## 3 Results and Discussion  
Mixed convection heat transfer has been numerically studied in a channel filled with a porous medium, and whose thickness of the upper and bottom walls are negligible. The results obtained from the study are given in terms of streamlines, isotherms, and local and average Nusselt numbers. The calculations have been carried out for different Rayleigh and Peclet numbers. The Rayleigh and Peclet numbers have been found to be effective parameters on flow and heat transfer. In Figure 2, the heat source close to the down-stream, the streamline (above), and the isotherms (below) that emerge in the enclosed volume at different values of the Rayleigh number for \(Pe=0.5\) are seen. In Figure 2a, for \(Ra=100\), there are several centers of rotation seen at the top of the heater. Because the Rayleigh number is low, the temperature curves are distributed properly. Flow field is symmetric and single cell is formed in the heat source. Right cells rotate in clockwise direction with \(\Psi_{\rm min}=-3.26\). In other words, there is a weak clockwise circulation due to domination of conduction. The other small cell sits at the left of the heat source with \(\Psi_{\rm max}=3.56\). In Figure 2b, \(Ra=250\) increases the number of rotation centers, thus causing the temperature curves to fluctuate. The values of \(\Psi_{\rm min}=-8.58\) and \(\Psi_{\rm max}=9.62\). With the increase of the Rayleigh number in Figure 2c, the heated fluid on the lower surface diffuses into the closed volume, and hence \begin{table}
\begin{tabular}{c c c c} \hline \hline _Pe_ & 1 & 5 & 10 \\ \hline
**[PERSON] and [PERSON] [10]** & 5.7 & 5.8 & 5.85 \\
**[PERSON] and [PERSON] [20]** & 4.1 & 4.5 & 5.8 \\
**Present Study** & 4.23 & 4.27 & 4.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison of average \(Nu\) number for \(Ra=100\)[3]. Figure 2: Streamlines (above) and isotherms (below) profiles for different \(Ra\) numbers and \(Pe=0.5\), (\(c=0.33\)), (**a**) \(Ra=100\), (**b**) \(Ra=250\), (**c**) \(Ra=500\), (**d**) \(Ra=1000\). penetrates further. As is seen from the relevant temperature curves, the heat transfer through the transmission is reduced, while the heat transfer through the convection starts to dominate. This situation is more apparent in Figure 2d. The increase in the Rayleigh number intensifies the flow, increases the center of rotation within the volume, and causes the temperature curves to further penetrate the edges of the surface. For higher Rayleigh numbers, Figure 2c,d, the convection is vigorous with \(\Psi_{\min}=-12.9\), \(\Psi_{\max}=15.2\) and \(\Psi_{\min}=-21.7\), \(\Psi_{\max}=23.5\), respectively. However, the absolute values of minimum and maximum flow values increase. In Figure 3, the flow-temperature profiles have been given depending on the varying Rayleigh numbers for \(Pe=1\) in the case that the heat source is close to the down-stream. For small values of Peclet number (=0.5), an oscillation in the temperature and flow fields is observed for a short time, after which a steady-state is reached as shown in Figure 2. For \(Pe=1\), an oscillation in the temperature and flow fields is observed, and in this value of the Peclet number there will be no steady-state flow but periodic convection state, as shown in Figure 3. As is shown in Figure 2, the flow is concentrated around the heat source, whereby numerous centers of rotation have formed for all of the Rayleigh numbers; in addition, the minimum and maximum flow function values are also given. Two or more centers of rotation have been acquired among all of the Rayleigh numbers. Some of the cells formed on the heaters rotate clockwise, whereas others rotate counterclockwise. The values of stream function are close to each other as \(\Psi_{\min}=-1.56\), \(\Psi_{\min}=-4.73\), \(\Psi_{\min}=-7.08\) and \(\Psi_{\min}=-10.4\), on the Figure 3a,d. As shown from the figure, values of \(\Psi_{\max}=1.85\), \(\Psi_{\max}=5.07\), \(\Psi_{\max}=8.07\) and \(\Psi_{\max}=12.1\).