Power law memory of natural convection flow of hybrid nanofluids with constant proportional Caputo fractional derivative
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© Indian Academy of Sciences
Power law memory of natural convection flow of hybrid nanofluids with constant proportional Caputo fractional derivative due to pressure gradient RIZWAN ALI1 , ALI AKGÜL2
,∗
and MUHAMMAD IMRAN ASJAD1
1 Department
of Mathematics, University of Management and Technology, Lahore, Pakistan of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey ∗ Corresponding author. E-mail: [email protected] 2 Department
MS received 17 March 2020; revised 22 June 2020; accepted 1 July 2020 Abstract. In this work, influence of hybrid nanofluids on heat transfer flow of a viscous fluid due to pressure gradient is discussed with innovative constant proportional Caputo fractional derivative. For this purpose, we consider an infinite vertical wall which is exponentially moving in the x-direction with variable temperature. Nanosized particles of Cu and Al2 O3 are suspended in water, the base fluid. The governing equations of the problem are converted into dimensionless form. Further, we develop the constant proportional Caputo fractional model with a new operator with power law kernel which can be used to study the fluid behaviour for different values of fractional parameter at the present time. We applied the Laplace transform method to obtain the solutions and to see the impact of hybrid nanofluids and fractional parameter α respectively. We compared the present results with the recently published work (Nehad et al, Adv. Mech. Eng. 11(7): 1 (2019)) with Caputo fractional derivative. As a result, we have found that the present solutions are best to describe the memory concept of temperature and velocity. For small values of fractional parameter, temperature and velocity have maximum values and for larger values of fractional parameter, temperature and velocity have minimum values. Further, rate of heat transfer and skin friction are also computed in tabular forms and it is found that Nusselt number with CPC is much less than that is computed with Caputo fractional derivative for greater values of fractional parameter α. Keywords. Hybrid nanofluids; Newtonian fluid; pressure gradient; channel flow; power law kernel. PACS Nos 44.05; 44.30
1. Introduction In the past years, from all branches of engineering, technology and sciences the opinion about non-local operators of differentiation has been used by many researchers because of their abilities of adding complicated natural methods in mathematical equations. Three important laws were suggested which are Mittag–Leffler law, exponential decay law and power law [1–6]. For the differentiation of these laws, some of the frequently used operators are Atangana–Baleanu for non-local and nonsingular type, Caputo–Fabrizio for non-singular local type, Riemann–Liouville and Caputo fractional operators for power law or non-local and singular kernel type. It was significant that, the function of kernel Mittag– Leffler is more general than power law and exponential decay function. Therefore, both Riemann–Liouville and 0123456789().: V,-vol
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