in turn leads to stabilization of the system. The critical wave number is observed to have a remarkable influence on Damköhler number as well as power-law index.     

Instabilities of horizontal magnetoconvection with rotating fluid in a sparsely packed porous media

We studied linear and nonlinear instabilities of horizontal magnetoconvection with rotating fluid in a sparsely packed porous media. We studied the critical Rayleigh number and traced marginal stability curves at different parameters , , , and . We obtained Takens‐Bogdanov and co‐dimension two bifurcation points. The Newell‐Whitehead multiple scheme was employed to derive amplitude equations at Pitchfork and Hopf bifurcation. At the onset of Pitchfork bifurcation we identified Eckhaus and Zigzag instability regions and studied Nusselt number. The system of coupled Landau Ginzburg equations were derived at the onset of Hopf bifurcation and identified secondary instability regions for fixed parameters, steady state mode shifted to standing and traveling waves as increases.     

Effects of viscous dissipation on forced convective heat transfer in a channel embedded in a power-law fluid saturated porous medium

The convective heat transfer analysis in a channel embedded in a power-law fluid saturated porous medium subject to uniform heat flux is presented and compared with a Newtonian fluid concerning the effects of viscous dissipation. Governing momentum and energy equations for non-Newtonian fluids which accounts for the viscous dissipation effects are solved numerically. The temperature profiles of the non-Newtonian fluids are found to relate closely to the velocity profiles. When viscous dissipation is taken account of, Nusselt numbers for non-Newtonian fluid are found to deviate more from Newtonian fluid with increasing Brinkman number for a certain range of the Darcy number.     

Free convection of composite porous medium in a vertical channel

A fully developed free convection flow of immiscible fluids in a vertical channel filled with a porous medium is analyzed in the presence of source/sink. The flow is modeled using the Darcy–Brinkman–Forchheimer equation model. The viscous and Darcy dissipation terms are included in the energy equation. The channel walls are maintained at two different constant temperatures. The transport properties of both fluids are assumed to be constant. Continuous conditions for velocity, temperature, shear stress, and heat flux of both fluids at the interface are employed. The resulting coupled nonlinear equations are solved analytically using regular perturbation method and numerically using finite difference method. The velocity and temperature profiles are obtained in terms of porous parameter, Grashof number, viscosity ratio, width ratio, conductivity ratio, and heat generation or heat absorption coefficient. It is found that the presence of porous matrix and heat absorption reduces the flow field. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20340     

Characterization of a porous medium employing numerical tools: Permeability and pressure-drop from Darcy to turbulence

A large set of microscopic flow simulations in the Representative Elementary Volume (REV) of a porous medium formed by staggered square cylinders is presented. For each Reynolds number selected, 10 different porosities are simulated in the 5–95% range. The Reynolds number is varied from Re = 10⁻³ to Re = 10⁵, covering the Stokes flow regime, the laminar flow regime and the turbulence flow regime. Low and moderate Reynolds number flow solutions (Re  200) are achieved by numerically solving the 2D Navier–Stokes equations. Reynolds Averaged Navier–Stokes equations are employed to simulate the turbulence regime. Numerical results allow the investigation of the microscopic features of the flow as a function of the porosity and Reynolds number. Based on these microscopic results, the permeability of the porous medium is computed and a porosity-dependent correlation is developed for this macroscopic parameter. The Darcy–Forchheimer term or, equivalently, the friction factor, is also computed to characterize the porous medium for the complete range of porosity and Reynolds number simulated. The Forchheimer coefficient is found to be weakly dependent on the Reynolds number and strongly dependent on the porosity if the flow is fully turbulent. A porosity-dependent correlation is proposed for this quantity for high Reynolds numbers.     

Doubly diffusive unsteady mixed convection flow over a vertical plate embedded in a porous medium

The unsteady laminar incompressible boundary layer mixed convection flow over a semi-infinite vertical plate with a magnetic field situated in a porous medium has been studied where the buoyancy forces result both from temperature and concentration gradients. The effects of the presence of an isotropic solid matrix and dissipation have been included in the analysis. The governing coupled nonlinear partial differential equations with three independent variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The effects of the permeability of the medium, magnetic field and dissipation on the skin friction, heat transfer and mass transfer are found to be more pronounced for large value of the streamwise distance. Also, for assisting flow, the skin friction, heat transfer and mass transfer increase with the buoyancy forces, but the effect is exactly opposite for opposing flow.     

Numerical simulation of mixed convection in a porous medium filled with water/Al2 O3 nanofluid

The present work encloses the application of a Brinkman–extended Darcy model in a problem concerning mixed convection ina lid–driven porous cavity using nanofluids. The transport equations are solved numerically by the finite volume method on a co–located grid arrangement using the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme. The effects of governing parameters, namely, Grashof number (Gr), Darcy number (Da), and solid volume fraction $(\chi )$ , on the streamlines and the isotherms are studied. The present results are validated by favorable comparisons with previously published results and are in good agreement with them. The present numerical results show that the addition of nanoparticles to a base fluid has produced an augmentation of the heat transfer coefficient and it is found to increase significantly with an increase of the particle volume concentration. It is observed from the results that at the higher value of the Grashof number (Gr = 10⁴ ), the average Nusselt number increases with an increase in the Darcy number for a constant solid volume fraction. The detailed results are reported by means of streamlines, isotherms, and Nusselt numbers. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21029     

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.     

Theoretical study of superheated vapor effect on liquid film condensation in a saturated porous medium using Forchheimer's model

In this work, the superheated vapor effect on liquid film condensation in a saturated porous medium using Forchheimer's model has been investigated analytically and numerically. The applied governing equations, the continuity equation, the Forchheimer equation, and the energy equation were transformed using the similarity transformation technique into a dimensionless form using a set of suitable variables and then solved numerically using the Runge–Kutta method. Results obtained were graphically drawn to illustrate the effects of superheated vapor and subcooled liquid on the liquid film condensation, temperature, and heat transfer rate through the porous medium. It was found that the film thickness is a function of dimensionless parameters related to the degree of subcooling and Grashof number without a superheating effect. Consequently, the Nusselt number depends on the square root of the Rayleigh number, the Grashof number, and the dimensionless film thickness. It was also found that if superheating exists, the liquid film thickness then depends on four dimensionless parameters related to the Grashof number, the degree of subcooling of the liquid, the extent of the superheating of the surrounding vapor, and a property ratio of the liquid and the vapor phase.     

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.     

Double‐diffusive natural convection in a partially heated enclosure using a nanofluid

This paper analyzes numerically the effect of double‐diffusive natural convection of a water–Al₂O₃ nanofluid in a partially heated enclosure with Soret and Dufour coefficients. The top horizontal wall has constant temperature T_c, while the bottom wall is partially heated T_h, with T_h > T_c . The concentration in the top wall is maintained higher than the bottom wall C_c < C_h. The remaining bottom wall and the two vertical walls are considered adiabatic. Water is considered as the base fluid. The governing equations are solved by the Penalty Finite Element Method using Galerkin's weighted residual scheme. The effect of the parameters, namely, Rayleigh number and solid volume fraction of the nanoparticles on the flow pattern and heat and mass transfer has been depicted. Comprehensive average Nusselt and Sherwood numbers, average temperature and concentration, and mid‐height horizontal and vertical velocities at the middle of the cavity are presented as functions of the governing parameters mentioned above. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21010