Analysis of newly developed fractal-fractional derivative with power law kernel for MHD couple stress fluid in channel embedded in a porous medium

Muhammad Arif; Poom Kumam; Wiyada Kumam; Ali Akgul; Thana Sutthibutpong

doi:10.1038/s41598-021-00163-3

Analysis of newly developed fractal-fractional derivative with power law kernel for MHD couple stress fluid in channel embedded in a porous medium

Sci Rep. 2021 Oct 21;11(1):20858. doi: 10.1038/s41598-021-00163-3.

Authors

Muhammad Arif^{1

2}, Poom Kumam^{3

4

5}, Wiyada Kumam⁶, Ali Akgul⁷, Thana Sutthibutpong^{2

8}

Affiliations

¹ Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand.
² Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand.
³ Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand. poom.kum@kmutt.ac.th.
⁴ Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand. poom.kum@kmutt.ac.th.
⁵ Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan. poom.kum@kmutt.ac.th.
⁶ Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani, 12110, Thailand.
⁷ Department of Mathematics, Faculty of Arts and Sciences, Siirt University, TR-56100, Siirt, Turkey.
⁸ Theoretical and Computational Physics Group, Department of Physics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140, Thailand.

Abstract

Fractal-fractional derivative is a new class of fractional derivative with power Law kernel which has many applications in real world problems. This operator is used for the first time in such kind of fluid flow. The big advantage of this operator is that one can formulate models describing much better the systems with memory effects. Furthermore, in real world there are many problems where it is necessary to know that how much information the system carries. To explain the memory in a system fractal-fractional derivatives with power law kernel is analyzed in the present work. Keeping these motivation in mind in the present paper new concept of fractal-fractional derivative for the modeling of couple stress fluid (CSF) with the combined effect of heat and mass transfer have been used. The magnetohydrodynamics (MHD) flow of CSF is taken in channel with porous media in the presence of external pressure. The constant motion of the left plate generates the CSF motion while the right plate is kept stationary. The non-dimensional fractal-fractional model of couple stress fluid in Riemann-Liouville sense with power law is solved numerically by using the implicit finite difference method. The obtained solutions for the present problem have been shown through graphs. The effects of various parameters are shown through graphs on velocity, temperature and concentration fields. The velocity, temperature and concentration profiles of the MHD CSF in channel with porous media decreases for the greater values of both fractional parameter [Formula: see text] and fractal parameter [Formula: see text] respectively. From the graphical results it can be noticed that the fractal-fractional solutions are more general as compared to classical and fractional solutions of CSF motion in channel. Furthermore, the fractal-fractional model of CSF explains good memory effect on the dynamics of couple stress fluid in channel as compared to fractional model of CSF. Finally, the skin friction, Nusselt number and Sherwood number are evaluated and presented in tabular form.