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ABSTRACT
A two-phase model based on the double-diffusive approach is used to perform a numerical study on the natural convection of water-based nanofluids in differentially heated square cavities, inclined with respect to gravity so that the heated wall is positioned below the cooled wall, assuming that Brownian diffusion and thermophoresis are the only slip mechanisms by which the solid phase can develop a significant relative velocity with respect to the liquid phase. The system of the governing equations of continuity, momentum, and energy for the nanofluid, and continuity for the nanoparticles, is solved through a computational code that incorporates three empirical correlations for the evaluation of the effective thermal conductivity, the effective dynamic viscosity, and the thermophoretic diffusion coefficient, all based on several sets of literature experimental data. Pressure–velocity coupling is handled by the way of the SIMPLE-C algorithm. Numerical simulations are executed for tilting angles in the range 50−70 deg, such that the nonuniform distribution of the suspended solid phase gives rise to a nonnegligible solutal buoyancy force, whose effects are investigated using the nanoparticle diameter and average volume fraction, the cavity width, the nanofluid average temperature, and the temperature difference imposed across the cavity, as independent variables. It is found that the competition between the solutal and thermal buoyancy forces results in an oscillatory flow, with an oscillation amplitude that increases on increasing the cavity size and the imposed temperature difference. Moreover, the impact of the nanoparticle dispersion into the base liquid is found to be higher at higher average temperatures, whereas, by contrast, the other variables have moderate or negligible effects.
Nomenclature
| c | = | specific heat at constant pressure, J/(kg K) |
| DB | = | Brownian diffusion coefficient, m2/s |
| DT | = | thermophoretic diffusion coefficient, m2/s |
| dp | = | nanoparticle diameter, m |
| g | = | gravity vector, m/s2 |
| I | = | unit tensor |
| Jp | = | nanoparticle diffusion mass flux, kg/(m2 s) |
| k | = | thermal conductivity, W/(m K) |
| kB | = | Boltzmann constant = 1.38066 · 10−23 J K−1 |
| m | = | nanoparticle mass fraction |
| Nu | = | Nusselt number |
| p | = | pressure, Pa |
| Pr | = | Prandtl number |
| Q | = | heat transfer rate, W |
| q | = | heat flux, W/m2 |
| Ra | = | Rayleigh number |
| ST | = | thermophoresis parameter |
| T | = | temperature, K |
| t | = | time, s |
| U | = | x-wise velocity component, m/s |
| V | = | velocity vector, m/s |
| V | = | y-wise velocity component, m/s |
| W | = | width of the enclosure, m |
| x,y | = | Cartesian coordinates, m |
| Greek symbols | = | |
| φ | = | nanoparticle volume fraction |
| μ | = | dynamic viscosity, kg/(m s) |
| θ | = | tilting angle of the enclosure, deg |
| ρ | = | mass density, kg/m3 |
| τ | = | stress tensor, kg/(m s2) |
| Ω | = | period of oscillation, s |
| Subscripts | = | |
| av | = | average |
| c | = | cooled wall, at the temperature of the cooled wall |
| f | = | base fluid |
| h | = | heated wall, at the temperature of the heated wall |
| max | = | maximum value |
| min | = | minimum value |
| n | = | nanofluid |
| s | = | solid phase |
Nomenclature
| c | = | specific heat at constant pressure, J/(kg K) |
| DB | = | Brownian diffusion coefficient, m2/s |
| DT | = | thermophoretic diffusion coefficient, m2/s |
| dp | = | nanoparticle diameter, m |
| g | = | gravity vector, m/s2 |
| I | = | unit tensor |
| Jp | = | nanoparticle diffusion mass flux, kg/(m2 s) |
| k | = | thermal conductivity, W/(m K) |
| kB | = | Boltzmann constant = 1.38066 · 10−23 J K−1 |
| m | = | nanoparticle mass fraction |
| Nu | = | Nusselt number |
| p | = | pressure, Pa |
| Pr | = | Prandtl number |
| Q | = | heat transfer rate, W |
| q | = | heat flux, W/m2 |
| Ra | = | Rayleigh number |
| ST | = | thermophoresis parameter |
| T | = | temperature, K |
| t | = | time, s |
| U | = | x-wise velocity component, m/s |
| V | = | velocity vector, m/s |
| V | = | y-wise velocity component, m/s |
| W | = | width of the enclosure, m |
| x,y | = | Cartesian coordinates, m |
| Greek symbols | = | |
| φ | = | nanoparticle volume fraction |
| μ | = | dynamic viscosity, kg/(m s) |
| θ | = | tilting angle of the enclosure, deg |
| ρ | = | mass density, kg/m3 |
| τ | = | stress tensor, kg/(m s2) |
| Ω | = | period of oscillation, s |
| Subscripts | = | |
| av | = | average |
| c | = | cooled wall, at the temperature of the cooled wall |
| f | = | base fluid |
| h | = | heated wall, at the temperature of the heated wall |
| max | = | maximum value |
| min | = | minimum value |
| n | = | nanofluid |
| s | = | solid phase |