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# A New Idea of Fractal-Fractional Derivative with Power Law Kernel for Free Convection Heat Transfer in a Channel Flow between Two Static Upright Parallel Plates

Dolat Khan1, Gohar Ali1, Arshad Khan2, Ilyas Khan3, *, Yu-Ming Chu4, 5, Kottakkaran Sooppy Nisar6

1 Department of Mathematics, City University of Science and Information Technology, Peshawar, 25000, Pakistan.
2 Institute of Computer Sciences and Information Technology, The University of Agriculture, Peshawar, Pakistan.
3 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh, 72915, Vietnam.
4 Department of Mathematics, Huzhou University, Huzhou, 313000, China.
5 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha, 410114, China. 6 Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, 11991, Saudi Arabia.

* Corresponding Author: Ilyas Khan. Email: .

Computers, Materials & Continua 2020, 65(2), 1237-1251. https://doi.org/10.32604/cmc.2020.011492

## Abstract

Nowadays some new ideas of fractional derivatives have been used successfully in the present research community to study different types of mathematical models. Amongst them, the significant models of fluids and heat or mass transfer are on priority. Most recently a new idea of fractal-fractional derivative is introduced; however, it is not used for heat transfer in channel flow. In this article, we have studied this new idea of fractal fractional operators with power-law kernel for heat transfer in a fluid flow problem. More exactly, we have considered the free convection heat transfer for a Newtonian fluid. The flow is bounded between two parallel static plates. One of the plates is heated constantly. The proposed problem is modeled with a fractal fractional derivative operator with a power-law kernel and solved via the Laplace transform method to find out the exact solution. The results are graphically analyzed via MathCad-15 software to study the behavior of fractal parameters and fractional parameter. For the influence of temperature and velocity profile, it is observed that the fractional parameter raised the velocity and temperature as compared to the fractal operator. Therefore, a combined approach of fractal fractional explains the memory of the function better than fractional only.

## Keywords

D. Khan, G. Ali, A. Khan, I. Khan, Y. Chu et al., "A new idea of fractal-fractional derivative with power law kernel for free convection heat transfer in a channel flow between two static upright parallel plates," Computers, Materials & Continua, vol. 65, no.2, pp. 1237–1251, 2020.

## Citations

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