The proper process of applying heat to many technological devices is a significant challenge. There are many nanofluids of different sizes used inside the system. The current study combines this potential to improve convection effects, considering numerical simulations of natural convection using Cu/water nanofluids in a square enclosure with bottom blocks embedded in baffles. The enclosure consists of two vertical walls with isothermal boundary conditions; the left wall is the sinusoidal heat source, whereas the right wall is cooled. The investigations dealt with the influences of nanoparticle concentration, Rayleigh number, baffle length, and thermal conductivity ratioon isotherms, stream functions, and average Nusselt number. The results present that, when the Rayleigh number rises, the fluid flow velocity increases, and the heat transfer improves. Furthermore, the baffle length case (L_b = 0.3) provides higher heat transfer characteristics than other baffle height cases.

The time-fractional derivative based on the Grunwald–Letnikove derivative of the 2D-ISPH method is applying to emulate the dual rotation on MHD natural convection in a hexagonal-shaped cavity suspended by nano-encapsulated phase change material (NEPCM). The dual rotation is performed between the inner fin and outer hexagonal-shaped cavity. The impacts of a fractional time derivative $$\alpha$$α $$\left( {0.92 \le \alpha \le 1} \right)$$0.92≤α≤1 , Hartmann number Ha $$\left( {0 \le Ha \le 80} \right)$$0≤Ha≤80 , fin length $$\left( {0.2 \le L_{Fin} \le 1} \right)$$0.2≤LFin≤1 , Darcy parameter Da $$\left( {10^{ - 2} \le Da \le 10^{ - 4} } \right)$$10-2≤Da≤10-4 , Rayleigh number Ra $$\left( {10^{3} \le Ra \le 10^{6} } \right)$$103≤Ra≤106 , fusion temperature $$\theta_{f}$$θf $$\left( {0.05 \le \theta_{f} \le 0.8} \right)$$0.05≤θf≤0.8 , and solid volume fraction $$\varphi$$φ $$\left( {0 \le \varphi \le 0.06} \right)$$0≤φ≤0.06 on the velocity field, isotherms, and mean Nusselt number $$\overline{Nu}$$Nu¯ are discussed. The outcomes signaled that a dual rotation of the inner fin and outer domain is affected by a time-fractional derivative. The inserted cool fin is functioning efficiently in the cooling process and adjusting the phase change zone within a hexagonal-shaped cavity. An increment in fin length augments the cooling process and changes the location of a phase change zone. A fusion temperature $$\theta_{f}$$θf adjusts the strength and position of a phase change zone. The highest values of $$\overline{Nu}$$Nu¯ are obtained when $$\alpha = 1$$α=1 . An expansion in Hartmann number $$Ha $$Ha reduces the values of $$\overline{Nu}$$Nu¯ . Adding more concentration of nanoparticles is improving the values of $$\overline{Nu}$$Nu¯ .