An Analytical Study for Caputo Fractional Derivative on Unsteady Casson Fluid with Thermal Radiation Effect

Zaiton Mat Isa,Muhammad Saqib,Yeaou Jiann Lim,Sharidan Shafie,Ahmad Qushairi Mohamad,Ridhwan Reyaz

doi:10.13189/ms.2021.090602

Abstract

Studies on Casson fluid are essential in the development of the manufacturing and engineering fields since it is widely used there. Meanwhile, fractional derivative has been known to be a constructive paradox that can be beneficial in the future. In this study, the development fractional derivative on Casson fluid flow is investigated. A fractional Casson fluid model with effect of thermal radiation is derived together with momentum and energy equations. The Caputo definition of fractional derivative is used in the mathematical formulation. Casson fluid with constant wall temperature over an oscillating plate in the presence of thermal radiation is considered. Solutions were obtained by using Laplace transform and are presented in the form of Wright function. Graphical analysis on velocity and temperature profiles was conducted with variations in parametric values such as fractional parameter, Grashof number, Prandtl number and radiation parameter. Numerical computations were carried out to investigate behaviours of skin friction and Nusselt number. It is found that when the fractional parameter is increased, the velocity and temperature profiles will also increase. Existence of fractional parameter in both velocity and temperature profiles shows the transitional phenomenon of both profiles from an unsteady state to steady state, providing a new perspective on Casson fluid flow. An increment in both profiles is also observed when the thermal radiation parameter is increased. The present results are also validated with published results, and it is found that they are in agreement with each other.

Highlights

The concept of fractional calculus first surfaced when L’Hospital wrote a letter to Leibniz, who invented the notation, asking the outcome if is [1]
Graphical results for velocity and temperature profiles as well numerical results for skin friction and Nusselt number are generated with the aid Mathcad-15 software
Accuracy of the model and final solution for velocity and temperature profiles is evaluated by comparing the current results with that of Ali et al [12]

Summary

Introduction

The concept of fractional calculus first surfaced when L’Hospital wrote a letter to Leibniz, who invented the notation, asking the outcome if is [1]. Heaviside, Abel and Liouville developed the fundamental applications and theory of fractional calculus [8,9,10,11]. Fractional has proven to be more accurate compared to conventional ordered derivative models [1213]. Debnath [14] wrote a review on applications of fractional calculus in fields of engineering and science. Many definitions of fractional integral and derivative have been developed and applied. Dalir [15] and De Oliveira [16] did an extensive review on the applications of different definitions of fractional calculus in different fields of engineering and science. Caputo derivative is one of the many definitions for fractional derivatives. Solving a fractional model with the Caputo derivative will result in a special Wright function. It has been widely used to solve fractional ordered partial difference problem

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Conclusion