Abstract
In this paper, the effects of **surface** **radiation** on **heat** **transfer** and **natural** **convection** in a **cavity** **containing** a **horizontal** **porous** **layer** have been studied numerically. The governing equations for the momentum and **heat** **transfer** in both free fluid and **porous** medium were solved by the finite element method. The radiative **heat** **transfer** is calculated by making use of the radiosity of the surfaces that assumed to be grey. Comparisons with experimental and numerical results in the literature have been carried out. Effects of **thermal** **radiation** on **natural** **convection** and **heat** **transfer** in both free fluid and **porous** medium were analyzed. It was found that **surface** **thermal** **radiation** can significantly change the temperature fields in both the regions of free flow and **porous** medium. The mean temperature at the interface decreases and the temperature gradients are created on the upper two corners of the **porous** medium region as Ra increases.

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heated **porous** lid-driven **cavity** was investigated by Oztop [2] . He concluded that **heat** **transfer** decreases as the Richardson number (Ri) increases, but that increments in the Darcy number (Da) cause **heat** **transfer** to rise. Sameh E. Ahmed [3] studied mixed **convection** from a discrete **heat** source in enclosures with two adjacent moving walls and filled with micropolar nanofluids. Numerical modeling of **thermal** characteristics in a microstructure filled **porous** **cavity** with mixed **convection** conducted by Bhuiyan et al. [4] . Javaherdeh et al. [5] studied **natural** **convection** **heat** and mass **transfer** in MHD fluid flow past a moving vertical plate with variable **surface** temperature and concentration in a **porous** medium. Khanafer and Vafai [6] carried out a numerical study of mixed **convection** **heat** and mass transport in a lid-driven square **cavity** filled with a non-Darcian fluid-saturated **porous** medium, and reported that the buoyancy ratio, Reynolds number (Re), Darcy number, and Richardson number have a profound **effect** on **heat** **transfer**. A numerical study on **natural** **convection** in **porous** media-filled an inclined triangular enclosure with **heat** sources using nanofluid in the presence of **heat** generation **effect** is con- ducted by Mansour and Ahmed [7] . Agarwal et al. [8] investigated double diffusive mixed **convection** in a lid-driven **porous** **cavity**. It was observed that **convection** flow was significant up to Darcy number of 0.1. Nasseddine Ouertatani et al. [9] studied the intri- cate three-dimensional flow structure and **heat** **transfer** rate in a

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Abstract— The flow progress in a water-filled square **cavity** which is suddenly heated and cooled on the opposing sidewalls is noticed by using the CFD software package Fluent 6.3. Two approximations were used for CFD simulation of **radiation** in a liquid **cavity**, namely, Rosseland approximation the methods of spherical harmonic functions. Comparison of three different approaches to description of the **radiation** mechanism of energy **transfer** has made it possible to recognize the special features of these models. Temperature and stream distribution functions are acquired in a large range of the governing criterion. The **horizontal** boundaries of the **cavity** are insulated, while the heated and cooled vertical left and right walls are conducting. The present thesis continues the explanation of the flow growth until a steady state is reached. A clear visualization of the interaction between the second wave group and the laminar interruption flow across the roof of the **cavity** is given. The consequent collapse of the **horizontal** interruption cause the **thermal** **layer** in the **cavity** core and the adjustment of the vertical boundary **layer** are also observed. The final state is represented by waves which continuously travel along the boundary layers.

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This study investigates the **effect** of electromagnetic field and **radiation** absorption on **natural** **convection** in a **horizontal** shallow **porous** **cavity** filled with an electrically conducting binary fluid. subjected to cross fluxes of **heat** and mass. The Darcy model, Rosseland approximation for the radiative flux and the Boussinesq approximation for density variations are used in the formulation of the problem. In the limit of shallow **cavity**, parallel proximation is adopted and the result established that the flow intensity, **heat** and mass **transfer** are considerably affected by **radiation** absorption and magnetic field depending on whether the tical **radiation** value, depending on the **thermal** Rayleigh number, for the onset of unstable **convection** is also established in this study. The

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Partha et al. (2006) examined the Soret and Dufour effects in a non-Darcy **porous** medium. Lakshmi Narayana and Murthy (2008) considered both the Soret and Dufour effects on a free **convection** boundary **layer** of a **horizontal** flat plate in a Darcy **porous** medium. Cheng (2009) reported the Soret and Dufour effects on **natural** **convection** **heat** and mass trans- fer from a vertical cone in a **porous** medium with uniform wall temperature and concentration (UWT/UWC). The Soret and Dufour effects on **heat** and mass **transfer** by **natural** convec- tion from a vertical truncated cone in a fluid-saturated **porous** medium with variable wall temperature and concentration were studied by Cheng (2010) . Makinde (2011) studied the mixed **convection** flow with Soret and Dufour effects past a vertical plate embedded in a **porous** medium. Makinde et al. (2012) used a numerical method to study of chemically- reacting hydromagnetic boundary **layer** flow with Soret/ Dufour effects and a convective **surface** boundary condition.

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Gebhart and Pera [15], using the similarity method, investigated the laminar ows which arise in uids due to the interaction of the force of gravity and density dierences caused by the simultaneous diusion of **thermal** energy and chemical species. Bejan [16] presented a fundamental study of laminar **natural** **convection** in a rectangular enclosure with **heat** and mass **transfer** from the side. He used scale analysis to determine the scales of the ow, temperature and concentration elds in boundary **layer** ow for all values of Prandtl and Lewis numbers. He investigated the case of N = 0 to study the **heat**-**transfer**-driven ows. Wee et al. [17] investigated numerically and experimentally the same problem for both **horizontal** and vertical **cavity**. The experimental technique employed two **porous** plastic plates as two **cavity** walls allowing the imposition of simultaneous moisture and temperature gradients.

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There are many opportunities for further theoretical work to explore the thermosolutal reaction–**convection** system described here. **Natural** extensions of the stability analysis include considering other boundary conditions at the upper and lower surfaces (for exam- ple, **heat** flux conditions or chemical disequilibrium), or studying the role of effects such as anisotropic permeability or nonlinear equilibrium solubility curves. In a future study, we intend to investigate further the nonlinear dynamics of the system, with a particular focus on behaviour in the subcritical SS regime (which Mamou & Vasseur (1999) found to be particularly rich in the case of pure DDC), on quantifying **heat** and mass **transfer** across the **layer**, and on the long-term evolution of the **porous** matrix as precipitation and dissolution affects its porosity and permeability. It is likely that in this paper we have done no more than to scratch the **surface** of an interesting and complicated problem.

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2. Mathematical Formulation
We consider a **horizontal** **layer** of Brinkman **porous** medium of thickness d . The lower
**surface** is held at constant temperature T , while the upper **surface** is at l T (< u T ). A Cartesian l
coordinate system ( , , ) x y z is chosen such that the origin is at the bottom of the **porous** **layer** and the z-axis vertically upward in the presence of gravitational field. The solid and fluid phases of the **porous** medium are assumed to be in local **thermal** non-equilibrium (LTNE) with a two-field model for temperatures. The solid temperature equation is modified to allow the **heat** **transfer** via a Cattaneo **heat** flux theory, while the usual Fourier **heat**-**transfer** law is used in the fluid. The basic equations governing the flow of an incompressible fluid saturating a **layer** of Brinkman **porous** medium with LTNE and Cattaneo **effect** in the solid are [2, 16]

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4.1. The 12×6 arrangements
The influence of staggered arrangement of blockages on the flow and **heat** **transfer** is analysed by comparing the results with in-line arrangements. Figure 3 shows the temperature profile at mid-width and near the bottom part of the **cavity**. A nearly similar pattern is observed for the top part and hence is omitted to avoid repetition. The change in temperature between the two configurations is significant especially near the passive **horizontal** walls, with a maximum of about 2.5 ᴼC and is found to occur within the boundary **layer**. The **effect** due to staggering is more prominent up to y/H=0.15 after which a stable core region can be seen. The turbulence intensity for the two configurations is shown in Fig. 4. It can be observed that the **effect** is very prominent on turbulence intensity. The average Nusselt number for the two configurations presented in Table 1 shows that the staggering reduces the **heat** **transfer** by a modest 7%.

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conditions. An analytical and numerical study of **natural** **convection** **heat** and mass **transfer** through a vertical **porous** **layer** subjected to a concentration difference and a temperature difference in the **horizontal** direction has been studied by Trevisan and Bejan [27]. Many physical systems were modelled as a two-dimensional **cavity** with the vertical walls held at fixed but different temperatures or concentrations and the connecting **horizontal** walls considered as adiabatic or impermeable Angirasa et al. [28] were reported a numerical study of combined **heat** and mass **transfer** by **natural** **convection** adjacent to vertical surfaces situated in fluid saturated **porous** media. Akbal and Baytas [29] have investigated a radioactive gas **transfer** depending on the decay of the gas, Schmidt and concentration Grashof numbers by **natural** **convection** in a fluid saturated **porous** medium. Merkin and Mahmood [30] have investigated a model for the convective flow in a fluid-saturated **porous** medium **containing** a reactive component. Goyeau et al. [31] have studied the double diffusive **natural** **convection** using Darcy—Brinkman formulation in a **porous** **cavity** with impermeable boundaries. Bahloul et al. [32] have investigated the double diffusive **convection** in a long vertical **cavity** heated from the below and imposed concentration gradient from the sides both analytically and numerically.

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Complex shape enclosures and wavy geometries are utilized in various engineering designs to enhance transport performance (Sheremet et al., 2016). In this regard, many of literature studies addressed the **natural** **convection** **heat** **transfer** (Khanafer, 2014). For instance, Adjlout et al. (2002) theoretically addressed the influence of a wavy wall of the convective **heat** **transfer** for various value of Rayleigh number, **cavity** inclination angles. The outcomes reveal that the geometry of the **cavity** walls influences the flow and **heat** **transfer** rate in the **cavity**. Mahmud et al. (“Free **convection** in an enclosure with vertical wavy walls”, 2002) studied the **effect** of amplitude and aspect ratio of a wavy wall on the convective **heat** **transfer** characteristics in an enclosure. Yu and Xu (2018) investigated the **effect** of various **thermal** boundary conditions on the **heat** **transfer** in a **cavity** and revealed using a finite element method. The **natural** **convection** over vertical plates is also studied by Ahmed and Mahdy (2016) and Ahmed (2017).

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4.1. The 12×6 arrangements
The influence of staggered arrangement of blockages on the flow and **heat** **transfer** is analysed by comparing the results with in-line arrangements. Figure 3 shows the temperature profile at mid-width and near the bottom part of the **cavity**. A nearly similar pattern is observed for the top part and hence is omitted to avoid repetition. The change in temperature between the two configurations is significant especially near the passive **horizontal** walls, with a maximum of about 2.5 ᴼC and is found to occur within the boundary **layer**. The **effect** due to staggering is more prominent up to y/H=0.15 after which a stable core region can be seen. The turbulence intensity for the two configurations is shown in Fig. 4. It can be observed that the **effect** is very prominent on turbulence intensity. The average Nusselt number for the two configurations presented in Table 1 shows that the staggering reduces the **heat** **transfer** by a modest 7%.

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In all these investigations, the **radiation** effects are neglected. For some industrial applications such as glass production and furnace design and in space technology applications, such as cosmical flight aerodynamics rocket, propulsion systems, plasma physics and spacecraft re – entry aerothermodynamics which operate at higher temperatures, **radiation** effects can be significant. Alagoaet al. [24] studied radiative and free **convection** effects on MHD flow through **porous** medium between infinite parallel plates with time – dependent suction. Bestman and Adjepong [25]analyzed unsteady hydromagnetic free **convection** flow with radiative heattransfer in a rotating fluid.Promise Mebine and Emmanuel MunakuroghaAdigio [26]investigates the effects of **thermal** **radiation** on transientMHD free **convection** flow over a vertical **surface** embedded in a porousmedium with periodic temperature. Analytical solutions are obtainedfor the governing coupled dimensionless partial differential equations ofvelocity and temperature.An unsteady, two – dimensional, hydromagnetic, laminar free convective boundary – **layer** flow of an incompressible, Newtonian, electrically conducting and radiating fluid past an infinite heated verticalporous plate with **heat** and mass **transfer** is analyzed by Ramachandra Prasad and Bhaskar Reddy [27], by takinginto account the **effect** of viscous dissipation. The dimensionlessgoverning equations for this investigation are solved analyticallyusing two – term harmonic and non – harmonic functions.The influence of viscous dissipation and **radiation** on an unsteady MHD freeconvection flow past an infinite heatedvertical plate in a **porous** medium with time – dependent suction was studied by Israel – Cookeyet al.[28].

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Some important conclusions from the study are provided as follows: It is found that when the nanoparticle volume fraction is applied, the circulation intensity increases due to the increase in **thermal** conductivity of the nanofluid. The conduction **heat** **transfer** pushes the isotherms within the **porous** **layer** to take almost a diagonal shape, while the **convection** mode **heat** **transfer** forces the isotherms within the nanofluid **layer** to appear almost **horizontal** to the sloping walls. A higher nanoparticle volume fraction (φ = 0.2) leads to a higher overall Nusselt number due to higher **thermal** conductivity. The smaller **porous** **layer** thickness (S = 0.3) has stronger **effect** on the **heat** **transfer** rate, which has the higher average Nusselt number due to lower **thermal** conductivity of pure fluid compared to that of nanofluid. Qualitatively, the enhanced-**heat** **transfer** situation is seen in all the three nanofluids compared to that of the base fluid but the following general result holds: Nu water–Ag > Nu water–Cu > Nu water–TiO2 . The ramification of this

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In this paper, the effects of Rayleigh numbers on **natural** **convection** and **heat** **transfer** with **thermal** **radiation** in a **cavity** partially filled with a **porous** medium have been studied numerically. The governing equations for the momentum and **heat** **transfer** in both free fluid and **porous** medium were solved by the finite element method. The radiative **heat** **transfer** is calculated by making use of the radiosity of the surfaces that assumed to be grey. Comparisons with experimental and numerical results in the literature have been carried out. Effects of Rayleigh number on **natural** **convection** and **heat** **transfer** in both free fluid and **porous** medium were analyzed. It was found that Rayleigh numbers can significantly change the temperature fields in both the regions of free flow and **porous** medium. The mean temperature at the interface decreases and the temperature gradients are created on the upper two corners of the **porous** medium region as Ra increases.

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very few studies have been done for the **natural** convec- tion of nanofluids in **porous** media. Nield and Kuznetsov [24] studied the Cheng-Minkowycz problem for **natural** **convection** boundary **layer** flow in a **porous** medium satu- rated by a nanofluid. In the modeling of the problem, they used nanofluids by incorporating the effects of Brownian motion and thermophoresis. For the **porous** medium, the Darcy model was taken. Aziz et al. [25] found the numer- ical solution for the free **convection** boundary **layer** flow past a **horizontal** flat plate embedded in **porous** medium filled by nanofluid **containing** gyrotactic microorganisms. Recently, Rana et al. [26] found the numerical solution for steady-mixed **convection** boundary **layer** flow of a nano- fluid along an inclined plate embedded in a **porous** medium. In the studies of **natural** **convection** of nanofluids in **porous** media, the authors did the parametric study only. However, they did not account any **effect** of para- meters influencing the **thermal** conductivity and dynamic viscosity, such as particle concentration, particle size, temperature, nature of base fluid, and the nature of nano- particle, which satisfy the experimental data for the ther- mal conductivity and dynamic viscosity of the nanofluids. In the best knowledge of the authors of this article, no such study has been done with regard to the **natural** con- vection of nanofluids in **porous** media. It is known that **heat** **transfer** in a fluid depends upon the temperature dif- ference in fluid and heated **surface** and the thermophysical properties of the fluid. **Heat** **transfer** also depends upon the fluid flow rate, which depends upon the viscosity of the fluid.

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Free **convection** **heat** **transfer** from inclined wavy **surface** has received attention because of its vast applications. Some of these applications include ground water flows, oil recovery processes, **thermal** insulation engineering food processing etc. Extensive literature on the topic is availed for **porous** media, Slimi et. al. 1998[1] studied two – dimensional and transient fluid flow and **heat** **transfer** by **natural** **convection** in a vertical cylinder opened at both ends filled with a saturated **porous** medium and heated with a uniform lateral **heat** flux. The study was carried out using the forchheimer – extended Darcy flow model. Taofik et.al. 1999[2] studied unsteady **natural** **convection** which occurs in a vertical cylindrical enclosure opened at both ends, filled with a fluid saturated **porous** medium with a periodic lateral **heat** flux density. The study was carried out by the use of the Darcy law and it takes in to account **heat** conduction in the wall. The set of equations was solved numerically by the standard finite volume method. Saaed 2000[3] proposed a simple numerical expression for average Nusselt numbers over isothermal **horizontal** cylinder for all Rayleigh by using Darcy flow model. And also Khalid Abd.Al-hussein 2001[4] obtained a simple relation for Nusselt number which is a strong function of modified Rayleigh number, time, radius ratio, and aspect ratio. AL-Najar 2004[5] used the finite difference method to investigate the steady free **convection** from a two separated **horizontal** cylinders embedded in saturated **porous** media bounded by rectangular **cavity**. The cylinders kept isothermally hot while the bounded **cavity** is isothermally. It found that the large **heat** gained to the **cavity** been at the upper **horizontal** wall above cylinders. Saleh 2008[6] studied numerically unsteady **natural** **convection** **heat** **transfer** through a fluid–saturated **porous** media in inclined pipe enclosure. The temperature at cylindrical sidewall Tw was

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