In the modern era, diathermic oils have been gotten the great attention from researchers due to its notable and momentous applications in engineering, mechanics and in the industrial field. The aim of this paper is to model the problem to augment the heat transfer rate of diathermic oils, specifically, Engine-oil (EO) and Kerosene-oil (KO) are taken. The present work is dedicated to examine the shape impacts of molybdenum-disulfide (MoS2) nanoparticles in the free convection magnetohydrodynamic (MHD) flow of Brinkman-type nanofluid in a rotating frame. The problem is modeled in terms of partial differential equations with oscillatory boundary conditions. The integer-order model is transformed to fractional-order model in time (Caputo-Fabrizio). The exact solutions are obtained using the Laplace transform technique. Figures are drawn to compare the different non-spherically shaped molybdenum-disulfide nanoparticles on secondary and primary velocities. The Nusselt number is computed in the tabular form and discussed in detail. It is worth noting that platelet and blade shape of MoS2 nanoparticle has more tendency to improve the heat transfer rate of both fluids as compared to nanoparticles with brick and cylinder shapes. It is also shown that the rate of heat transfer enhances 13.51% by adding MoS2 in engine oil which improved its lubrication properties.