Dynamics of Jeffrey fluid flow and heat transfer: A Prabhakar fractional operator approach

The primary goal of current study is to formulate a novel mathematical model employing the Prabhakar fractional operator, aimed at examining the dynamic behaviour of Jeffrey fluid flow and heat transfer phenomena under ramped wall temperature conditions. Our investigation entails a meticulous investigation of magnetohydrodynamic (MHD) natural convective flow within Jeffrey fluids, where we derive precise analytical solutions utilizing the Prabhakar fractional derivative, characterized by a non-singular type kernel. Furthermore, our study integrates fundamental principles such as Fick's and Fourier's laws into the model, leveraging a multi-parameter Mittag-Leffler kernel associated with the fractional operator. We delve into the intricacies of fluid flow near an infinitely vertical plate, considering characteristics such as velocity u0. To address the complexities of the problem, we express it in context the of partial differential equations along side generalized boundary conditions, employing a set of appropriate variables to transform these equations into a dimensionless form. Utilizing Laplace transform, we analyze the equations of fractional system, presenting outcomes both in the form of series and through specialized functions. We systematically explore the influence of key parameters α, Pr, β, Gm, Sc, γ, Gr on fluid flow dynamics, unveiling significant insights. Our comparative analysis reveals the superior performance of the Prabhakar-like non-integer approach over existing operators, substantiated by graphical representations of the results. Additionally, we extend our investigation to various limiting cases, including Newtonian fluids and second-grade, in both fractionalized and classical forms, thus highlighting the versatility and applicability of our proposed model within fluid dynamics research.