## 3 Mathematical Modeling  
### Physical Hydrodynamic Problem  
The numerical investigation of nanofluid flow and heat transfer is governed by Navier-Stokes equations (momentum conservation) supported by the continuity equation (mass conservation) and the energy conservation equation [15; 16; 17]. Some assumptions are considered when elaborating the mathematical modeling of the investigated nanofluid hydrodynamic problem:  
* The nanofluid is considered Newtonian, incompressible and in thermal equilibrium.
* The nanofluid flow is considered 2D, steady, and laminar.
* The Boussinesq approximation is adopted for the density variation in the buoyancy-driven flow.  
**Continuity equation** \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\] (1)  
**x-Momentum** \[\begin{array}{l}\rho_{nf}u\frac{\partial u}{\partial x}+\rho_{nf}v\frac{ \partial u}{\partial y}=\\ \end{array}\begin{array}{l}-\frac{\partial p}{\partial x}+\frac{\partial}{ \partial x}\left(\mu_{nf}\frac{\partial u}{\partial x}\right)+\frac{\partial} {\partial y}\left(\mu_{nf}\frac{\partial u}{\partial y}\right)\\ \end{array}\] (2)  
**y-Momentum** \[\begin{array}{l}\rho_{nf}u\frac{\partial u}{\partial x}+\rho_{nf}v\frac{ \partial u}{\partial y}=\\ \end{array}\begin{array}{l}-\frac{\partial p}{\partial x}+\frac{\partial}{ \partial x}\left(\mu_{nf}\frac{\partial u}{\partial x}\right)+\frac{\partial} {\partial y}\left(\mu_{nf}\frac{\partial u}{\partial y}\right)\\ \end{array}\] (3)  
**Energy conservation equation** \[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha_{nf} \bigg{(}\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^ {2}}\bigg{)}\] (4)  
The effective nanofluid mechanical properties are determined based on the mechanical properties of base fluid and nanoparticles as follows:  
\[\rho_{nf}=(1-\phi)\rho_{f}+\phi\rho_{p} \tag{5}\]  
\[\left(\rho C_{p}\right)_{nf}=(1-\phi)\left(\rho C_{p}\right)_{f}+\phi\left( \rho C_{p}\right)_{p} \tag{6}\]  
\[\left(\rho\beta\right)_{nf}=(1-\phi)\left(\rho\beta\right)_{f}+\phi\left(\rho \beta\right)_{p} \tag{7}\]  
\[\alpha_{nf}=\frac{k_{nf}}{\left(\rho C_{p}\right)_{nf}} \tag{8}\]  
\[\mu_{nf}=\frac{\mu_{f}}{\left(1-\phi\right)^{\frac{5}{2}}} \tag{9}\]  
\[k_{nf}=k_{f}\frac{k_{p}+2k_{f}+2\phi\left(k_{p}-k_{f}\right)}{k_{p}+2k_{f}+ \phi\left(k_{f}-k_{p}\right)} \tag{10}\]  
**Boundary conditions**  
The boundary conditions, applied on the investigated nanofluid, are set as follows:  
* On the top wall (absorber plate): \[\left|\begin{array}{c}u(x,h)\ =\ v(x,h)\ =\ 0\\ T(x,h)\ =\ T_{h}\end{array}\right.\]
* On the bottom wall: \[\left|\begin{array}{c}u(x,0)\ =\ v(x,0)\ =\ 0\\ \frac{\partial T}{\partial y}\Big{|}_{y=0}\ =\ 0\end{array}\right.\]
* On the left wall: \[\left|\begin{array}{c}u(0,y)\ =\ v(0,y)\ =\ 0\\ T(0,y)\ =\ T_{c};\frac{h}{4}\leq y\leq\frac{3h}{4}\\ \frac{\partial T}{\partial x}\Big{|}_{x=0}\ =\ 0;y<\frac{h}{4}\ \text{and}\ y> \frac{3h}{4}\end{array}\right.\]
* On the right wall: \[\left|\begin{array}{c}u(l,y)\ =\ v(l,y)\ =\ 0\\ T(l,y)\ =\ T_{c};\frac{h}{4}\leq y\ \leq\ \frac{3h}{4}\\ \frac{\partial T}{\partial x}\Big{|}_{x=l}\ =\ 0;y<\frac{h}{4}\ \text{and}\ y> \ \frac{3h}{4}\end{array}\right.\]