Singular fractional technique for free convective Casson hybrid nanofluid with optically thick medium and shape effects

The transmission of heat in a time-dependent flow of a viscid non-Newtonian hybrid nanofluid comprising magnetite and copper oxide nanoparticles persuaded by an upright plate has been explored in regards to the effect of heat radiation and nanoparticle shape factors. The fluid flow phenomenon of the problem is constructed using the derivative of the Caputo fractional order 0 1  . As a hybrid method, the dimensionless governing fractional partial differential equation was solved analytically using transforms such as Laplace and Fourier sine. With the Mittag-Leffler function, analytical solutions are achieved for fluid flow, energy distribution, rate of heat transmission, and shear stress. Moreover, limit-case solutions for classical PDEs were given for the derived governing flow model. Graphical depictions, tables, and bar graphs are constructed using "MATLAB" for a thorough examination of the problem. The graphical findings suggest that the efficiency of hybrid nanofluids is substantially better with the Caputofractional order approach than with ordinary derivatives. Finally, a comparison with existing literature results is performed and determined to be good.