Inertia effect on free convection over horizontal surface in a porous medium with a variable wall heat flux and variable wall temperature containing Internal heat generation

Numerous phenomena involve free convection driven by internal heat generation. Free convection, as a process of vital significance, has been of interest to researchers in both industry and academia, such as a recent numerical work by Liu et al. [1]. Two important modelling of free convection are related to nuclear energy and combustion, specifically in the development metal waste from spent nuclear fuel and the storage of spent nuclear fuel, see Horvat et al. [2], Jahn and Reinke [3]. A couple of recent papers have been devoted to the subject of similarity solutions for free with internal heat generation in porous media for several geometric configurations, see Bagai [4], Postelnicu and Pop [5], Postelnicu et al. [6,7]. Several recent papers, such as Magyari et al. [8,9], Mealey and Merkin [10], Merkin [11] throw new lights on the internal heat generation in porous media, remaining in the frame of similar solutions.


Introduction
Numerous phenomena involve free convection driven by internal heat generation. Free convection, as a process of vital significance, has been of interest to researchers in both industry and academia, such as a recent numerical work by Liu et al. [1]. Two important modelling of free convection are related to nuclear energy and combustion, specifically in the development metal waste from spent nuclear fuel and the storage of spent nuclear fuel, see Horvat et al. [2], Jahn and Reinke [3]. A couple of recent papers have been devoted to the subject of similarity solutions for free with internal heat generation in porous media for several geometric configurations, see Bagai [4], Postelnicu and Pop [5], Postelnicu et al. [6,7]. Several recent papers, such as Magyari et al. [8,9], Mealey and Merkin [10], Merkin [11] throw new lights on the internal heat generation in porous media, remaining in the frame of similar solutions.
Turning now to the non-Darcy formulations, with focus on vertical and horizontal configurations in porous media, we present below a short review of the literature dedicated on these topics.

a) Vertical configuration
• Thermal dispersion, which becomes prevalent in the non-Darcy regime, and viscous dissipation effects are studied by Murthy [12]) for a vertical flat plate configuration, with inertia effect included in the momentum equation. Constant wall temperature conditions were considered in that study.

•
Nonsimilarity solutions for the non-Darcy mixed convection from a vertical flat plate embedded in a saturated porous medium for a power-law variation of the wall temperature were reported by Kodah and Duwairi [13].
• A number of non-Darcian effects, including the high-flowrate inertia were analysed by Hung and Chen [14] for a free convection along a impermeable vertical plate embedded in a thermally stratified fluid-saturated porous medium. Variable heat flux conditions are imposed in that study.

b) Horizontal configurations
• The presence of inertia (Forchheimer form-drag) in the steady free convection boundary layer over an upward-facing horizontal embedded in a porous medium was studied by Rees [15].
• This work was extended later by Hossain and Rees [16] by taking the surface temperature as a power-law variation in terms of the distance from the leading edge.
• Duwairi et al. [17] analyzed the combined free and forced convection regimes within a non-Darcy model for a horizontal configuration.
In view of the above literature survey, it seems that the effect of inertia on free convection from a horizontal surface embedded in a porous medium with internal heat generation was not considered till now, at our best knowledge. Two cases are considered in the paper: variable wall heat flux and variable wall temperature. Distinctive from the previous quoted studies where inertia effects have been considered, in the present paper there are obtained similarity solutions, in the spirit of studies dealing with internal heat generation.

Analysis
Consider the natural convection , laminar, two dimensional boundary layer flow over horizontal surfaces in porous media with internal heat generation of wall temperature Tw and uniform ambient temperature T∞,where Tw > T∞. The x-coordinate is measured along the surface and the y-coordinate normal to it. Under the Boussinesq and boundary layer approximations, the basic equations are continuity equation, momentum equation and energy equation where the stream function Ψ was introduced through the relationships such as the continuity equation Is identically satisfied.

Case of variable wall heat flux
The boundary conditions for the model are We now introduce the following dimensionless variables Where the local Rayleigh number is defined as We then consider the internal heat generation (IHG) of the form Equations (1) and (2) Where ( ) Is the inertia parameter. The corresponding boundary conditions are Quantities of engineering interest are the skin friction coefficient and Nusselt number, defined as where U c is a characteristic velocity, taken here as Performing the calculations, we get ( ) Nu Ra (17) Case of variable wall temperature 1.
The boundary conditions in this case are u=0, T=T ∞ , as y→∞ (18b) We now introduce the following dimensionless variables Where the local Rayleigh number is defined as Now, the internal heat generation term is taken in the following form in order to obtain similarity solutions.
The problem becomes, in transformed variables, That must be solved along the boundary conditions The expression of the skin friction coefficient remains the same, so that ( ) While for the Nusselt number we get ( )

a)
A well-known finite difference method, namely the Kellerbox method, incorporated in our own code b) Using the dsolve routine from MAPLE [18].
The results given by these methods were compared and in each case the maximum value of η, say η max , was adjusted in order to conciliate the outputs and get grid independence in case a). As a rule, the dynamic and thermal boundary layers become thicker as G increases and also the convergence of the numerical procedure is achieved with more difficulty. Basically, η max was raised till 20, when G was increased. Results are reported separately for the two cases considered in this study: Figures 1-6 Figures 1 and 2 respectively considering with and without IHG. From Figure 1, it is seen that in both the cases (with and without IHG) the skin friction is more significant and induced more flow than that of without IHG, hence give rise to the velocity gradient leads to the increase of the skin friction. On the other hand, one can easily remark from Figure 2 that the Nusselt number with IHG is less intensive than the presence of IHG.       We can see that the velocity profiles decreases with increasing G (and m). The reverse happens as one move far away from the surface indicating that the effects of the parameters are more dominant near the surface. Further it is also seen that the momentum boundary layer thickness has influenced greatly by IHG. From Figures  5 and 6, it is seen that the temperature profiles increases and tends to zero at the edge of the thermal boundary layer satisfying the boundary condition θ()=0. Also the thermal boundary layer thickness increases when the effect of IHG is considered. The dimensionless velocity and temperature profiles as well as ski friction coefficient and Nusselt number for some values of G and λ in case of VWT are presented in Figures 7-12. We can draw the same conclusions as in the case of VHF.5.

Conclusions
At our best knowledge, the available studies on free convective boundary layers in porous media with inertia effects included in non-Darcy models do not report variations of the skin friction and Nusselt number with G, due to the fact that only non-similarity solutions have been obtained. In such circumstances, the authors concentrated themselves on the variation of these two quantities along the non-     . We are here in another situation, due to the existence of similarity solutions, so that we found that for both cases, VHF and VWT, a maximum value G=100 are enough to achieve an almost constant value of the skin friction and Nusselt number.