## 1 Introduction  
The term \"nanofluid\" was first coined by [PERSON] and [PERSON] [1; 2] in 1995. Originally, the term described nanometer-sized copper particles dispersed in water to improve their thermal conductivity. Today, a nanofluid can be more accurately intended as an engineered colloidal suspension of nanoparticles into a base fluid. Nanofluids have shown a promising future as thermal fluids for various heat transfer applications [3]: in solar collectors [4; 5], to enhance the heat-transfer coefficient of solar water heaters or to improve the capacity of thermal energy storage systems; and in the area of refrigeration, to enhance the performance of refrigeration systems. Later, many authors (see for example refs. [6; 7; 8; 9; 10]) studied the nanofluid flow and discussed their heat transfer performance. In the literature, a widely employed approach to describe the features of a nanofluid is the so-called Buongiorno model, which relies on the assumptions: (i) the flow is incompressible, (ii) there are no chemical reactions at nanofluids, (iii) external forces are negligible, (iv) the mixture is considered to be diluted, (v) viscous dissipation is negligible, (vi) radiation heat transfer is negligible, (vii) nanoparticles and base fluid are located in thermal equilibrium. More specifically, [PERSON] presented the following seven mechanisms of slip between nanoparticle and fluid [11]: Inertia, Brownian diffusion, Thermophoresis, Diffusiophoresis, Magnus effect, Fluid Drainage, and Gravity settling. It is worth mentioning that the above-mentioned phenomena can produce a relative velocity between the base fluid and nanoparticles, that is, water. He also suggested a two-component four-equation nonhomogeneous equilibrium model for three basic equations: mass, momentum, and heat transfer. Furthermore, a comparison of length scales and the nanoparticle turbulent eddy time demonstrated that, in the presence of turbulent eddies, the nanoparticle moves with the base fluid homogeneously. As a consequence, the turbulence intensity effect is also double. The presence of turbulence in the flow, which prevents the agglomeration of particles, decreases the discrepancy between experimental and numerical results. Hence, the results obtained from an analysis of [PERSON]'s two-phase model are likely to be more accurate. In this context, [PERSON] et al. [12]recount the effects of the rib shape with constant height, width and pitch, and nanoparticle diameter on natural convection heat transfer of low turbulence flow of AL2O3-water nanofluid inside a square cavity based on [PERSON]'s model. The steady Cu-water nanofluid flow in the presence of a magnetic field has been numerically investigated by [PERSON] et al. [13]under the effects of mixed convection, thermal radiation, and chemical reaction. Besides this, the modeled nanofluid problem is characterized by a single-phase nanofluid model (i.e., [PERSON]'s model) and a two-phase mixture nanofluid model (i.e., [PERSON]'s model) together with the Cattaneo-Christov Double diffusion model. In addition, according to [PERSON]'s model, many papers have examined the convection of a nanofluid in a cavity or in a channel with an embedded cavity. In this regard, the forced convection heat transfer of a nanofluid in a channel has been widely investigated in refs. [14; 15], while the mixed convection of a nanofluid (with various types of nanoparticles) in an open cavity has been pointed out by [PERSON] et al. in ref. [16]. The single phase model has been employed to investigate the Couette forced convection of a nanofluid in ref. [17]: the governing PDEs are reduced to ODEs using a similarity transformation, and then analytically solved. The recent literature has paid attention, especially, to mixed and free convention, or, concerning forced convection, to the presence of MHD effects, as in refs. [18; 19; 20]. On the contrary, recently, the analysis of vortex formation in the case of laminar flow of a Newtonian fluid, flowing in a duct with an embedded cavity has been studied in ref.