## 2 Mathematical Model  
Let us refer to the geometry sketched in Figure 1, that is, a channel with an embedded open cavity. The characteristic lengths of the geometry under consideration are evidenced in the figure (the geometry was built according to the size of the inlet channel H). In accordance with [PERSON]'s model [11], let us consider the nanofluid as a two-component mixture (base fluid plus nanoparticles) with the flowing assumption: (i) incompressible flow, (ii) no chemical reactions, (iii) negligible external forces, (iv) dilute the mixture, (v) negligible viscous dissipation, (vi) negligible radiative heat transfer, (vii) nanoparticles and base fluid locally in thermal equilibrium. Definwith [PERSON] and [PERSON] [18], the transport model for the nanofluid is described by the following system of equations:  
_Continuity equation_  
\[\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0 \tag{1}\]  
_Momentum equations in X and Y direction_  
\[\rho\left(U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}\right) =-\frac{\partial P}{\partial X}+\mu\left(\frac{\partial^{2}U}{ \partial X^{2}}+\frac{\partial^{2}U}{\partial Y^{2}}\right) \tag{2}\]  
\[\rho\left(U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}\right) =-\frac{\partial P}{\partial Y}+\mu\left(\frac{\partial^{2}V}{ \partial X^{2}}+\frac{\partial^{2}V}{\partial Y^{2}}\right) \tag{3}\]  
_Energy equation_  
\[U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}=\alpha\left( \frac{\partial^{2}T}{\partial X^{2}}+\frac{\partial^{2}T}{\partial Y^{2}} \right)+\delta\left\{D_{B}\left(\frac{\partial\phi}{\partial X}\frac{ \partial T}{\partial X}+\frac{\partial\phi}{\partial Y}\frac{\partial T}{ \partial Y}\right)+\frac{D_{T}}{T_{c}}\left[\left(\frac{\partial T}{\partial X }\right)^{2}+\left(\frac{\partial T}{\partial Y}\right)^{2}\right]\right\} \tag{4}\]  
_Nanoparticles equation_  
\[U\frac{\partial\phi}{\partial X}+V\frac{\partial\phi}{\partial Y}=D_{B}\left( \frac{\partial^{2}\phi}{\partial X^{2}}+\frac{\partial^{2}\phi}{\partial Y^{2 }}\right)+\frac{D_{T}}{T_{c}}\left(\frac{\partial^{2}T}{\partial X^{2}}+\frac {\partial^{2}T}{\partial Y^{2}}\right) \tag{5}\]  
In Equations (1)-(5), (\(U\), \(V\)) are the X and Y-component of the nanofluid velocity vector, \(P\) is the nanofluid pressure, \(T\) is the nanofluid temperature, \(\phi\) is the nanoparticles concentration. These dependent variables are given as a function of the position vector (\(X\), \(Y\)). Furthermore, \(\rho\) is the nanofluid density, \(\mu\) is the dynamic viscosity, \(T_{c}\) is the cold temperature, \(\alpha=k/(\rho\ C_{p})\) is the thermal diffusivity (\(k\) is the thermal conductivity of the fluid and \(C_{p}\) is the fluid-specific heat constant), \(\delta=(\rho\ C_{p})_{p}/(\rho\ C_{p})_{f}\) (\(p\) and \(f\) denotes nanoparticle  
Figure 1: Domain and boundary conditions. and the base fluid). Let us introduce the Brownian motion coefficient \(D_{B}\), defined by the Einstein-Stokes equation, and the thermophoresis coefficient \(D_{T}\), that is:  
\[D_{B}=\frac{k_{B}\ T}{3\ \pi\ \mu\ d_{p}} \tag{6}\]  
\[D_{T}=0.26\frac{k}{2k+k_{p}}\frac{\mu}{\rho}\phi \tag{7}\]  
where \(k_{B}\) is Boltzmann's constant equal to \(1.38064852\times 10^{-23}\ \text{(m}^{2}\ \text{kg)}/\text{(s}^{2}\ \text{K)}\), \(d_{p}\) is the nanoparticles diameter, \(k_{p}\) is the particle thermal conductivity. For the thermal energy Equation (4), non-linearities take place with both Brownian diffusion and thermophoretic diffusion.