In the pump of different machines, the vacuum pump oil (VPO) is used as a lubricant. The heat rate transport mechanism is a significant requirement for all industries and engineering. The applications of VPO in discrete fields of industries and engineering fields are uranium enrichment, electron microscopy, radio pharmacy, ophthalmic coating, radiosurgery, production of most types of electric lamps, mass spectrometers, freeze-drying, and, etc. Therefore, in the present study, the nanoparticles are mixed up into the VPO base liquid for the augmentation of energy transportation. Further, the MHD flow of a couple stress nanoliquid with the applications of Hall current toward the rotating disk is discussed. The Darcy-Forchheimer along with porous medium is examined. The prevalence of viscous dissipation, thermal radiation, and Joule heating impacts are also considered. With the aid of Cattaneo-Christov heat-mass flux theory, the mechanism for energy and mass transport is deliberated. The idea of the motile gyrotactic microorganisms is incorporated. The existing problem is expressed as higher-order PDEs, which are then transformed into higher-order ODEs by employing the appropriate similarity transformations. For the analytical simulation of the modeled system of equations, the HAM scheme is utilized. The behavior of the flow profiles of the nanoliquid against various flow parameters has discoursed through the graphs. The outcomes from this analysis determined that the increment in a couple-stress liquid parameter reduced the fluid velocity. It is obtained that, the expansion in thermal and solutal relaxation time parameters decayed the nanofluid temperature and concentration. Further, it is examined that a higher magnetic field amplified the skin friction coefficients of the nanoliquid. Heat transport is increased through the rising of the radiation parameter.