In the present study, the flow, heat transfer, and entropy generation of aqueous copper oxide-silver hybrid nanofluid over a down-pointing rotating vertical cone in the presence of a cross magnetic field has been investigated numerically. The modeling is done by using of the mass-based algorithm and the entropy generation analysis will be implemented in combination with the mentioned model. Five shapes of nanoparticle including the sphere, brick, cylinder, platelet, and disk as well as two types of boundary conditions containing the linear surface temperature (LST) and linear surface heat flux (LSHF) have been considered to present a collection of critical results. The governing PDEs are converted to the dimensionless ODEs using the similarity transformation method and the simplified ODEs are solved numerically by Runge–Kutta–Fehlberg approach coupled with a shooting procedure for some values of the emerged parameters. After achieving the excellent agreement of the computational results with previous reports, the results are presented for the behavior analysis of the hydrodynamics/thermal boundary layers, the skin friction and Nusselt number as well as entropy generation and Bejan number. The highest Nusselt number has been computed for a state of hybrid nanofluid, which shows a remarkable heat transfer augmentation relative to the states of mono-nanofluid and regular fluid; however, the skin friction values always should be considered and controlled. The entropy generation number enhances with the mass of the second nanoparticle (silver), while the reverse trend is demonstrated for the Bejan number. Further, the lowest value of entropy generation number belongs to the cylindrical shape of nanoparticles in the LST case. Consequently, reliable treatment of the mass-based algorithm for the entropy analysis is a substantial achievement of the present research.