## 3 Results and Discussion
Since conductive heat transfer is effective, as is observed in the isotherms, it appears that there is no extreme change in the lower Rayleigh numbers and, moreover, there are isotherms distributing in fungi-shaped on the horizontal heater of the high Rayleigh numbers. Figure 3: Streamlines (above) and isotherms (below) profiles for different \(Ra\) numbers and \(Pe=1\), (\(c=0.33\)), (**a**) \(Ra=100\), (**b**) \(Ra=250\), (**c**) \(Ra=500\), (**d**) \(Ra=1000\). In Figures 4 and 5, the flow-temperature profiles are shown depending upon the different numbers of Peclet and Rayleigh numbers in the cases where the heat source is close to the up-stream. Similar to the other positions of the heat source, the flow is concentrated around the heat source, and numerous centers of rotation have formed for all Rayleigh numbers; the minimum and maximum flow function values are also given. It can be seen from Figure 4 that \(\Psi_{\max}=3.5\), \(\Psi_{\min}=-3.2\) (Figure 4a), \(\Psi_{\max}=9.6\), \(\Psi_{\min}=-8.5\) (Figure 4b), \(\Psi_{\max}=15.2\), \(\Psi_{\min}=-12.9\) (Figure 4c), \(\Psi_{\max}=23.5\), \(\Psi_{\min}=-21.3\) (Figure 4d). Depending on the increase in the Rayleigh number, a temperature distribution in the form of a flame, especially on the heater, is acquired. In cases where the Rayleigh number is low, the fungi-shaped temperature distribution also appears to be present. The flame-shaped temperature refers to a higher convection heat transfer. It can be seen from this Figure 5 that \(\Psi_{\max}=1.84\) when \(Ra=100\) (Figure 5a) and it increases to \(\Psi_{\max}=5.07\) when Rayleigh is increased to 250 (Figure 5b). Increasing the Rayleigh number from 500 to 1000 leads to further increase in \(\Psi_{\max}\) from 8.06 to 12.1 in Figure 5. The values of stream function are found as: \(\Psi_{\min}=-1.54\), \(\Psi_{\min}=-4.72\), \(\Psi_{\min}=-7.04\) and \(\Psi_{\min}=-10.1\), and shown in Figure 5a,d. Figure 4: Streamlines (above) and isotherms (below) profiles for different \(Ra\) numbers and \(Pe=0.5\), (\(c=0.66\)), (**a**) \(Ra=100\), (**b**) \(Ra=250\), (**c**) \(Ra=500\), (**d**) \(Ra=1000\). In Figure 6, it can be seen that the change of the local Nusselt number is close to the down-stream, depending on the different Peclet and Rayleigh numbers. The local Nusselt numbers for all parameters represent a sinusoidal distribution. The location of the points at the highest value in the sinusoidal change can be explained by correlating it in relation to the change of flow lines in the flow and temperature profiles. The points which represent the center of rotation is influenced by the flow. Similarly the highest value of local Nusselt number is obtained at the point which is the center of rotation. The symmetrical behavior within the closed volume is disrupted as the Rayleigh number increases. In this case, the flow penetrates more into the closed volume, and the flow thus becomes activated. The increase in the Rayleigh number lets the local Nusselt number increase as well. Figure 5: Streamlines (above) and isotherms (below) profiles for different \(Ra\) numbers and \(Pe=1\), (\(c=0.66\)), (**a**) \(Ra=100\), (**b**) \(Ra=250\), (**c**) \(Ra=500\), (**d**) \(Ra=1000\). Figure 7 shows the change of the local Nusselt number depending on the different numbers of Peclet and Rayleigh, in the case that the heat source is close to the up-stream. The local Nusselt numbers for all of the parameters represent a sinusoidal distribution parallel to the figures given above. What is more, the location of the points at the highest value in the sinusoidal change can be explained by correlating a relation with the change of streamlines in the flow and temperature profiles. Those points with a center of rotation are influenced by the flow; and therefore the highest value in the local Nusselt number is also obtained where the center of rotation is found. Figure 6: Variation of local Nusselt number along the channel (\(c=0.33\)), (**a**) \(Pe=0.5\), (**b**) \(Pe=1\), (**c**) \(Pe=5\).