A two-phase model based on the double-diffusive approach is used to perform a numerical study of natural convection of alumina-water nanofluids in differentially heated vertical slender cavities. In the mathematical formulation, Brownian diffusion and thermophoresis are assumed to be 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 relying on the SIMPLE-C algorithm for the pressure-velocity coupling. The effective thermal conductivity and dynamic viscosity of the nanofluid, and the coefficient of thermophoretic diffusion of the suspended solid phase, are evaluated using three empirical correlations based on a high number of experimental data available from diverse sources, and validated by way of literature data different from those used in generating them. Numerical simulations are executed for different height-to-width aspect ratios of the enclosure, as well as different average temperatures of the nanofluid. The heat transfer performance of the nanoparticle suspension relative to that of the base fluid is found to increase as the nanofluid average temperature is increased and, at low to moderate temperatures, the aspect ratio of the enclosure is decreased. Moreover, at temperatures higher than room temperature, a peak at an optimal particle loading is found to exist for any investigated configuration.