A macroscopic and mesoscopic model of Newtonian and non-Newtonian nanofluids with a two-energy equation method

We present an updated comprehensive macroscopic model of nanofluids, considering a revisited local thermal non-equilibrium (LTNE) condition to study the temperature difference between carrier fluid and nanoparticles. A new relation for thermal conductivity of solid and liquid phases in the LTNE condition is introduced which considers the possible particle aggregation. This model is thermodynamically consistent and covers the non-Newtonian models of nanofluids, including power-law and viscoplastic ones. A mesoscopic scheme based on the lattice Boltzmann method (LBM) which satisfies the presented macroscopic equations is introduced and derived. This investigation is a further development of our recent studies[G. H. R. Kefayati and A. Bassom, “A lattice Boltzmann method for single and two phase models of nanofluids: Newtonian and non-Newtonian nanofluids,” Phys. Fluids 33, 102008 (2021); G. H. R. Kefayati, “A two- and three-dimensional mesoscopic method for an updated non-homogeneous model of Newtonian and non-Newtonian nanofluids,” Phys. Fluids 34, 032003 (2022).] for simulating and analyzing nanofluids by a two-phase model. To assess the present numerical method, it is studied for a benchmark problem of natural convection in a cavity. The dimensional and non-dimensional macroscopic equations for the mentioned benchmark are defined and the implemented non-dimensional relations of LBM are shown. The present approach is verified with the obtained results of the mixture approach and a previous two-phase model, which demonstrated the accuracy of the presented method. The results including the temperature distributions of the solid and fluid phases, the nanoparticles distributions, and fluid flow behavior as well as the yielded/unyielded sections for the viscoplastic nanofluids are shown and discussed for the defined non-dimensional parameters. It was also demonstrated that the previous proposed thermal conductivity model of nanofluids in the LTNE approach generates a significantly different value compared to experimental results, and the current suggested model produces reliable results to the experimental ones.