Abstract
Fractional-order mathematical modelling of physical phenomena is a hot topic among various researchers due to its many advantages over positive integer mathematical modelling. In this context, the appropriate solutions of such fractional-order physical modelling become a challenging task among scientists. This paper presents a study of unsteady free convection fluid flow and heat transfer of Maxwell fluids with the presence of Clay nanoparticle modelling using fractional calculus. The obtained model was transformed into a set of linear nondimensional, partial differential equations (PDEs). The finite difference scheme is proposed to discretize the obtained set of nondimensional PDEs. The Maple code was developed and executed against the physical parameters and fractional-order parameter to explain the behavior of the velocity and temperature profiles. Some limiting solutions were obtained and compared with the latest existing ones in literature. The comparative study witnesses that the proposed scheme is a very efficient tool to handle such a physical model and can be extended to other diversified problems of a complex nature.
Highlights
Many physical phenomena of the natural sciences can be illustrated by differential equations with suitable initial and boundary conditions such as the flow of Maxwell fluids
The finite difference method is proposed and applied to inspect the numerical solution of the problem presented in Equations (12) and (13) for ∆t = 0.02 and ∆x = 0.005
For μ = 1 dual behavior of the velocity profile is attained against Gr. These results are expected because the Grashof number is the ratio between inertia and viscous forces
Summary
Many physical phenomena of the natural sciences can be illustrated by differential equations with suitable initial and boundary conditions such as the flow of Maxwell fluids. Heat transfer has various commercial and industrial applications such as plastic film manufacturing, fiber coating and artificial fiber, and other chemical processing tools [20,21,22,23] As a result, it is a main interest of researchers to give a new direction to novel fractional modeling rather than that of a classical study of the positive integer order of PDEs, as it gives a generalized result by various fractional differentiation operators [24,25,26] (See [27]).
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