he natural convection heat transfer and fluid flow attracts many interests due to many applications in the engineering applications and industrial projects. Some these applications can be mentioned as solar collectors, passive cooling, double pane windows, ventilation systems, lead-acid batteries, furnaces, etc. (***).

in the cavities filled with two immiscible fluids of sulfuric acidwater (25-75%) and air with different profiles of heat flux from side walls. The heatline visualization approach is employed to detect the heat energy path in the cavity from heat sources to heat sinks. The operating fluid is sulfuric acid-water (25-75%) at the bottom region and air at the top of the cavity. The Navier-Stokes equations are solved based on two-dimensional form, and finite volume approach is utilized. The side walls are heated variable heat flux and no-slip condition is applied to them. The top and bottom walls are cooled by environment temperature with no-slip condition. The influences of different governing parameters of Rayleigh number (?10?^3<Ra<?10?^5), and four different variable distributions of heat flux on the flow structure, temperature field, velocity and temperature distributions, average Nusselt number, skin friction coefficient and heatlines have been presented comprehensively.

The natural convection phenomenon is investigated my many researchers who analyzed the influence of different governing parameters on the fluid flow and heat transfer. In this context, Alshuraiaan and Khanafer [1] examined the effects of the heated thin porous fin position on the laminar natural convection in a square cavity. It was found that the average Nusselt number enhances with the presence of fin compared with heated cavity without fin. Kefayati and CheSidik[2] simulated the natural convection and entropy generation in inclined square cavity filled by non-Newtonian nanofluid using Buongiorno's mathematical model. It was observed that the augmentation of the power-law increases various entropy generations in different inclined angles and Rayleigh number. Khatamifar et al. [3] studied the conjugate natural convection in a square cavity which is divided into two parts and heated partially for different Rayleigh number in range of 10 5 to 10 9 . The results indicate that the average Nusselt number reduces with increasing of partition thickness and enhances with Rayleigh number. Sathe and sammakia [4] obtain a review on developments in practical studied of air-cooled electronic packages. They studied various cases of cavities of single heat source, two heat sources with same dissipation rate and two heat sources with different dissipation rates. In all cases, the heat sources were located at vertical wall. They analyzed different parameters which are important in the natural convection such as Rayleigh number, the ratio between heat sources dissipation rates, the distance between heat sources and the aspect ratio of the cavities. Ye [5] investigated the thermal and hydraulic performance of natural convection in a rectangular storage cavity. The fluid flow and temperature field were analyzed to identify the type of heat transfer mechanism during different stages of melting process. It was shown that the conduction mechanism was dominant heat transfer mode at the first levels of melting process. Subsequently, by transition from conduction mechanism to convection mechanism, the convection heat transfer became dominant mode of heat transfer. Mahdavi et al. [6] carried out an experimental and numerical investigation on the hydrodynamic and thermal characteristics of laminar free convection in a rectangular cavity filled by different fluids such as water, ethylene glycol-water and air. They presented the three-dimensional distribution of local Nusselt number on the surfaces of active walls. It was found that the impact of adiabatic walls on the Nusselt number is significant when the studied fluid is air compared with other type of fluids, in addition; the convective terms were dominant compared with thermal diffusion. Teamah and Shehata[7] The main aim of this article is to consider the natural convection in a rectangular enclosure filled with two immiscible fluids (air and sulfuric acid) heated from side walls with different profile of heat flux and investigate the effect of different governing parameters such as heat flux profiles and Rayleigh number on heat transfer characteristics and flow pattern.

In this study, the side walls are exposed to four special profile of heat flux, while, the part of walls that located the air phases are exposed to constant heat flux. The equation profiles of these heat fluxes are:

In this study the governing equations are solved in two dimensional forms. The two dimensional form of continuity, momentum and energy equations for laminar and steady natural convection can be written as follows:

By introduction of the following dimensionless parameters:

The dimensionless form of the governing equations can be obtained as follows:

The local Nusselt number of fluid on the side walls can be obtained from following equation:

The total heat flux entry to side walls (?? ? ) is same for all heat flux profiles and ?? ?? at the equation ( 10 The heat line can be used to visualize the pathline and intensity of heat flow which is similar to streamlines. The heatlines are applicable to visualize and identify the heat flow from heat sources to heat sinks in the cavities. The heat filed within a twodimensional cavity for convective transport process was mathematically studied by Kimura and Bejan [9]. The heatlines are represented by heat functions (? ? ) which can be applied to plot the heatlines in the cavity and also obtained from the conductive heat fluxes

To make dimensionless form of heatfunctions??, the above-mentioned dimensionless variables are used:

This can be written in a single equation as follows:

It should be noted that the counter clockwise circulation is represented by positive sign of??, and the clockwise circulation is represented by negative sign of??.

IV.

The above-mentioned equations have been solved based on the finite volume approach and the SIMPLE algorithm has been employed. To discrete the convection terms, the second-order upwind approach is employed. Also, the central differencing scheme is used to discrete the diffusive terms. In order to perform the grid independency analysis, 5 different structured grids are used. The value of average Nusselt number for left wall with constant heat flux in ???? = 10 5 is evaluated and shown in Table .1. The grid distribution of 80×40 is selected for further simulations. Moreover, the numerical result obtained by Oztop and Abunada[10] is selected in order to verify the obtained results in the present investigation. It can be observed that there are close consistencies between obtained mean Nusselt number of present code and the previous one in a range of Rayleigh number (10 3 < ???? < 10 5 ). V.

In the natural convection phenomenon, the fluid flow depends on many different parameters which have influences on the flow structure in confined environments. Some these effective parameters are gravity acceleration, thermo-physical properties of operating fluid, thermal boundary condition, physical boundary condition, geometry of considered enclosure, external forces like magnetic force, etc. As a result, it can be concluded that to analyze the natural convection heat transfer, the influences of these parameters must be identified and considered in the investigation. In this study, the cavity is filled with two immiscible fluids, liquid and gas, and heated from side walls with different kind of temperature distribution. The temperature of the electrolyte stream at the adjacent of side walls enhances. As a result of this matter, the density of the electrolyte stream at these regions, sides of the cavity, reduces causing ascending the electrolyte stream forced by buoyancy force in presence of gravity acceleration. So, the electrolyte stream goes up along the side walls to reach the interface of liquid and gas. At this part, the electrolyte stream transfers its heat energy to the gas phase causing decreasing the temperature of the electrolyte stream. The density of electrolyte stream enhances again resulted by lower temperature, and the electrolyte stream descending from the collision point of two ascending electrolyte stream from sides at the surface of interface. This process occurs again and again which creating two main clockwise and counterclockwise circulations at the left and right sides of the cavity, respectively. The similar process occurs at the gas phase creating two main circulations.

In the engineering applications, it is necessary to find the different parameters to control the temperature distributions in the confined environment such as cavity. For example, in some industrial applications, it is desirable to hold the temperature distribution uniform throughout the enclosure which prevents creating very high temperature stream in some regions and using special strong alloys which is not cost effective. The temperature fields within the cavity for different linear and non-linear heat flux profiles of heated side walls and different aspect ratio of cavities are depicted in Fig. 5. It is obvious that the heat flux profiles at the side walls have considerable influence on the temperature field. The cold regions in case with heat flux profile of ???? = ???? 2 is wider with respect to other cases since the amount of heat flux offloaded at the above regions of the cavity is more than other cases, although the total amount of heat flux is constant for all cases.

The influences of heat flux profile of the cavities on the heatlines for both air phase and electrolyte phase at different Rayleigh numbers are presented graphically in Fig. 9. It can be observed that the heat flux profiles at side walls have no pronounced effects on the heatlines as a result of limited space of air phase. On the contrary, the influences of Rayleigh numbers on the heatlines are significant. As Rayleigh number enhances, the fluid flow becomes stronger as a consequence of increases temperature difference. As a result of this matter, the fluid flow will be able to transfer the heat energy in the fluid media. As Rayleigh number enhances, the heatline maps at the air phase have no considerable differences. It is due to the fact that the volume of air phase is limited. On the other hand, in the electrolyte phase, the fluid flow becomes stronger as Rayleigh number increase and the main circulations will be able to transfer the heat energy. As a result of this matter, it can be observed that the heat energy is entrapped in the circulations at high Rayleigh number. The values of skin friction coefficient with respect to dimensionless height of the cavity as a function of different profiles of heat flux on side heated walls are presented in Fig. 7. It should be noted that the skin friction coefficient is calculated for the electrolyte phase. The value of skin friction coefficient or the similar quantities such as wall shear stress are important in some engineering applications. For an instance of an industrial application needing to enhance the skin friction coefficient, in the lead-acid batteries, it is important to design the gap distance of the cells to increase the skin friction coefficient in order to omit the created bubbles at the surface of the anodes and cathodes.Because, the effective area of electrodes is reduce when the bubbles located on the surface of them. Then, the performance of battery will be reduced. The results show that the profiles of the heat fluxes at the side walls have pronounced influences of the distributions of the skin friction quantity. It should be noted that the distribution of the skin friction coefficient has direct relationship with the profile of the heat fluxes. It can be observed that the distribution of the skin friction coefficient for the case with linear heat flux profile (???? = ????????????????) is almost linear except at the corners which are caused by the secondary eddies and weak electrolyte stream at these regions. For ???? = ????, the distribution of the skin friction coefficient is linear with constant slope as same as the heat flux profile. For other case, ???? = ???? 0.5 and ???? = ???? 2 , the distribution of the skin friction coefficient and heat flux are similar as well. So, it can be concluded that the heat flux profile of the side walls is an effective approach to control the distribution of the skin friction coefficient.

The natural convection heat transfer in cavities with different profiles of heat flux and filled with two immiscible fluids of sulfuric acid-water (25-75%) and air has been studied. The heatline visualization approach has been utilized to detect the heat energy path within the cavity. The physical and thermal boundary conditions are as follows: temperature filed, Nusselt number and heatlines have been analyzed systematically. The results can be summarized as follows:

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