Abstract
This paper presents a Caputo–Fabrizio fractional derivatives approach to the thermal analysis of a second grade fluid over an infinite oscillating vertical flat plate. Together with an oscillating boundary motion, the heat transfer is caused by the buoyancy force induced by temperature differences between the plate and the fluid. Closed form solutions of the fluid velocity and temperature are obtained by means of the Laplace transform. The solutions of ordinary second grade and Newtonian fluids corresponding to time derivatives of integer and fractional orders are obtained as particular cases of the present solutions. Numerical computations and graphical illustrations are used in order to study the effects of the Caputo–Fabrizio time-fractional parameter \(\upalpha \), the material parameter \(\alpha _2 \), and the Prandtl and Grashof numbers on the velocity field. A comparison for time derivative of integer order versus fractional order is shown graphically for both Newtonian and second grade fluids. It is found that fractional fluids (second grade and Newtonian) have highest velocities. This shows that the fractional parameter enhances the fluid flow.
Highlights
Mahmood et al [13] determined the velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized Maxwell fluid between two infinite coaxial circular cylinders by means of the Laplace and Hankel transforms
Similar attempts for other viscoelastic fluids, namely the Oldroyd-B fluid, the Burgers fluid, and the generalized Burgers fluid were made by Khan et al [16,17,18], Fetecau et al [19], Qi and Jin [20], Jamil et al [21], Zheng et al [22], Liu et al [23], Tong [24], and Zheng et al [25]
Apart from the rate-type fluids, the idea of fractional derivatives is implemented on differential type fluids, the second grade fluid; see for instance [26,27,28,29]
Summary
Very limited investigations were carried out for fractional models of non-Newtonian fluids, more exactly for exact solutions Such investigations are even more scarce when the non-Newtonian fluid flow is considered in the presence of convection heat transfer. With this motivation, Vieru et al [30] in a recent investigation used the idea of a fractional derivative and studied the free convection flow of a viscous fluid past a vertical infinite plate with Newtonian heating and constant mass diffusion conditions. Vieru et al [30] in a recent investigation used the idea of a fractional derivative and studied the free convection flow of a viscous fluid past a vertical infinite plate with Newtonian heating and constant mass diffusion conditions Such investigations are not available for any subclass of non-Newtonian fluids. Some important formulas used in this paper are presented in the appendix
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