## 1 Introduction  
Heat transfer is one of the most important processes in many industrial and heating-cooling applications, such as microelectronics, transportation, manufacturing, metrology, defense, and energy supply industries [1, 2]. However, the inherent low thermal conductivity of conventional fluids, such as water, oils, and ethylene glycol, is a primary limitation in developing efficient heat transfer systems. The [PERSON]'s theory [3] showed that an enhancement of the thermal conductivity may be achieved by dispersing millimeter or micrometer-sized solid particles into a base fluid. However one major drawback associated with the use of such large size particles is their rapid settling, which may result into a complete separation of the two phases along with the clogging of heat exchangers due to the sedimentation of the solid aggregates formed by the large size particles. This type of solid-fluid suspensions requires also the addition of a large number of particles resulting in significantly greater pressure drop, hence increased pumping power, corrosion of the walls and a noticeable increase in the wall shear stress. Thus, [PERSON] and [PERSON] [4] suggested a novel approach to enhance heat transfer processes in industrial applications by exploiting the properties of nanoparticles and their dispersion in a host fluid. These metallic or non-metallic nanoparticles have an equivalent diameter \(d_{np}\) lower than 100 nm. As opposed to milli- or microsized suspensions, very stable suspensions may be achieved by introducing nanoparticles. Moreover, nanoparticles benefit from a \(10^{3}\) times larger surface/volume ratio than that of microparticles and exhibit much higher thermal conductivity than that of base fluids. For examples, the thermal conductivities of copper or alumina at room temperature are about 670 and 70 times greater than that of water, respectively [5]. On the contrary, it leads most of the time to a decrease in the heat capacity [6; 7] and an increase in the dynamic viscosity of the mixture [6]. A compromise must be then found between the increase in thermal conductivity without loosing too much heat storage capacity and consuming too much power for pumping. If well stabilized, nanofluids represent nowadays a major technological and economical challenge and should offer very interesting perspectives for any heat transfer process. The exponential increase in the number of publications about nanofluids [8] prevents from making an exhaustive state-of-the-art review on the topic. Many authors concentrated on measuring the thermophysical properties of various nanofluids showing that their properties depend on a large number of parameters such as the type of nanoparticle, their size, their mass or volume fraction, the type and the concentration of the surfactant, the pH of the mixture, the Brownian motion and the thickness of the interfacial nanolayer among other parameters (see in [6; 7; 8; 9; 10]). Others developed experimental set-ups to measure the convective heat transfer and temperature profiles in pipes [11; 12], coaxial [13] or plate [14] heat exchangers among other geometries. Most authors focused on measuring global thermal quantities due to the difficulty to measure velocity and temperature profiles in such insulated systems. It has relatively slowed down the development of accurate models dedicated to nanofluid flows, especially regarding the agglomeration and sedimentation processes. Only a few in-house solvers have been developed to investigate convective nanofluid flows. Most of them assume the flow as being a single-phase flow with constant or variable nanofluid properties in canonical configurations. For example, [PERSON] et al. [15] numerically investigated the entropy generation and the mixed convection heat transfer of copper/water-based nanofluids in an inclined open cavity with uniform heat flux at the wall.