Abstract
The incompressible smoothed particle hydrodynamics (ISPH) scheme is advanced to handle the conformable fractal differential equations of natural convection for a tall fin inside a cavity barred by nano encapsulated phase change material (NEPCM). The bottom circular cylinder is occupied by a porous medium plus a nanofluid, while the upper circular cylinder is filled by a nanofluid only. The conformable fractal operators generalize the classic idea of differentiability and allow for the generation of new and universal rates of variation. So, the study's novelty is revealing the validity of these operators in creating a new environment to look for more extended natural convection systems. The pertinent parameters are dimensionless time parameter τ(0-0.325), Rayleigh number Ra(103-106), Darcy parameter Da(10-2-10-5), cold source length LCold0.5-2.5, Stefan parameter Ste0.2-0.8, conformable fractal parameter α(0.93-1), and fusion temperature θf(0.05-0.95). The main outcomes showed the responsibility of α in speeding up the phase change process and the transition procedure of the convection flow. The nanofluid speed is reduced in the porous layer of a cavity, especially at the lower Darcy parameter. Different thermal conditions are altering the phase change material and strength of isotherms within a cavity.