Non-Darcy free convective flow along a vertical cylinder embedded in a porous medium with surface mass flux

The nonsimilar non-Darcian free-convection flow about a vertical cylinder with impermeable surface embedded in a saturated porous medium, where surface temperature of the cylinder varies as x^m, a power function of distance from the leading edge, has been studied by employing the implicit finite-difference method together with the Newton's quasilinearization technique. In the present investigation, effects of the surface mass flux together with the inertial effects on the rate of heat transfer at the surface, on the velocity distribution, and on the temperature distribution are shown graphically.     

Transient non-Darcy free convection between parallel vertical plates in a fluid saturated porous medium

Transient non-Darcy free convection between two parallel vertical plates in a fluid saturated porous medium is investigated using the generalized momentum equation proposed by Vafai and Tien. The effects of porous inertia and solid boundary are considered in addition to the Darcy flow resistance. Exact solutions are found for the asymptotic states at small and large times. The large time solutions reveal that the velocity profiles are rather sensitive to the Darcy number Da when Da<1. It has also been found that boundary friction alters the velocity distribution near the wall, considerably. Finite difference calculations have also been carried out to investigate the transient behaviour at the intermediate times in which no similarity solutions are possible. This analytical and numerical study reveals that the transient free convection between the parallel plates may well be described by matching the two distinct asymptotic solutions obtained at small and large times.Nomenclature C empirical constant for the Forchheimer term - f velocity function for the small time solution - F velocity function for the large time solution - g acceleration due to gravity - Gr* micro-scale Grashof number - H a half distance between two infinite plates - K permeability - Nu Nusselt number - Pr Prandtl number - t time - T temperature - u, v Darcian velocity components - x, y Cartesian coordinates - effective thermal diffusivity - coefficient of thermal expansion - porosity - dimensionless time - similarity variable - dimensionless temperature - viscosity - kinematic viscosity - density - the ratio of heat capacities     

Nonlinear stability of a convective motion in a porous layer driven by a horizontally periodic temperature gradient

The nonlinear global exponential pointwise stability of a vertical steady flow driven by a horizontal periodic temperature gradient in a porous layer is performed. It is shown that the stability threshold depends on the supremum of a quadratic functional, having non constant coefficients, and new in the literature on the convection problem. In solving the variational problem, a suitable functional transformation is used.Received: 27 January 2003, Accepted: 10 March 2003, Published online: 12 September 2003 Correspondence toF. Capone     

Laminar MHD natural convection of nanofluid containing gyrotactic microorganisms over vertical wavy surface saturated non-Darcian porous media

Magnetohydrodynamic (MHD) bioconvection of an incompressible electrically conducting nanofluid near a vertical wavy surface saturated porous medium containing both nanoparticle and gyrotactic microorganisms is investigated. The nanofluid is represented by a model that includes both Brownian motion and thermophoresis effects. A suitable set of non-dimensional variables are used to transform the governing boundary layer equations into a dimensionless form. The resulting nonlinear system is mapped to the vertical flat plate domain, and a non-similar solution is used to the obtained equations. The obtained non-similar system is then solved numerically using the fourth-order Runge-Kutta method. The influence of various physical parameters on the local Nusselt number, the local Sherwood number, the local density number of the motile microorganisms, the dimensionless velocity, the dimensionless temperature, and the rescaled density of motile microorganisms is studied. It is found that the local Nusselt number, the local Sherwood number, and the local density number of the motile microorganisms decrease by increasing either the Grashof number or the magnetic field parameter.     

Effects of anisotropy on the free convection from a vertical plate embedded in a porous medium

The effects of anisotropy on the steady laminar boundary-layer free convection over a vertical impermeable surface are analysed by using the method of integral relations. If the permeability in the direction orthogonal to the plate is greater than the permeability along the plate, then there is an increase in the temperature field.     

Mixed convection of non-newtonian fluids from a vertical plate embedded in a porous medium

This paper investigates mixed free and forced convection of non-Newtonian fluids from a vertical isothermal plate embedded in a homogenous porous medium. A mathematical model is developed based on the modified Darcy's law and boundary-layer approximations, and the exact similarity solution is obtained as well as an integral solution. These two solutions agree within 3% for aiding flows and 10% for opposing flows. It is found that, non-Newtonian characteristics of fluids have appreciable influences on velocity profiles, temperature distributions and flow regimes.     

Effects of thermal dispersion and stratification on combined convection on a vertical surface embedded in a porous medium

The effects of thermal dispersion and thermal stratification on mixed convection about a vertical surface in a porous medium are studied. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations are solved on the basis of the local similarity approximation. The results indicate that both dispersion and stratification effects have considerable influence on the heat transfer rate.     

Magnetohydrodynamics convective flow in a vertical slot through a porous medium

An analysis is presented with magnetohydrodynamics natural convective flow of a viscous Newtonian fluid saturated porous medium in a vertical slot. The flow in the porous media has been modeled using the Brinkman model. The fully-developed two-dimensional flow from capped to open ends is considered for which a continuum of solutions is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. The asymptotic behaviour and the volume flux are analyzed and incorporated graphically for the three-parameter family of solution.     

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.     

Stability of free convection in a rotating porous layer distant from the axis of rotation

The stability and onset of convection in a rotating fluid saturated porous layer subject to a centrifugal body force and placed at an offset distance from the center of rotation is investigated analytically. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number for different values of the parameter representing the dimensionless offset distance from the center of rotation. At the limit of an infinite distance from the center of rotation the results are identical to the convection resulting from heating a porous layer from below subject to the gravitational body force. At the other limit, when the parameter controlling the offset distance approaches zero, the results converge to previously found solutions for the convection in a porous layer adjacent to the axis of rotation. The results provide the stability map for all positive values of the parameter controlling the offset distance from the center of rotation, hence bridging the gap between the two extreme limit cases.     

Capillary end effect in a water saturated porous layer

The uni-directional propagation of oil injected into water flowing through a water wetted porous slab of a finite length is investigated. The inlet and outlet edges of the slab are impermeable to the oil flux. Hence, the oil accumulates within the slab, thereby leading to a saturation build-up-capillary end effect. This phenomenon is studied analytically on the basis of a nonlinear equation describing oil-water transport in porous media. A dimensionless criterion is derived, which governs the appearance and relative strength of the capillary end effect. For weak oil-water interfacial tension (large capillary number) and long porous slabs the above effect is not observed and the temporal evolution of the oil saturation is described by the Buckley-Leverett solution. Short porous slabs are found to be almost entirely subjected to the capillary end effect. Intermediate situations are identified and quantitatively described, in which the downstream part of the slab may be divided into two zones: one-characterized by the capillary end effect, and the other being a Buckley-Leverett zone.It is shown, that the oil flux injected into the slab is limited by a maximum value which depends upon the location of the injection point. The partition of the inlet flux between the upstream and downstream directions is investigated. In the upstream side of the porous slab the oil moves under the action of free imbibition only. It is found that the upstream flux is limited by the value, which is independent of the slab's length and of the location of the injection point.     

On the existence of self-similar solutions of the equations governing unsteady flow through a porous medium

In this paper the conditions for the existence of self-similar solutions of the equations governing unsteady flows through a porous medium are presented and discussed. The first two sections deal with the case of non-Newtonian fluids of power-law behavior; the third section analyzes the case of non-Darcy gas flows. The boundary and initial conditions occuring currently in a large class of fluid mechanics problems, of practical interest in engineering, are considered.     

Mixed convection about a non-isothermal vertical surface in a porous medium

The problem of mixed convection about non-isothermal vertical surfaces in a saturated porous medium is analysed using boundary layer approximations. The analysis is made assuming that the surface temperature varies as an arbitrary function of the distance from the origin. A perturbation technique has been applied to obtain the solutions. Using the differentials of the wall temperature, which are functions of distance along the surface, as perturbation elements, universal functions are derived for various values of the governing parameter Gr/Re. Both aiding and opposing flows are considered. The universal functions obtained can be used to estimate the heat transfer and fluid velocity inside the boundary layer for any type of wall temperature variation. As a demonstration of the method, heat transfer results have been presented for the case of the wall temperature varying as a power function of the distance from the origin. The results have been studied for various combinations of the parameters Gr/Re and the power index m, taking both aiding and opposing flows into consideration. On comparing these results with those obtained by a similarity analysis, the agreement is found to be good.     

Boundary-layer non-Newtonian flow over vertical plate in porous medium saturated with nanofluid

The free convective heat transfer to the power-law non-Newtonian flow from a vertical plate in a porous medium saturated with nanofluid under laminar conditions is investigated. It is considered that the non-Newtonian nanofluid obeys the mathematical model of power-law. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The partial differential system governing the problem is transformed into an ordinary system via a usual similarity transformation. The numerical solutions of the resulting ordinary system are obtained. These solutions depend on the power-law index n, Lewis number Le, buoyancy-ratio number N _r, Brownian motion number N _b, and thermophoresis number N _t. For various values of n and Le, the effects of the influence parameters on the fluid behavior as well as the reduced Nusselt number are presented and discussed.     

Natural convection in a horizontal porous layer with anisotropic thermal diffusivity

The present paper is concerned with free convection in a horizontal porous layer with anisotropic thermal diffusivity. It is assumed that the diffusivity has rotational symmetry, with a symmetry axis making an arbitrary angle against the vertical. The critical Rayleigh number and wave number at marginal stability are calculated and the steady motion occurring at convection onset is examined. It is found that there are two different types of convection cells, depending on whether the longitudinal diffusivity is larger than the transverse diffusivity or not. In the former case, the convection cells are rectangular with vertical lateral walls. In the latter case, however, the lateral cell walls are tilted as well as curved.     

Two-phase boundary layer formation in a semi-infinite porous slab

Similarity profiles of pressure and saturation are analysed which result from one-dimensional planar withdrawal of fluid from a porous region initially containing a two phase mixture of steam and water. Approximate expressions are derived for the evolution of pressure and saturation profiles, and boundary-layer changes in saturation are identified. The existence of a similarity variable implies that the saturation conditions for the reservoir tend with time either to having both phases flowing; or to a single phase vapour. In particular, the nonlinear nature of the governing equations implies that infinitesimal changes in pressure can produce finite changes in saturation. The two mechanisms responsible for saturation changing with time involve local changes in energy storage in rock and fluid; together with spatial variations in flowing enthalpy. The latter mechanism occurred relatively slowly in the examples treated, and was responsible for boundary-layer formation when one phase was initially immobile. Dimensional analysis reveals that when a boundary layer develops, the underlying equations involve essentially only one dimensionless parameter which is typically small, being associated with the ratio of the energy density of the mobile phase relative to the total energy density.     

Investigation of convection in a horizontal cylindrical layer of a saturated porous medium

The structures of the convective motions and the nature of the heat transfer in a horizontal cylindrical layer are studied numerically for the Forchheimer model of a porous medium in the Boussinesq approximation. New asymmetric solutions of the equations of convection flow through a porous medium are found. Their development, domains of existence, and stability are investigated. One consists of a multivortex structure with asymmetric vortices in the near-polar region. Another asymmetric solution is realized at large Grashof numbers in the form of a convective plume deflected from the vertical. The threshold Grashof number of formation of the asymmetric motions depends on the Prandtl number and the cylindrical layer thickness.     

Effect of double dispersion on non-Darcy mixed convective flow over vertical surface embedded in porous medium

A numerical study of a non-Darcy mixed convective heat and mass transfer flow over a vertical surface embedded in a dispersion, melting, and thermal radiation is porous medium under the effects of double investigated. The set of governing boundary layer equations and the boundary conditions is transformed into a set of coupled nonlinear ordinary differential equations with the relevant boundary conditions. The transformed equations are solved numerically by using the Chebyshev pseudospectral method. Comparisons of the present results with the existing results in the literature are made, and good agreement is found. Numerical results for the velocity, temperature, concentration profiles, and local Nusselt and Sherwood numbers are discussed for various values of physical parameters.     

A boundary layer analysis of combined heat and mass transfer by natural convection from a concentrated source in a saturated porous medium

A boundary layer analysis was carried out to investigate the coupled phenomena of heat and mass transfer by natural convection from concentrated heat and mass sources embedded in saturated porous media. Both line and point source problems were treated. The boundary layer equations based on Darcy's law and Boussinesq approximation were solved by means of similarity transformation to obtain the details of velocity, temperature and concentration distributions above a concentrated heat source. Two important parameters, namely the Lewis number Le and the buoyancy ratioN were identified to conduct a series of numerical integrations. For the case of small Le, a substance diffuses further away from the plume centerline, such that the mass transfer influences both velocity and temperature profiles over a wide range. For large Le, on the other hand, the substance diffuses within a narrow range along the centerline. Naturally, the influence of mass transfer is limited to the level of the centerline velocity, so that a peaky velocity profile appears for positiveN whereas a velocity defect emerges along the centerline for negativeN. For such cases of large Le, the temperature profiles are found to be fairly insensitive to Le.     

Onset of electroconvective instability of Oldroydian viscoelastic liquid layer in Brinkman porous medium

The problem of the onset of electrohydrodynamic instability in a horizontal layer of Oldroydian viscoelastic dielectric liquid through Brinkman porous medium under the simultaneous action of a certical ac electric field and a vertical temperature gradient is analyzed. Applying linear stability theory, we derive an equation of eight order. Under somewhat suitable boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Both the cases of stationary and oscillatory instabilities are discussed if the liquid layer is heated from below or above. The effects of the porosity of porous medium, the medium permeability, the Prandtl number, the ratio of retardation time to relaxation time, the elastic number, in the presence or absence of Rayleigh number are shown graphically for both cases. Some of the known results are derived as special cases. The electrical force has been shown to be the sole agency causing instability of the considered system since it is much more important than the buoyancy force even if the medium is porous.