On the Chebyshev collocation spectral approach to stability of fluid flow in a porous medium

In this paper, the temporal development of small disturbances in a pressure‐driven fluid flow through a channel filled with a saturated porous medium is investigated. The Brinkman flow model is employed in order to obtain the basic flow velocity distribution. Under normal mode assumption, the linearized governing equations for disturbances yield a fourth‐order eigenvalue problem, which reduces to the well‐known Orr–Sommerfeld equation in some limiting cases solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. 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. It is found that a decrease in porous medium permeability has a stabilizing effect on the fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Buoyant Flow with Viscous Heating in a Vertical Circular Duct Filled with a Porous Medium

Combined, forced, and free flow in a vertical circular duct filled with a porous medium is investigated according to the Darcy–Boussinesq model. The effect of viscous dissipation is taken into account. It is shown that a thermal boundary condition compatible with fully developed and axisymmetric flow is either a linearly varying wall temperature in the axial direction or, only in the case of uniform velocity profile, an axial linear-exponential wall temperature change. The case of a linearly varying wall temperature corresponds to a uniform wall heat flux and includes the uniform wall temperature as a special case. A general analytical solution procedure is performed, by expressing the seepage velocity profile as a power series with respect to the radial coordinate. It is shown that, for a fixed thermal boundary condition, i.e., for a prescribed slope of the wall temperature, and for a given flow rate, there exist two solutions of the governing balance equations provided that the flow rate is lower than a maximum value. When the maximum value is reached, the dual solutions coincide. When the flow rate is higher than its maximum, no axisymmetric solutions exist. E. Magyari is on leave from the Institute of Building Technology, ETH—Zürich.     

On the Non-Isentropic Sound Waves Propagation in a Cylindrical Tube Filled with Saturated Porous Media

A rigid frame, cylindrical capillary theory of sound propagation in porous media that includes the nonlinear effects of the Forchheimer type is laid out by using variational solutions. It is shown that the five main parameters governing the propagation of sound waves in a fluid contained in rigid cylindrical tubes filled with a saturated porous media are: the shear wave number, , the reduced frequency parameter, , the porosity, ε, Darcy number, , and Forchheimer number, . The manner in which the flow influences the attenuation and the phase velocities of the forward and backward propagating non-isentropic acoustic waves is deduced. It is found that the inclusion of the solid matrix increases wave’s attenuations and phase velocities for both forward and backward sound waves, while increasing the porosity and the reduced frequency number decreased attenuation and increased phase velocities. The effect of the steady flow is found to decrease the attenuation and phase velocities for forward sound waves, and enhance them for the backward sound waves. This work is done during a sabbatical leave year granted form the University of Jordan to Dr. Hamzeh Duwairi for the academic year 2007/2008 at the German Jordanian University.     

Fully Developed Mixed Convection Flow Between Inclined Parallel Plates Filled with a Porous Medium

A fully developed mixed convection flow between inclined parallel flat plates filled with a porous medium is considered through which there is a constant flow rate and with heat being supplied to the fluid by the same uniform heat flux on each plate. The equations governing this flow are made non-dimensional and are seen to depend on two dimensionless parameters, a mixed convection parameter λ and the Péclet number Pe, as well as the inclination γ of the plates to the horizontal. The velocity and temperature profiles are obtained in terms of λ, Pe and γ when the channel is inclined in an upwards direction as well as for horizontal channels. The limiting cases of small and large λ and small Pe are considered with boundary-layer structures being seen to develop on the plates for large values of λ.     

Mixed Convection Boundary-Layer Flow in a Porous Medium Filled with Water Close to its Maximum Density

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water close to its maximum density is considered for uniform wall temperature and outer flow. The problem can be reduced to similarity form and the resulting equations are examined in terms of a mixed convection parameter λ and a parameter δ which measures the difference between the ambient temperature and the temperature at the maximum density. Both assisting (λ > 0) and opposing flows (λ < 0) are considered. A value δ₀ is found for which there are dual solutions for a range λ_c < λ < 0 of λ (the value of λ_c dependent on δ) and single solutions for all λ ≥ 0. Another value of δ₁ of δ, with δ₁ > δ₀, is found for which there are dual solutions for a range 0 < λ < λ_c of positive values of λ, with solutions for all λ≤ 0. There is also a range δ₀ <  δ < δ₁ where there are solutions only for a finite range of λ, with critical points at both positive and negative values of λ, thus putting a finite limit on the range of existence of solutions.     

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.     

The Starting Flow in Ducts Filled with a Darcy–Brinkman Medium

Analytical solutions are found for the transient starting flow due to a sudden pressure gradient in cylindrical, rectangular, and parallel plate ducts fill with a Darcy– Brinkman porous medium. It is found that, for all geometries, the initial velocity front is flat. It eventually becomes more parabolic for small porous media parameter s but remains flat for large s. The boundary layer thickness is of order (1/s). The transient is also shorter (proportional to exp(−s ² t)) for large s.     

A Nonlinear Stability Analysis for Thermoconvective Magnetized Ferrofluid Saturating a Porous Medium

The generalized energy method is developed to study the nonlinear stability analysis for a magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. By introducing a suitable generalized energy functional, we perform a nonlinear energy stability (conditional) analysis. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, and medium permeability, Da, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter (M ₃) and Darcy number (Da), the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and magnetic forces in nonlinear energy stability analysis as well as in linear instability analysis.     

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.     

Fourier Representation of Intermittent Flow in a Porous Medium

The field measurements and numerical results for intermittent flow regime in a sandy soil show that the time distributions of the soil water flux q(z, t), and the soil water content θ(z, t)at various depths are periodic in nature, where t is time and z is the depth (i.e., at the surface z = 0 and at depths z = − 5, − 10, − 15 cm, etc). The period of q(z, t) and θ(z, t) variations are generally determined by the sum of the duration of pulse and the duration between the initiation of two consecutive pulses of water at the soil surface. Fourier series models have been given for q(z, t) and θ(z, t) variations. The predicted Fourier results for these variations have been compared with the experimentally verified numerical results—designated as observed values. The results show that the amplitudes of these variations were damped exponentially with depth, and the phase shift increased linearly with depth.     

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.     

Mixed Convection Boundary-Layer Flow Near the Stagnation Point on a Vertical Surface in a Porous Medium: Brinkman Model with Slip

The steady boundary-layer flow near the stagnation point on an impermeable vertical surface with slip that is embedded in a fluid-saturated porous medium is investigated. Using appropriate similarity variables, the governing system of partial differential equations is transformed into a system of ordinary differential equations. This system is then solved numerically. The features of the flow and the heat transfer characteristics for different values of the governing parameters, namely, the Darcy–Brinkman, Γ, mixed convection, λ, and slip, γ, parameters, are analysed and discussed in detail for the cases of assisting and opposing flows. It is found that dual solutions exist for assisting flows, as well as those usually reported in the literature for opposing flows. A stability analysis of the steady flow solutions encountered for different values of the mixed convection parameter λ is performed using a linear temporal stability analysis. This analysis reveals that for γ  =  0 (slip absent) and Γ  =  1 the lower solution branch is unstable while the upper solution branch is stable.     

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.     

Numerical Study of a Magneto-fluid Motion Through a Porous Medium Between Two Wavy Plates with Constant Suction

In this note, the problem of an incompressible viscous fluid moving through a porous medium (Brinkman model) between two wavy plates under the effects of a constant inclined magnetic field that makes an angle with the vertical axis and constant suction, are studied numerically by a method related to the method of Takabatake and Ayukawa in 1982. The present approach is not restricted by any of the parameters appearing in the problem such as Reynolds number, magnetic parameter, suction parameter, the wave number and amplitude ratio. The variations in velocity, flow rate and pressure gradient with the above governing parameters are presented. Moreover, the effect of varying the porous medium and the inclined angle is also studied.     

Finite Element Analysis of Convection Flow Through a Porous Medium in a Horizontal Channel

We analyse the convection flow of a viscous fluid through a horizontal channel enclosing a fully saturated porous medium. The Galerkin finite element analysis is used to discuss the flow and heat transfer through the porous medium using serendipity elements. The velocity, the temperature distributions and the rate of heat transfer are analysed for variations in the governing parameters. The profiles at different vertical levels are asymmetric curves, exhibiting reversal flow everywhere except on the midplane. In a given porous medium, for fixed G or N, the temperature in the fluid region at any position in fluids with a higher Prandtl number, is much higher than in fluids with a lower Prandtl number. Likewise, other parameters being fixed, lesser the permeability of the medium, lower the temperature in the flow field. Nu reduces across the flow at all axial positions, while it enhances along the axial direction of the channel. Nu reduces with decrease in the Darcy parameter D, and thus lesser the permeability of the medium, lesser the rate of heat transfer across the boundary at any axial position of the channel.     

Attenuated Wave Field in Fluid-Saturated Porous Medium with Excitations of Multiple Sources

This research addresses the investigation of an elastic wave field in a homogeneous and isotropic porous medium which is fully saturated by a Newtonian viscous fluid. A new methodology is developed for describing the wave field in the medium excited by multiple energy sources. To quantify the relative displacements between the fluid and solid of the medium, the governing equations of the elastic wave propagation are derived in the form of displacements specially. The velocities and attenuation of the waves are considered as functions of viscosity and frequency. Making use of the Hankel function and the moving-coordinate method, a model of the wave motion with multiple cylindrical wave sources is built. Making use of the model established in this research, the relative displacement between the fluid and the solid can be quantified, and the wave field in the porous media can then be determined with the given energy sources. Numerical simulations of cylindrical waves from multiple energy sources propagating in the porous medium saturated by viscous fluid are performed for demonstrating the practicability of the model developed.     

Modeling of Heat and Mass Transfer in a Fractured Porous Medium

While fractured formations are possibly the most important contributors to the production of oil worldwide, modeling fractured formations with rigorous treatments has eluded reservoir engineers in the past. To date, one of the most commonly used fractured reservoir models remains the one that was suggested by Warren and Root nearly four decades ago. In this paper, a new model for fractures embedded in a porous medium is proposed. The model considers the Navier-Stokes equation in the fracture (channel flow) while using the Brinkman equation for the porous medium. Unlike the previous approach, the proposed model does not require the assumption of orthogonality of the fractures (sugar cube assumption) nor does it impose incorrect boundary conditions for the interface between the fracture and the porous medium. Also, the transfer coefficient between the fracture and matrix interface does not need to be specified, unlike the cases for which Darcy's law is used. In order to demonstrate the usefulness of the approach, a two-dimensional model of a fractured formation is developed and numerical simulation runs conducted.

The proposed model is derived through a series of finite element modeling runs for various cases using the Navier-Stokes equation in the channel while maintaining the Brinkman equation in the porous medium. Various cases studied include different fracture orientations, fracture frequencies, and thermal and solutal constraints. The usefulness of the proposed model in modeling complex formations is discussed. Finally, a series of numerical runs also provided validity of the proposed model for the cases in which thermal and solutal effects are important. Such a study of double diffusive phenomena, coupled with forced convection, in the context of fractured formations has not been reported before.     

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 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.