Fully Developed Forced Convection Heat Transfer in a Porous Channel with Asymmetric Heat Flux Boundary Conditions

An analytical study is performed on steady, laminar, and fully developed forced convection heat transfer in a parallel plate channel with asymmetric uniform heat flux boundary conditions. The channel is filled with a saturated porous medium, and the lower and upper walls are subjected to different uniform heat fluxes. The dimensionless form of the Darcy–Brinkman momentum equation is solved to determine the dimensionless velocity profile, while the dimensionless energy equation is solved to obtain temperature profile for a hydrodynamically and thermally fully developed flow in the channel. Nusselt numbers for the lower and upper walls and an overall Nusselt number are defined. Analytical expressions for determination of the Nusselt numbers and critical heat flux ratio, at which singularities are observed for individual Nusselt numbers, are obtained. Based on the values of critical heat flux ratio and Darcy number, a diagram is provided to determine the direction of heat transfer between the lower or upper walls while the fluid is flowing in the channel.     

Darcy Flow in a Wavy Channel Filled with a Porous Medium

Flow in channels bounded by wavy or corrugated walls is of interest in both technological and geological contexts. This paper presents an analytical solution for the steady Darcy flow of an incompressible fluid through a homogeneous, isotropic porous medium filling a channel bounded by symmetric wavy walls. This packed channel may represent an idealized packed fracture, a situation which is of interest as a potential pathway for the leakage of carbon dioxide from a geological sequestration site. The channel walls change from parallel planes, to small amplitude sine waves, to large amplitude nonsinusoidal waves as certain parameters are increased. The direction of gravity is arbitrary. A plot of piezometric head against distance in the direction of mean flow changes from a straight line for parallel planes to a series of steeply sloping sections in the reaches of small aperture alternating with nearly constant sections in the large aperture bulges. Expressions are given for the stream function, specific discharge, piezometric head, and pressure.     

Convective Stability in a Rectangular Box of Fluid-Saturated Porous Medium with Constant Pressure Top and Constant Flux Bottom Heating

A fluid-saturated porous medium in a rectangular box is heated from below by constant flux. The top is open at constant pressure and the sides are insulated. Linear free convection stability analysis yields a complicated characteristic equation. It is found that the critical Rayleigh number and the incipient mode are sensitive to the dimensions of the box.     

Flow of a Weakly Conducting Fluid in a Channel Filled with a Porous Medium

We investigate the fully developed flow in a fluid-saturated porous medium channel with an electrically conducting fluid under the action of a parallel Lorentz force. The Lorentz force varies exponentially in the vertical direction due to low fluid electrical conductivity and the special arrangement of the magnetic and electric fields at the lower plate. Exact analytical solutions are derived for fluid velocity and the results are presented in figures. All these flows are new and are presented for the first time in the literature.     

Natural Convection Boundary-Layer Flow in a Porous Medium with Temperature-Dependent Boundary Conditions

The natural convection boundary-layer flow on a surface embedded in a fluid-saturated porous medium is discussed in the case when the wall heat flux is related to the wall temperature through a power-law variation. The flow within the porous medium is assumed to be described by Darcy’s law and the Boussinesq approximation is assumed for the density variations. Two cases are discussed, (i) stagnation-point flow and (ii) flow along a vertical surface. The possible steady states are considered first with the governing partial equations reduced to ordinary differential equations by similarity transformations and these latter equations further transformed to previously studied free-convection problems. This identifies values of the exponent N in the power-law wall temperature variation, N = 3/2 for stagnation-point flows and 3/2 ≤ N ≤ 3 for the vertical surface, where similarity solutions do not exist. Time development for stagnation-point flows is seen to depend on N, for N <  3/2 the steady state is approached at large times, for N ≥ 3/2 a singularity develops at finite time leading to thermal runaway. Numerical solutions for the vertical surface, where the temperature-dependent boundary condition becomes more significant as the solution develops, show that, for N < 3/2, the corresponding similarity solution is approached, whereas for N >  3/2 the solution breaks down at a finite distance along the surface.     

On Temporal Stability Analysis for Hydromagnetic Flow in a Channel Filled with a Saturated Porous Medium

In this paper, a linear stability analysis is presented to trace the time evolution of an infinitesimal, two-dimensional disturbance imposed on the base flow of an electrically conducting fluid in a channel filled with a saturated porous medium under the influence of a transversely imposed magnetic field. An eigenvalue problem is obtained and solved numerically using the Chebyshev collocation spectral method. The critical Reynolds number Re _c, the critical wave number α _c and the critical wave speed c _c are obtained for a wide range of the porous medium shape factor parameter S and Hartmann number H. It is found that an increase in the magnetic field intensity and a decrease in porous medium permeability have a stabilizing effect on the fluid flow.     

Mixed Convection Boundary-Layer Flow on a Vertical Surface in a Porous Medium with a Constant Convective Boundary Condition

The mixed convection boundary-layer flow on a vertical surface heated convectively is considered when a constant surface heat transfer parameter is assumed. The problem is seen to be chararterized by a mixed convection parameter $\gamma $ γ . The flow and heat transfer near the leading edge correspond to forced convection solution and numerical solutions are obtained to determine how the solution then develops. The solution at large distances is obtained and this identifies a critical value $\gamma _c$ γ c of the parameter $\gamma $ γ . For $\gamma > \gamma _c$ γ > γ c a solution at large distances is possible and this is approached in the numerical integrations. For $\gamma <\gamma _c$ γ < γ c the numerical solution breaks down at a finite distance along the surface with a singularity, the nature of which is discussed.     

Effects of Viscous Dissipation and Flow Work on Forced Convection in a Channel Filled by a Saturated Porous Medium

Fully developed forced convection in a parallel plate channel filled by a saturated porous medium, with walls held either at uniform temperature or at uniform heat flux, with the effects of viscous dissipation and flow work included, is treated analytically. The Brinkman model is employed. The analysis leads to expressions for the Nusselt number, as a function of the Darcy number and Brinkman number.     

Thermally Developing Forced Convection in a Porous Medium Occupied by a Rarefied Gas: Parallel Plate Channel or Circular Tube with Walls at Constant Heat Flux

An adaptation of the classical Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel or a circular tube filled by a porous medium saturated by a rarefied gas, with walls held at constant heat flux. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number Nu as functions of the dimensionless longitudinal coordinate and the Darcy number. It is found that an increase in the velocity slip coefficient generally increases Nu by a small or moderate amount (but the circular tube at large Darcy number is an exception) while an increase in the temperature slip coefficient reduces Nu by a more substantial amount. These trends are uniform as the longitudinal coordinate varies.     

Approximate Analytical Solutions for Flow of a Third-Grade Fluid Through a Parallel-Plate Channel Filled with a Porous Medium

The flow of a non-Newtonian fluid through a porous media in between two parallel plates at different temperatures is considered. The governing momentum equation of third-grade fluid with modified Darcy’s law and energy equation have been derived. Approximate analytical solutions of momentum and energy equations are obtained by using perturbation techniques. Constant viscosity, Reynold’s model viscosity, and Vogel’s model viscosity cases are treated separately. The criteria for validity of approximate solutions are derived. A numerical residual error analysis is performed for the solutions. Within the validity range, analytical and numerical solutions are in good agreement.     

Analytical Investigation of the Effect of Viscous Dissipation on Couette Flow in a Channel Partially Filled with a Porous Medium

An analytical study of viscous dissipation effect on the fully developed forced convection Couette flow through a parallel plate channel partially filled with porous medium is presented. A uniform heat flux is imposed at the moving plate while the fixed plate is insulated. In the fluid-only region the flow field is governed by Navier–Stokes equation while the Brinkman-extended Darcy law relationship is considered in the fully saturated porous medium. The interface conditions are formulated with an empirical constant β due to the stress jump boundary condition. Fluid properties are assumed to be constant and the longitudinal heat conduction is neglected. A closed-form solution for the velocity and temperature distributions and also the Nusselt number in the channel are obtained and the viscous dissipation effect on these profiles is briefly investigated.     

Analytical Study of Fluid Flow and Heat Transfer during Forced Convection in a Composite Channel Partly Filled with a Brinkman–Forchheimer Porous Medium

In this paper, the problem of fully developed forced convection in a parallel-plate channel partly filled with a homogeneous porous material is considered. The porous material is attached to the walls of the channel, while the center of the channel is occupied by clear fluid. The flow in the porous material is described by a nonlinear Brinkman–Forchheimer-extended Darcy equation. Utilizing the boundary-layer approach, analytical solutions for the flow velocity, the temperature distribution, as well as for the Nusselt number are obtained. Dependence of the Nusselt number on several parameters of the problem is extensively investigated.     

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.     

Effects of Time-Periodic Thermal Boundary Conditions and Internal Heating on Heat Transport in a Porous Medium

The effects of time-periodic boundary temperatures and internal heating on Nusselt number in the Bénard–Darcy convective problem has been considered. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small. By performing a weakly non-linear stability analysis, the Nusselt number is obtained in terms of the amplitude of convection, which is governed by the non-autonomous Ginzburg–Landau equation, derived for the stationary mode of convection. The effects of internal Rayleigh number, amplitude and frequency of modulation, thermo-mechanical anisotropies, and Vadasz number on heat transport have been analyzed and depicted graphically. Increasing values of internal Rayleigh number results in the enhancement of heat transport in the system. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system.     

Analytical Solution for Forced Convection in a Semi-Circular Channel Filled with a Porous Medium

Fully developed laminar forced convection inside a semi-circular channel filled with a Brinkman-Darcy porous medium is studied. Analytical solutions for flow and constant flux heat transfer are found using a mixture of Cartesian and cylindrical coordinates. The problem depends on a parameter s, which is proportional to the inverse square of the Darcy number. Velocity boundary layers exist when s is large. Both friction factor-Reynolds number product and Nusselt number are determined. Closed form expressions for the clear fluid () limit are found. Rare analytical solutions not only describe fundamental channel flows, but also serve as a check for more complicated numerical solutions.     

Free Convection from a Vertical Plate in a Porous Medium Subjected to a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

An analysis is made of the transient free convection from a vertical flat plate which is embedded in a fluid-saturated porous medium. It is assumed that for time a steady state temperature or velocity has been obtained in the boundary-layer which occurs due to a uniform flux dissipation rate . Then at time the heat flux on the plate is suddenly changed to and maintained at this value for 0$$ " align="middle" border="0"> . An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. These effects cause a new steady boundary layer. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from to the steady state, firstly by means of a step-by-step method and then by a matching technique. The transition between the two distinct solution methods is always observed to occur very near to the turning point of the plate surface temperature, a time at which the fluid temperature is close to its steady state profile. The solution obtained using the step-by-step method shows excellent agreement with the small time analytical solution. Results are presented to illustrate the occurrence of transients from both small and large increases and decreases in the levels of existing energy inputs.     

Free Convection Boundary Layer Flow from a Heated Upward Facing Horizontal Flat Plate Embedded in a Porous Medium Filled by a Nanofluid with Convective Boundary Condition

The steady laminar incompressible free convective flow of a nanofluid over a permeable upward facing horizontal plate located in porous medium taking into account the thermal convective boundary condition is studied numerically. The nanofluid model used involves the effect of Brownian motion and the thermophoresis. Using similarity transformations the continuity, the momentum, the energy, and the nanoparticle volume fraction equations are transformed into a set of coupled similarity equations, before being solved numerically, by an implicit finite difference numerical method. Our analysis reveals that for a true similarity solution, the convective heat transfer coefficient related with the hot fluid and the mass transfer velocity must be proportional to x ^−2/3, where x is the horizontal distance along the plate from the origin. Effects of the various parameters on the dimensionless longitudinal velocity, the temperature, the nanoparticle volume fraction, as well as on the rate of heat transfer and the rate of nanoparticle volume fraction have been presented graphically and discussed. It is found that Lewis number, the Brownian motion, and the convective heat transfer parameters increase the heat transfer rate whilst the thermophoresis decreases the heat transfer rate. It is also found that Lewis number and the convective heat transfer parameter enhance the nanoparticle volume fraction rate whilst the thermophoresis parameter decreases nanoparticle volume fraction rate. A very good agreement is found between numerical results of the present article for special case and published results. This close agreement supports the validity of our analysis and the accuracy of the numerical computations.     

Perturbation Analysis of the Local Thermal Non-equilibrium Condition in a Fluid-Saturated Porous Medium Bounded by an Iso-thermal Channel

This study focuses analytically on the local thermal non-equilibrium (LTNE) effects in the developed region of forced convection in a saturated porous medium bounded by isothermal parallel-plates. The flow in the channel is described by the Brinkman–Forchheimer-extended Darcy equation and the LTNE effects are accounted by utilizing the two-equation model. Profiles describing the velocity field obtained by perturbation techniques are used to find the temperature distributions by the successive approximation method. A fundamental relation for the temperature difference between the fluid and solid phases (the LTNE intensity) is established based on a perturbation analysis. It is found that the LTNE intensity ( $\Delta \textit{NE}$ ) is proportional to the product of the normalized velocity and the dimensionless temperature at LTE condition. Also, it depends on the conductivity ratio, Da number, and the porosity of the medium. The intensity of LTNE condition ( $\Delta \textit{NE}$ ) is maximum at the middle of the channel and decreases smoothly to zero by moving to the wall. Finally, the established relation for the intensity of LTNE condition is simple and fundamental for estimating the importance of LTNE condition and validation of numerical simulation results.     

Conjugate Phase Change Heat Transfer in an Inclined Compound Cavity Partially Filled with a Porous Medium: A Deformed Mesh Approach

In this paper, the melting process of a PCM inside an inclined compound enclosure partially filled with a porous medium is theoretically addressed using a novel deformed mesh method. The sub-domain area of the compound enclosure is made of a porous layer and clear region. The right wall of the enclosure is adjacent to the clear region and is subject to a constant temperature of T_c. The left wall, which is connected to the porous layer, is thick and thermally conductive. The thick wall is partially subject to the hot temperature of T_h. The remaining borders of the enclosure are well insulated. The governing equations for flow and heat transfer, including the phase change effects and conjugate heat transfer at the thick wall, are introduced and transformed into a non-dimensional form. A deformed grid method is utilized to track the phase change front in the solid and liquid regions. The melting front movement is controlled by the Stefan condition. The finite element method, along with Arbitrary Eulerian–Lagrangian (ALE) moving grid technique, is employed to solve the non-dimensional governing equations. The modeling approach and the accuracy of the utilized numerical approach are verified by comparison of the results with several experimental and numerical studies, available in the literature. The effect of conjugate wall thickness, inclination angle, and the porous layer thickness on the phase change heat transfer of PCM is investigated. The outcomes show that the rates of melting and heat transfer are enhanced as the thickness of the porous layer increases. The melting rate is the highest when the inclination angle of the enclosure is 45°. An increase in the wall thickness improves the melting rate.

     

Unsteady Natural Convection Flow in a Square Cavity Filled with a Porous Medium Due to Impulsive Change in Wall Temperature

Unsteady natural convection flow in a two-dimensional square cavity filled with a porous material has been studied. The flow is initially steady where the left-hand vertical wall has temperature T _h and the right-hand vertical wall is maintained at temperature T _c (T _h > T _c) and the horizontal walls are insulated. At time t > 0, the left-hand vertical wall temperature is suddenly raised to which introduces unsteadiness in the flow field. The partial differential equations governing the unsteady natural convection flow have been solved numerically using a finite control volume method. The computation has been carried out until the final steady state is reached. It is found that the average Nusselt number attains a minimum during the transient period and that the time required to reach the final steady state is longer for low Rayleigh number and shorter for high Rayleigh number.