In this paper, a quantitative analysis is performed to investigate the convective flow of Maxwell fluid along with an erect heated plate via Prabhakar-like energy transport. The governing equations for this mathematical model are attained by Prabhakar fractional derivative. Laplace transform is applied to obtain the generalized results for dimensionless velocity and temperature profiles. By applying the conditions of nanofluid flow, we develop the constantly accelerated, variables accelerated, and non-uniform accelerated solution of the model, respectively. Prabhakar fractional derivative for Maxwell fluid based on generalized Fourier’s thermal flux is determined for heat transfer. For volume fraction φ and fractional parameters α , β , and γ , different structures of graphs are obtained. Furthermore, fluid temperature and velocity can be increased by enhancing the fractional parameters’ values. The comparison of nanoparticles Ag and nanoparticles Cu with water-based nanoparticles for velocity and temperature profile is also examined. The behavior of second grade and Maxwell fluid results to become a viscous fluid is justified.

The unsteady flow of Jeffrey fluid along with a vertical plate is studied in this paper. The equations of momentum, energy, and generalized Fourier’s law of thermal flux are transformed to non-dimensional form for the proper dimensionless parameters. The Prabhakar fractional operator is applied to acquire the fractional model using the constitutive equations. To obtain the generalized results for velocity and temperature distribution, Laplace transform is performed. The influences of fractional parameters , thermal Grashof number , and non-dimensional Prandtl number upon velocity and temperature distribution are presented graphically. The results are improved in the form of decay of energy and momentum equations, respectively. The new fractional parameter contains the Mittag-Leffler kernel with three fractional parameters which are responsible for better memory of the fluid properties rather than the exponential kernel appearing in the Caputo–Fabrizio fractional operator. The Prabhakar fractional operator has advantage over Caputo–Fabrizio in the real data fitting where needed.

The flow and heat transfer induced by an exponentially shrinking sheet with hybrid nanoparticles are investigated comprehensively in this paper. Nanoparticles are considered due to their unusual characteristics such as extraordinary thermal conductivity, which is significant in advanced nanotechnology, heat exchangers, material sciences, and electronics. The main objective of this research is to enhance heat transportation. The flow model is first transformed and simplified to a system of ordinary differential equations utilizing non-dimensional quantities and similarity functions. Then, the desired system is solved with the help of the Runge–Kutta numerical method and the shooting technique in MATLAB script. The results show that a stronger porosity parameter raises the temperature while diminishing the velocity. Additionally, they emphasize that augmentations in the magnetic parameter, Eckert number, radiation parameter, and the volume fractions of titanium dioxide and silver nanoparticles are all proportional to the temperature profile.

The hybrid nanofluids are effective in terms of cooling where the range of temperature is high and includes a significant range of thermal applications, such as cooling electrical equipment, heat exchangers, automotive industry, heat pipes, manufacturing industry, and solar energy. As a result of these facts, this particular research emphasizes on the silver- titanium oxide /water hybrid nanofuid flow through an exponentially stretching sheet. The impacts such as viscous dissipation, magneto hydrodynamics, porosity, thermal radiation and heat generation have been also taken into account. By utilizing non-dimensional quantities and similarity functions, the flow model is transformed and simplified to a system of ordinary differential equations. With the help of the numerical method Runge-Kutta with shooting technique in Matlab script, the desired system is solved. It has been discovered that stronger the porosity parameter raises the temperature while diminishing the velocity. It additionally emphasizes that augmentations in the magnetic parameter, Eckert number, radiation parameter, and volume percentage of T i O 2 and A g nanoparticles proportional to the temperature profile. The results demonstrate satisfactory congruence’s when compared to the current literature.