Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives

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 α, the material parameter α 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.

Atangana–Baleanu and Caputo Fabrizio Analysis of Fractional Derivatives for Heat and Mass Transfer of Second Grade Fluids over a Vertical Plate: A Comparative Study

“…This comparison is shown in Figure 8. Clearly the solutions obtained by Shah and Khan [2] are in in excellent agreement with the present limiting solutions. This also confirms the accuracy of the present work.…”

“…The constraint of incompressibility is identically satisfied when such types of flow occur. Taking the usual Boussinesq approximation, the governing boundary layer equations are [1][2][3]:…”

“…In the literature, several models of non-Newtonian liquids have been launched by scientists and researchers for identifying the rheological properties and typical characteristics. Among them, the most popular model for non-Newtonian fluids is the second grade liquids model which enables prediction of differences in normal stresses [1][2][3][4][5][6][7][8][9][10]. Heat generation impacts are applicable abundantly; this is due to the thermal performance of working liquids.…”

“…Atangana et al [14] traced out RLC (resistor (R), an inductor (L), and a capacitor (C)) electrical circuit with an extension by implementing the time-fractional Caputo-Fabrizio derivative. Nehad et al [2] obtained analytical solutions for heat transfer of second grade fluids by applying fractional Caputo-Fabrizio derivatives over vertical oscillating plates. They also introduced newly published special functions for the heat transfer phenomenon of second grade fluids under the influence of Laplace transforms.…”

Bateman–Feshbach Tikochinsky and Caldirola–Kanai Oscillators with New Fractional Differentiation

“…There are few definitions of operators with fractional order, the Liouville-Caputo fractional derivative involving a kernel with singularity, and this definition is based on the power law and present singularity at the origin [9]. Recently, in order to solve the problem of singularity at the origin, Caputo and Fabrizio used the exponential decay law to construct a derivative with no singularity; however, the used kernel was local [10][11][12][13][14][15][16][17][18]. Thus, Atangana and Baleanu used the generalized Mittag-Leffler function to construct a derivative with no-singular and non-local kernel [19][20][21][22].…”