Transient, laminar natural convection in a trapezoidal cavity filled with a non-Newtonian nanofluid with sinusoidal boundary conditions on both sidewalls is studied numerically by using the finite element method. The sloping walls of the cavity are heated by sinusoidal temperature distributions, while the horizontal walls allow no heat transfer to the surrounding. Water-based nanofluids with Ag or Cu or Al2O3 or TiO2 nanoparticles are chosen for investigation. The governing parameters of this study are the Rayleigh number (10(4) <= Ra <= 10(6)), phase deviation (0 <= gamma <= pi), amplitude ratio (0 <= epsilon <= 1), power-law index (0.6 <= n <= 1.4), sidewall inclination angle (0 degrees <= phi <= 21.8 degrees), nanoparticle volume fraction (0 <= phi <= 0.2), and dimensionless time (0 <= tau <= 0.2). The results show that the heat transfer rate increases significantly by the addition of phase deviation. Strong heat transfer enhancements are obtained by higher sidewall inclination angles. However, for a square cavity, the heat transfer approaches the steady-state condition with the increment of the dimensionless time. (C) 2016 Elsevier B.V. All rights reserved.