## 2 Mathematical Model  
In this study, mixed convection heat transfer has been studied in a channel filled with a porous medium. The related geometry, boundary conditions, and coordinates are shown in Figure 1. This figure shows the coordinates and boundary conditions. The length of the bottom wall was \(L\), and the height of the cavity was \(H=L/10\). The rest of the bottom surface is adiabatic, while one third of the bottom of the channel was heated at a constant temperature. The top wall was kept at a constant cold temperature. The continuity, momentum, and energy equations that have become dimensionless must be solved simultaneously in order to solve problem of fluid motion and heat transfer in enclosed volumes. One can see that these are nonlinear partially differential equations, and that is impossible to solve them analytically. In order to solve obtained algebraic equations, the _Successive Under Relaxation_ (SUR) method of the iterative methods had been used. Given that the convergence criterion, \(10^{-5}\) had been selected for all of the dependent variables, the value of 0.1 is taken for under-relaxation parameter for all parameters. A 301 \(\times\) 31 equal pitched grid distribution has been used in the \(x\) and \(y\) directions. The dimensional governing equations for continuity, momentum and energy can be written as follow:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]  
Figure 1: Schematic configuration with coordinates and boundary conditions. \[\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}=-\frac{g\beta K}{\gamma} \frac{\partial T}{\partial x} \tag{2}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha\left( \frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}\right) \tag{3}\]  
In order to write Equations (1)-(3) in dimensionless form, the following dimensionless statements have been used;  
\(X=\frac{\pi}{L}\), \(Y=\frac{y}{L}\), \(U=\frac{\mu}{V_{0}}=\frac{\partial\Psi}{\partial Y}\), \(V=\frac{v}{V_{0}}=-\frac{\partial\Psi}{\partial X}\)  
\(\theta=\frac{T-T_{c}}{T_{h}-T_{c}}=\frac{g\beta K(T_{h}-T_{c})L}{\gamma a}\), \(Pe=\frac{\gamma_{0}L}{\alpha}\)  
which thus lead to the following dimensionless form of the governing equations used for the model;  
\[\frac{\partial^{2}\Psi}{\partial Y^{2}}+\frac{\partial^{2}\Psi}{\partial X^{ 2}}=-\frac{Ra}{Pe}\frac{\partial\theta}{\partial X} \tag{5}\]  
\[Pe\left[\frac{\partial\Psi}{\partial Y}\frac{\partial\theta}{\partial X}- \frac{\partial\Psi}{\partial X}\frac{\partial\theta}{\partial Y}\right]= \frac{\partial^{2}\theta}{\partial X^{2}}+\frac{\partial^{2}\theta}{\partial Y ^{2}} \tag{6}\]  
The finite difference method had been used for the solution to these equations. In the study, we can apply the boundary conditions for the solid wall and fluid interface as follows:  
\[(X,1)=0.1,\;\theta(X,1)=0 \tag{7}\]  
\[\Psi(0,Y)=Y,\;\theta(0,Y)=0 \tag{8}\]  
\[\frac{\partial\Psi(L,Y)}{\partial X}=0,\;\frac{\partial\theta(L,Y)}{\partial X }=0 \tag{9}\]  
Along the part where there is a heat source;  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]0  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]1  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]2  
The local and average \(Nu\) numbers can be given as follows:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]3  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]4  
Calculations have been carried out for various Peclet and Rayleigh numbers. The influence of the Peclet and Rayleigh numbers on the temperature distribution in the rigid substance has been previously pointed out. When making the necessary observations, the heat source had slid equally to both the up-stream and down-stream in a 1/3 ratio. For different Peclet and Rayleigh numbers, the flow and temperature profiles have been given in the case that the heat source is close to the down-stream, at the center, and close to the up-stream.