Eur. Phys. J. Plus (2019) 134: 535
https://doi.org/10.1140/epjp/i2019-12873-9

Regular Article

Numerical study of unsteady natural convection of variable-order fractional Jeffrey nanofluid over an oscillating plate in a porous medium involved with magnetic, chemical and heat absorption effects using Chebyshev cardinal functions

¹ Department of Mechanical Engineering, Faculty of Engineering, Fasa University, Fasa, Iran
² Department of Mathematics, Shiraz University of Technology, Shiraz, Iran
³ State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, China

^* e-mail: heydari@sutech.ac.ir

Received: 25 April 2019
Accepted: 6 July 2019
Published online: 25 October 2019

Abstract

The unsteady natural convection of an incompressible, magnetohydrodynamic (MHD), Jeffrey nanofluid in the vicinity of a vertical oscillating plate in a porous medium is examined. Several involving parameters such as thermal radiation, chemical reaction, Soret effect, thermal diffusion, magnetic field and heat absorption are taken into account. To gain an insight into the behavior of a memory-dependent fluid, the variable-order fractional calculus for heat, mass and concentration equations is implemented, which, to the best of the author's knowledge, has not been tackled by the researchers for the present comprehensive problem. The set of governing equations are reformed to the dimensionless form via invoking appropriate variables. A computational matrix method based on the Chebyshev cardinal functions (CCFs) is given to examine the behavior of the relevant problem. In fact, regarding to the established method, the unknown solutions are expanded by the CCFs. Then, the operational matrix of the variable-order fractional derivative is utilized to transfer the problem into solving an algebraic system of equations. The applicability and accuracy of the suggested method are investigated by solving some test problems. The numerical results confirm the high accuracy of the presented approach. The effect of several parameters such as Grashof, Hartmann, Prandtl, Soret and Schmidt numbers, in addition to oscillation frequency, retardation time, radiation, heat absorption and reaction rate are determined and presented graphically. According to the results, increasing the value of fractional order improves the natural convection mechanism as well as the peak value of the velocity profile.