Mixed Convection Stagnation-Point Flow Over a Vertical Plate with Prescribed Heat Flux Embedded in a Porous Medium: Brinkman-Extended Darcy Formulation

This article considers the problem of mixed convection stagnation-point flow towards a vertical plate embedded in a porous medium with prescribed surface heat flux. It is assumed that the free stream velocity and the surface heat flux vary linearly from the stagnation point. Using a similarity transformation, the governing system of partial differential equations is transformed into a system of ordinary differential equations, before being solved numerically by a finite-difference method. The features of the flow and the heat transfer characteristics are analyzed and discussed. It is found that dual solutions exist for both buoyancy assisting and opposing flows.     

Stability of Double-Diffusive Natural Convection in a Vertical Porous Layer

Transport in Porous Media - The stability of double-diffusive buoyant flow in a vertical layer of Darcy porous medium whose boundaries are held at different constant temperatures and solute...     

Natural Convection in a Spherical Porous Annulus: The Brinkman Extended Darcy Flow Model

The present study reports results of a numerical investigation of natural convection in a spherical porous annulus. The inner and outer surfaces are subjected to constant temperatures. The Brinkman extended Darcy flow model is considered in this study. The problem has been solved numerically employing successive accelerated replacement scheme. The effect of parameters like Rayleigh number, Darcy number and radius ratio on the fluid flow and heat transfer has been examined. There exists a critical radius ratio $(\mathrm{rr} = 3)$ at which the average Nusselt number attains a peak value. The average Nusselt number decreases as the Darcy number increases.     

Mixed Convection Boundary Layer Flow Past a Horizontal Circular Cylinder Embedded in a Bidisperse Porous Medium

Adopting a two-temperature and two-velocity model, appropriate to a bidisperse porous medium (BDPM) proposed by Nield and Kuznetsov (2008), the classical steady, mixed convection boundary layer flow about a horizontal, isothermal circular cylinder embedded in a porous medium has been theoretically studied in this article. It is shown that the boundary layer analysis leads to expressions for the flow and heat transfer characteristics in terms of an inter-phase momentum parameter, a thermal diffusivity ratio, a thermal conductivity ratio, a permeability ratio, a modified thermal capacity ratio, and a buoyancy or mixed convection parameter. The transformed partial differential equations governing the flow and heat transfer in the f-phase (the macro-pores) and the p-phase (the remainder of the structure) are solved numerically using a very efficient implicit finite-difference technique known as Keller-box method. A good agreement is observed between the present results and those known from the open literature in the special case of a traditional Darcy formulation (monodisperse system).     

Natural Convection from a Vertical Plate in a Porous Medium Using Brinkman's Model

A regular perturbation analysis is presented for the following laminar natural convection flows of Newtonian fluids with temperature-dependent effective viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface for porous medium. The temperature-dependent effective viscosity introduces nonsimilarity into the governing equations. Numerical results are presented for the flow and heat transfer characteristics.     

Natural Convection in a Nanofluid Saturated Rotating Porous Layer with Thermal Non-Equilibrium Model

In the present article, we study the effect of local thermal non-equilibrium on the linear and non-linear thermal instability in a nanofluid saturated rotating porous layer. The Darcy Model has been used for the porous medium, while the nanofluid layer incorporates the effect of Brownian motion along with thermophoresis. A three-temperature model is been used for the effect of local thermal non-equilibrium among the particle, fluid, and solid–matrix phases. The linear stability analysis is based on normal mode technique, while for nonlinear analysis a minimal representation of the truncated Fourier series analysis involving only two terms has been used.     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

Nonsimilar Boundary Layer Analysis of Free Convection Heat Transfer Over a Vertical Cylinder in Bidisperse Porous Media

This work presents a boundary layer analysis for the free convection heat transfer from a vertical cylinder in bidisperse porous media with constant wall temperature. A boundary layer analysis and the two-velocity two-temperature formulation are used to derive the nonsimilar governing equations. The transformed governing equations are solved by the cubic spline collocation method to yield computationally efficient numerical solutions. The effects of inter-phase heat transfer parameter, modified thermal conductivity ratio, and permeability ratio on the heat transfer and flow characteristics are studied. Results show that an increase in the modified thermal conductivity ratio and the permeability ratio can effectively enhance the free convection heat transfer of the vertical cylinder in a bidisperse porous medium. Moreover, the thermal nonequilibrium effects are strong for low values of the inter-phase heat transfer parameter.     

Mixed Convection Boundary Layer Flow from a Horizontal Circular Cylinder Embedded in a Porous Medium Filled with a Nanofluid

Steady mixed convection boundary layer flow from an isothermal horizontal circular cylinder embedded in a porous medium filled with a nanofluid has been studied for both cases of a heated and cooled cylinder. The resulting system of nonlinear partial differential equations is solved numerically using an implicit finite-difference scheme. The solutions for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction φ and the mixed convection parameter λ. Three different types of nanoparticles are considered, namely Cu, Al₂O₃ and TiO₂. It is found that for each particular nanoparticle, as the nanoparticle volume fraction φ increases, the magnitude of the skin friction coefficient decreases, and this leads to an increase in the value of the mixed convection parameter λ which first produces no separation. On the other hand, it is also found that of all the three types of nanoparticles considered, for any fixed values of φ and λ, the nanoparticle Cu gives the largest values of the skin friction coefficient followed by TiO₂ and Al₂O₃. Finally, it is worth mentioning that heating the cylinder (λ > 0) delays separation of the boundary layer and if the cylinder is hot enough (large values of λ > 0), then it is suppressed completely. On the other hand, cooling the cylinder (λ < 0) brings the boundary layer separation point nearer to the lower stagnation point and for a sufficiently cold cylinder (large values of λ < 0) there will not be a boundary layer on the cylinder.     

Weakly Nonlinear Convection in a Porous Layer with Multiple Horizontal Partitions

We consider convection in a horizontally uniform fluid-saturated porous layer which is heated from below and which is split into a number of identical sublayers by impermeable and infinitesimally thin horizontal partitions. Rees and Genç (Int J Heat Mass Transfer 54:3081–3089, 2010) determined the onset criterion by means of a detailed analytical and numerical study of the corresponding dispersion relation and showed that this layered system behaves like the single-sublayer constant-heat-flux Darcy–Bénard problem when the number of sublayers becomes large. The aim of the present work is to use a weakly nonlinear analysis to determine whether the layered system also shares the property of the single-sublayer constant-heat-flux Darcy–Bénard problem by having square cells, as opposed to rolls, as the preferred planform for convection.     

Natural Convection Induced by a Centrifugal Force Field in a Horizontal Annular Porous Layer Saturated with a Binary Fluid

The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, j{\varphi } , the radius ratio of the cavity, R, the normalized porosity, e{\varepsilon } , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations.     

Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries

An analysis of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates is presented. The flow is modeled using Brinkman equation model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel are considered. Analytical solutions for the governing ordinary differential equations are obtained by perturbation series method. In addition, closed form expressions for the Nusselt number at both the left and right channel walls are derived. Results have been presented for a wide range of governing parameters such as porous parameter, ratio of Grashof number and Reynolds number, viscosity ratio, width ratio, and conductivity ratio on velocity, and temperature fields. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature.     

Unsteady Natural Convection Within a Porous Enclosure of Sinusoidal Corrugated Side Walls

Numerically investigation of free convection within a porous cavity with differential heating has been performed using modified corrugated side walls. Sinusoidal hot left and cold right walls are assumed to receive sudden differentially heating where top and bottom walls are insulated. Air is considered as working fluid and is quiescent, initially. Numerical experiments reveal 3 distinct stages of developing pattern including initial stage, oscillatory intermediate, and finally steady-state condition. Implicit Finite Volume Method with TDMA solver is used to solve the governing equations. This study has been performed for the Rayleigh numbers ranging from 100 to 10,000. Outcomes have been reported in terms of isotherms, streamline, velocity and temperature plots and average Nusselt number for various Ra, corrugation frequency, and corrugation amplitude (CA). The effects of sudden differential heating and its resultant transient behavior on fluid flow and heat transfer characteristics have been shown for the range of governing parameters. The present results show that the transient phenomena are enormously influenced by the variation of the Rayleigh Number with CA and frequency.     

Natural Convection in a Nanofluid Saturated Rotating Porous Layer: A Nonlinear Study

The present paper deals with linear and nonlinear analysis of thermal instability in a rotating porous layer saturated by a nanofluid. Momentum equation with Brinkman term, involving the Coriolis term and incorporating the effect of Brownian motion along with thermophoresis has been considered. Linear stability analysis is done using normal mode technique, while for nonlinear analysis, a minimal representation of the truncated Fourier series, involving only two terms, has been used. Stationary and oscillatory modes of convection have been studied. A weak nonlinear analysis is used to obtain the concentration and thermal Nusselt numbers. The behavior of the concentration and thermal Nusselt numbers is investigated by solving the finite amplitude equations using a numerical method. Obtained results have been presented graphically and discussed in details.     

Natural Convection in a Nanofluid-Saturated Rotating Porous Layer: A More Realistic Approach

The present work aims at studying the thermal instability in a rotating porous layer saturated by a nanofluid based on a new boundary condition for the nanoparticle fraction, which is physically more realistic. The model used for nanofluid combines the effect of Brownian motion along with thermophoresis, while for a porous medium Brinkman model has been used. A more realistic set of boundary conditions where the nanoparticle volume fraction adjusts itself including the contributions of the effect of thermophoresis so that the nanoparticle flux is zero at the boundaries has been considered. Using linear stability analysis, the expression for critical Rayleigh number has been obtained in terms of various non-dimensional parameters. The effect of various parameters on the onset of instability has been presented graphically and discussed in detail.     

Onset of Thermal Convection in a Vertical Porous Cylinder with a Partly Conducting and Partly Penetrative Cylinder Wall

The onset of thermal convection in a vertical porous cylinder in three dimensions is investigated analytically. Top and bottom of the cylinder are set to be perfectly heat conducting and impermeable, and is uniformly heated from below. The convection problem is solved for a cylinder wall that is partly conducting and partly penetrative. The expressions for semi-conduction and semi-penetration are based on a porous medium separated from its surroundings by a thin wall. The eigenvalue problem is split into two Helmholtz equations, and the results are expressed by Bessel functions in the radial direction. Comparisons are made with existing solutions for the limit cases of a closed cylinder wall that is either conducting or insulating. Two different models are compared for the kinematic limit condition of an open boundary.     

Numerical Modeling of Unsteady Two-Dimensional Convection in a Finite-Length Horizontal Layer Heated from Below

The development of two-dimensional thermo-gravitational convection in an elongated horizontal layer bounded by solid surfaces with the bottom instantaneously heated is investigated. The characteristics of the transition from the heat conduction regime to the convective regime are considered. The flow pattern and the heat transfer properties are described from the initial instant, which corresponds to the isothermal fluid at rest, up to the attainment of the steady-state roll-convection regime. A criterial dependence between the Rayleigh number and the nondimensional time of onset of the influence of thermo-gravitational convection on heat transfer is obtained.     

Unsteady Free Convection Boundary Layer Flows of a Bingham Fluid in Cylindrical Porous Cavities

Transport in Porous Media - We consider two unsteady free convection flows of a Bingham fluid when it saturates a porous medium contained within a vertical circular cylinder. The cylinder is...     

Variable Permeability Effects on Natural Convection in a Vertical Porous Layer with Uniform Heat Flux from the Side

Transport in Porous Media - Natural convection heat transfer in a rectangular cavity filled with a saturated porous medium with variable permeability is investigated analytically and numerically....     

Free Convection Heat Transfer over a Vertical Cylinder in a Saturated Porous Medium Using a Local Thermal Non-equilibrium Model

In this article, free convection heat transfer over a vertical cylinder with variable surface temperature distributions in a porous medium is analyzed. It is assumed that the fluid and solid phases are not in local thermal equilibrium and, therefore, a two-temperature model of heat transfer is applied. The coupled momentum and energy equations are presented and then they are transformed into ordinary differential equations. The similarity equations are solved numerically. The resulting velocity, streamlines, temperature distributions for fluid and solid phases are shown for different values of parameters entering into the problem. The calculated values of the local Nusselt numbers for both solid and fluid phases are also shown.