LAUSR

The natural convection resultant from the uniform circular rotation of the paddle wheel in a cross-shaped porous cavity filled by $$\hbox {Al}_{2}\hbox {O}_{3}{-}\hbox {H}_{2}\hbox {O}$$ was simulated by the ISPH method. A cross-shaped cavity’s two vertical area is saturated with a homogeneous porous medium, whereas the entire horizontal area is saturated with a heterogeneous porous medium. The paddle wheel rotates with a uniform circular velocity around the cavity’s center. The paddle wheel’s entire integrated body has temperature $$T_\mathrm{h}$$ . The temperature is set on the inside walls of a cross-shaped cavity $$T_\mathrm{c}$$ . The present geometry can be used to analyze and comprehend the thermo-physical behaviors of electronic motors. Angular velocity is set to $$\omega =7.15$$ , and thus, the natural convection case is only evaluated due to the low speed of inner rotating shape. The simulation results are graphically represented for temperature distributions, velocity fields, and tabular representations for the average Nusselt number. The important parameter ranges are the Rayleigh number $$(10^{3} \le \hbox {Ra} \le 10^{6})$$ , paddle wheel length $$(2.5 \le L_\mathrm{P} \le 14)$$ , nanoparticles parameter $$(0 \le \phi \le 0.05)$$ , and Darcy parameter $$(10^{-3} \le \hbox {Da} \le 10^{-5})$$ . The results show that increasing the length of the paddle wheel increases heat transfer and nanofluid movements within a cross-shaped cavity. In addition, increasing the Rayleigh number improves heat transfer and the nanofluid speed inside a cross-shaped cavity. When the Darcy parameter is reduced, the fluid flow is restricted to the rotating inner shape. The value of $${\overline{\hbox {Nu}}}$$ powers as the length of the paddle wheel and $$\phi $$ are increasing.