▎ 摘　　要

The two-dimensional magnetohydrodynamic unsteady movement and transmission of energy confined to finite domain layer of the second-grade nanofluid embedded with graphene nanoparticles on an expanding space are studied. Graphene nanoparticles have continuous electrical conductivity because the charge carrier movement in graphene bears extremely peak points compared to the available nanomaterials. The well-known system of equations for movement and energy of the second-grade nanofluid film accompanying the additional information have been transformed into the fourth-order coupled differential systems accompanying the auxiliary facts on behalf of simplifying substitutions. The simplified systems are evaluated via an efficient approach through homotopy analysis method which provides very clear relations for the motion and energy representatives. All the potential factors of the output are debated and portrayed pictorially. The results are useful in the analysis, design of coating, and cooling/heating processes.