## 1 Introduction
They show that the performance of this system is around 69% relative to the nano fluid evaporation and the transient phase change depends on the nanoparticles surface chemistry. In addition, they use the Fourier law to model the thermal attitude of the nano fluid at microscale. In the same field, [PERSON] et al. [23] used gold nanoparticles to generate steam under solar heating. They demonstrate that a cooled bulk fluid causes an important surface of evaporation and they evaluate the photothermal performance by integrating a new method. They demonstrate, experimentally, that a 0.04% of gold in volume increases the evaporation rate by 95%. [PERSON] et al. [24] analyzed the absorption and desorption of molecules of fluid in a porous media formed by nanoparticles. Their numerical results are consistent with experimental data and prove that we can modify the properties of the energy storage by exothermic and endothermic adsorption and desorption. [PERSON] [25] studied the influence of a constant magnetic field in the heat and mass transfer through a micro channel in presence of cu-water nano fluid. The Brinkman model allowed him to investigate the problem and to transform the nonlinear equations to the simple linear system. [PERSON] et al. [26] gave a theoretical study of the convective transfer of heat taking into account the pressure drop. They consider the effect of the nanoparticle volume rate by using the Buongiorno model. [PERSON] [27] studied the influence of the (MHD) field on the developed forced convection of heat with nano fluid in pipes. [PERSON] and [PERSON] [28] looked mathematically for the effect of an external magnetic field on the nanoparticles motion. Their study was during a mixed gas convection inside micro-channels. They consider the effect of the nanoparticle volume rate by using the Buongiorno model. The same authors [29] related the effect of the nano particles migration on the heat exchangers efficiency. With changing the migration modes of the Al\({}_{2}\)O\({}_{3}\) inside water and under the mixed convection of gas in vertical micro channels, they demonstrate the influence of the thermophysical variations on the heat transfer. [PERSON] et al. [30] present a study of magnetic field direction on transfers. They vary this direction in the case of the (MHD) nano fluids in isothermal micro tubes. This (MHD) phenomena was investigated when the nano fluid is electrically conductive in micro annuli [31]. [PERSON] et al. [32] analysis the entropy relative a Maxwell nano fluid by modifying the heat and mass transfer models. Also, the generalize the law of Fick. [PERSON] et al. [33] showed that position and time affect the magnetic field influence on the temperature of nano fluid. [PERSON] et al. [34] studied the unsteady peristaltic motion of a non-Newtonian nano fluid. They show the influence of a magnetic field and the currents of Hall. In addition, they take into the account the chemical reaction. Regarding the previous studies, no study has given interest to the behavior of nano fluids in a semi-confined area like vertical channel. It will be the objective of this present work when no external heat is provided (heated flown plate case).