Exact solutions via Prabhakar fractional approach to investigate heat transfer and flow features of hybrid nanofluid subject to shape and slip effects

Abstract

AbstractThe core devotion of this study is to develop a generalized model by means of a recently proposed fractional technique in order to anticipate the enhancement in the thermal efficiency of engine oil because of the dispersion of graphene and magnesia nanoparticles. In addition to investigating the synergistic attributes of the foregoing particles, this work evaluates shape impacts for column, brick, tetrahedron, blade, and lamina-like shapes. In the primary model, the flow equation is coupled with concentration and energy functions. This classical system is transmuted into a fractional environment by generalizing mathematical expressions of thermal and diffusion fluxes by virtue of the Prabhakar fractional operator. In this study, ramped flow and temperature slip conditions are simultaneously applied for the first time to examine the behavior of a hybrid nanofluid. The mathematical analysis of this problem involves the incorporation of dimension-independent parameters into the model and the execution of the Laplace transform for the consequent equations. By doing so, exact solutions are derived in the form of Mittag–Leffler functions. Multiple illustrations are developed by dint of exact solutions to chew over all aspects of temperature variations and flow dynamics. For the preparation of these illustrations, the details of parametric ranges are as follows: $$0.00 \le \varUpsilon \le 0.04$$ 0.00 ≤ Υ ≤ 0.04 , $$2.0 \le Gr_1 \le 8.0$$ 2.0 ≤ G r 1 ≤ 8.0 , $$0.5 \le Sc \le 2.0$$ 0.5 ≤ S c ≤ 2.0 , $$0.1 \le \uptau \le 4.0$$ 0.1 ≤ τ ≤ 4.0 , $$0.1 \le d \le 0.6$$ 0.1 ≤ d ≤ 0.6 , $$0.2 \le \lambda _1 \le 1.5$$ 0.2 ≤ λ 1 ≤ 1.5 , and $$1.0 \le Gr_2 \le 7.0$$ 1.0 ≤ G r 2 ≤ 7.0 . The contribution of differently shaped nanoparticles, volume proportions, and fractional parameters in boosting the heat-transferring attributes of engine oil is also anticipated. In this regard, results for Nusselt number are provided in tabular form. Additionally, a brief analysis of shear stress is carried out for fractional parameters and various combinations of magnesia, graphene, and engine oil. This investigation anticipates that engine oil’s hybridization with magnesia and graphene would result in a 33% increase in its thermal performance, which evidently improves its industrial significance. The enhancement in Schmidt number yields an improvement in the mass transfer rate. An increment in collective volume fraction leads to raising the profile of the thermal field. However, the velocity indicates a decreasing behavior. Nusselt number reaches its highest value ($$Nu=8.1363$$ N u = 8.1363 ) for the lamina shape of considered particles. When the intensity of the buoyancy force augments, it causes the velocity to increase.