Effect of Viscous Dissipation on a Quasi-Parallel Free Convection Boundary-Layer Flow over a Vertical Flat Plate in a Porous Medium

Transport in Porous Media -     

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid Past a Vertical Plate Immersed in a Porous Medium

The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy–Brinkman–Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy–Brinkman–Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state.     

The Development of the Asymptotic Viscous Dissipation Profile in a Vertical Free Convective Boundary Layer Flow in a Porous Medium

The effect of viscous dissipation on the development of the boundary layer flow from a cold vertical surface embedded in a Darcian porous medium is investigated. It is found that the flow evolves gradually from the classical Cheng–Minkowycz form to the recently discovered asymptotic dissipation profile which is a parallel flow.     

Analytical Solution for the Effect of Viscous Dissipation on Mixed Convection in Saturated Porous Media

A new analytical solution is introduced for the effect of viscous dissipation on mixed convection flow and heat transfer about an isothermal vertical wall embedded in Darcy and non-Darcy porous media with uniform free stream velocity. The effect of viscous dissipation on mixed convection in both regimes has been analyzed for both the aiding and opposing flows using Gebhart number, Ge _x =gx/c _p. The governing parameters are Re, Ra, Pe and Ge _x . The case of Re=0 corresponds to Darcy mixed convection region and Re/Pe is identified as the mixed convection governing parameter, Ra=0 leading to pure forced convection. A good agreement was found between the numerical and analytical solutions. It was found from the Nusselt number results that viscous dissipation lowers the heat transfer rate in both Darcy and Forchheimer flow regimes for aiding as well as opposing flows.     

Backward Free Convection Boundary Layers in Porous Media

The well known steady free convection forward boundary layer (FBL) flows ascending over a heated upwards projecting semi-infinite flat plate embedded in a fluid saturated porous medium are compared in this paper to their less well known backward (BBL) counterparts descending over a cooled (also upwards projecting!) semi-infinite flat plate. The circumstance that the definite edge of the plate (x = 0) in the former case is a leading edge and in the latter one a trailing edge, leads to substantially different mathematical and physical features of the FBL and BBL flows, respectively. The paper considers under this aspect the case of similar flows corresponding to surface temperature distributions which are power-law functions of the distance x from the definite edge. For permeable plates the effect of an adequate lateral suction and injection of the fluid is also taken into account. The detailed investigation, however, is restricted to the particular values m = +1 and m = –1/3 of the power-law exponent m, where both FBL and BBL solutions are available in exact analytic form. For each of these values, both exponentially and algebraically decaying BBL solutions were found. In addition, the existence of an exact algebraic BBL solution valid for any value of m is reported.     

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.     

Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work

Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with respect to the Oberbeck–Boussinesq approximation is proposed.     

Vortex Instability of the Asymptotic Dissipation Profile in a Porous Medium

In this note we consider the thermoconvective stability of the recently-discovered asymptotic dissipation profile (ADP). The ADP is a uniform thickness, parallel-flow boundary layer which is induced by a cold surface in a warm saturated porous medium in the presence of viscous dissipation. We have considered destabilisation in the form of stream-wise vortex disturbances. The critical wavenumber and Rayleigh number for the onset of convection have been determined for all angles of the cooled surface between the horizontal and the vertical for which the ADP exists. The paper closes with a presentation of some strongly nonlinear computations of steady vortices.     

Aiding and Opposing Mixed Convection Flow in Melting From a Vertical Flat Plate Embedded in a Porous Medium

The problem of melting from a vertical flat plate embedded in a porous medium is studied. The main focus is to determine the effect of mixed convection flow in the liquid phase on the melting phenomenon. Both aiding and opposing flows are considered. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The governing equations are solved numerically. Numerical results are obtained for the temperature and flow fields in the melting region. The melting phenomenon decreases the local Nusselt number at the solid–liquid interface.     

Unsteady Free Convection along an Infinite Vertical Flat Plate Embedded in a Stably Stratified Fluid-Saturated Porous Medium

The problem of unsteady free convection heat transfer from a one-dimensional (parallel) flow along an infinite vertical flat plate embedded in a thermally stratified fluid-saturated porous medium is considered. Flows are induced by a sudden change in the arbitrary temporal plate temperature. By a formal reduction of the corresponding boundary value problems to well-known Fourier heat conduction problems, analytical solutions of the Darcy and energy equations are obtained. Several special cases are discussed in detail.     

Effect of the Source Term on Steady Free Convection Boundary Layer Flows over an Vertical Plate in a Porous Medium. Part I

The Darcy free convection boundary layer flow over a vertical flat plate is considered in the presence of volumetric heat generation/absorption. In the present first part of the paper it is assumed that the heat generation/absorption takes place in a self-consistent way, the source term q ^′′′≡ S of the energy equation being an analytical function of the local temperature difference T − T _∞. In a forthcoming second part, the case of the externally controlled source terms S = S(x,y ) will be considered. It is shown that due to the presence of S, the physical equivalence of the up- and downflows gets in general broken, in the sense that the free convection flow over the upward projecting hot plate (“upflow”) and over its downward projecting cold counterpart (“downflow”) in general become physically distinct. The consequences of this circumstance are examined for different forms of S. Several analytical solutions are given. Some of them describe algebraically decaying boundary layers which can also be recovered as limiting cases of exponentially decayingones. This asymptotic phenomenon is discussed in some detail.     

Reply to Comments on “Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid over a Flat Plate Embedded in a Porous Medium”

In this note, we reply to the comments by Rees and Magyari (2007) on our article (Aydin and Kaya 2007). They mainly stated that the thermal boundary conditions we defined at the edge of the boundary layer were incompatible with the energy equation. This is questionable and therefore we will discuss it below. They disclosed that our results were in error. However, this is quite misleading. Scientifically, they cannot reach such a conclusion without comparing our results with what they thought to be correct. In fact, this misleading and unproven statement will be shown not to be correct in the following.     

Thermal Capacity Effect on Transient Free Convection Adjacent to a Vertical Surface in a Porous Medium

In this paper we analyse how the presence of the thermal capacity of a vertical flat plate of finite thickness, which is embedded in a porous medium affects the transient free convection boundary-layer flow. At the time t = 0, the plate is suddenly loaded internally with a constant heat flux rate q, so that a transient boundary-layer flow is initiated adjacent to the plate. Initially, the transient effects due to the imposition of the uniform heat flux rate at the plate are confined to a thin fluid region near to the surface and are described by a small time solution. These effects continue to penetrate outwards and eventually evolve into a new steady state flow. Analytical solutions have been derived for these transient (small time) and steady state (large time) flow regimes, which are then matched by a numerical solution of the full boundary-layer equations. It has been found that the non-dimensional fluid temperature (or fluid velocity) profiles are reduced when the thermal capacity effects, described by a parameter Q ^*, are reduced. For small values of Q ^*, the approach of these profiles to their steady state values is monotonic. However, for large values of Q ^*, the temperature profiles are observed to locally exceed (pass through a maximum value) the final steady state values at certain distances from the plate. In general, the maxima in the temperature profiles increase in size as Q ^* increases and the time taken to approach the steady state solutions increases significantly.     

Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid over a Flat Plate Embedded in a Porous Medium

In this study, laminar boundary layer flow over a flat plate embedded in a fluid-saturated porous medium in the presence of viscous dissipation, inertia effect and suction/injection is analyzed using the Keller box finite difference method. The flat plate is assumed to be held at constant temperature. The non-Darcian effects of convection, boundary and inertia are considered. Results for the local heat transfer parameter and the local skin friction parameter as well as the velocity and temperature profiles are presented for various values of the governing parameters. The non-Darcian effects are shown to decrease the velocity and to increase the temperature. It is also shown that the local heat transfer parameter and the local skin friction parameter increase due to suction of fluid while injection reverses this trend. It is disclosed that the effect of the viscous dissipation for negative values of Ec (T _w < T _∞) is to enhance the heat transfer coefficient while the opposite is true for positive values of Ec (T _w > T _∞). The results are compared with those available in the existing literature and an excellent agreement is obtained.     

Radiation Effect on MHD Free Convection Flow of a Gas Past a Semi-infinite Vertical Plate with Variable Viscosity

The radiation effect in the presence of a uniform transverse magnetic field on steady free convection flow with variable viscosity is investigated. The fluid viscosity is assumed to vary as the reciprocal of a linear function of temperature. Boundary layer equations are derived. The resulting approximate non-linear ordinary differential equations are solved linearly and nonlinearly by shooting methods. The velocity and temperature profiles are shown, and the skin friction on the plate and heat transfer coefficient are presented and discussed. The results of the present study show that in the presence of magnetic field, as the radiation parameter increases the temperature increases, but the velocity decreases.     

Temperature Variations of Forced Convection in Porous Media for Heating and Cooling Processes: Internal Heating Effect of Viscous Dissipation

An analytical study on fully developed forced convection in a homogeneous porous medium is reported. Incorporating the internal heating effect of viscous dissipation, closed form solutions of the temperature distributions in the transverse direction are obtained and analyzed for both heating and cooling processes. Variations of Nusselt number as a function of Darcy number and Brinkman number and the existence of singularity in Nusselt number are also discussed. An erratum to this article can be found at     

An Analytical Study of Free Convective Boundary-Layer Flow in Porous Media: The Effect of Anisotropic Diffusitivity

The effect of an anisotropic thermal diffusivity tensor on the free convective boundary-layer flow in porous media is studied. Convection is induced by a generally inclined, uniformly heated surface embedded in a fluid-saturated medium. A third-order boundary-layer theory is presented in order to obtain accurate information on the effect of anisotropy on the rate of heat transfer into the porous medium. It is shown that the thickness of the resulting leading order boundary-layer flow depends on the precise nature of the anisotropy. On the other hand, the anisotropic diffusivity does not induce a fluid drift in the spanwise direction, a result which is different from that obtained in our earlier study of the effects of an anisotropic permeability. It is found that the second order temperature field does not contribute to the overall rate of heat transfer. Finally, we show that the third-order correction to the leading-order rate of heat transfer is given in terms of an explicit formula.     

The Effect of the Local Inertial Term on the Transient Free Convection from a Vertical Plate Inserted in a Semi-Infinite Domain Partly Filled with Porous Material

The transient hydrodynamics and thermal behaviours of the free convection from a vertical plate inserted in a semi-infinite domain partly filled with porous material are investigated. The role of the local macroscopic inertial term in the porous domain momentum equation is studied. It is found that the effect of the local inertial term on the domain behaviour is insignificant when Da < 1 × 10^–5 for all operating conditions. Also, the effect of the macroscopic inertial term is insignificant at large values of _R, where _R > 2.0, and over the entire ranges of _R, K_R and Pr (thermal diffusivity ratios, thermal conductivity ratios and Prandtl number, respectively).     

The Effect of Time-Periodic Surface Temperature Oscillations on Free Convection from a Vertical Surface in a Porous Medium

In this paper, we consider the unsteady free convection boundary layer flow which is induced by time-periodic variations in the surface temperature of a vertical surface embedded in a porous medium. The basic steady flow is that of a power-law distribution where the surface temperature varies as the nth power of the distance from the leading edge. Small-amplitude time-periodic disturbances are added to this basic distribution. Both the low- and high-frequency limits are considered separately, and these are compared with a full numerical solution obtained by using the Keller-box method. Attention is restricted to the cases n1; when n=1, the flow is locally self-similar for any prescribed frequency of modulation.