Local and Nonlocal Differential Operators: A Comparative Study of Heat and Mass Transfers in MHD Oldroyd-B Fluid With Ramped Wall Temperature

Abstract

Study of heat and mass transfers is carried out for MHD Oldroyd-B fluid (OBF) over an infinite vertical plate having time-dependent velocity and with ramped wall temperature and constant concentration. It is proven in many already published articles that the heat and mass transfers do not really or always follow the classical mechanics process that is known as memoryless process. Therefore, the model using classical differentiation based on the rate of change cannot really replicate such dynamical process very accurately, thus, a different concept of differentiation is needed to capture such process. Very recently, a new class of differential operators were introduced and have been recognized to be efficient in capturing processes following the power-law, the decay law and the crossover behaviors. For the study of heat and mass transfers, we applied the newly introduced differential operators to model such flow and compare the results with integer-order derivative. Laplace transform and inversion algorithms are used for all the cases to find analytical solutions and to predict the influences of different parameters. The obtained analytical solutions were plotted for different values of fractional orders [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] on the velocity field. In comparison, Atangana–Baleanu (ABC) fractional derivatives are found to be the best to explain the memory effects than the classical, Caputo (C) and Caputo–Fabrizio (CF) fractional derivatives. Some calculated values for Nusselt number and Sherwood number are presented in tables. Moreover, from the present solutions, the already published results were found as limiting cases.

On new exact solutions of the generalized Fitzhugh–Nagumo equation with variable coefficients

“…This procedure is well-known as the NCSM that major difficulty arises when we attempt to solve the determined equations with tedious computations. In order to overcome this difficulty we used the Heir equation method for Equation (2). Furthermore, to the author best knowledge, extracted exact solutions of GFNE with variable coefficients, thanks to the applications of this method, have not been reported previously.…”

“…Many approaches are proposed to solve different kinds of differential equations such as ordinary, partial and fractional differential equations. Different kinds of fractional derivatives are investigated in literature [1][2][3][4][5][6]. Two types of methods are generally exist.…”

Functional Application of G-Functions of Lorenzo and Hartley on the Free Convection Flow of Oldroyd-B Fluid with Ordinary and Fractional Techniques

“…In the absence of a pressure gradient in the flow direction and utilizing the typical Boussinesq's approximation, the governing equations for the unsteady free-convection flow of OBF through the vertical channel communicated by the subsequent equations [50,51] are the linear momentum equation,…”

On solutions of time‐fractional advection–diffusion equation

“…Within the literature, there are various kinds of fractional differential operators including, Caputo-Fabrizio, Caputo, Riemann-Liouville, Atangana-Baleanu, and so on. The list of researches for FC is endless, see for examples [4][5][6][7][8][9][10][11][12][13][14]. Of particular interest for us is the so-called Atangana-Baleanu derivative (AB) [15].…”