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Advancements in Numerical Solutions: Fractal Runge-Kutta Approach to Model Time-Dependent MHD Newtonian Fluid with Rescaled Viscosity on Riga Plate

by Muhammad Shoaib Arif^1,2,*, Kamaleldin Abodayeh¹, Yasir Nawaz²

1 Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
2 Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan

* Corresponding Author: Muhammad Shoaib Arif. Email:

(This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)

Computer Modeling in Engineering & Sciences 2024, 141(2), 1213-1241. https://doi.org/10.32604/cmes.2024.054819

Received 08 June 2024; Accepted 12 August 2024; Issue published 27 September 2024

Abstract

Fractal time-dependent issues in fluid dynamics provide a distinct difficulty in numerical analysis due to their complex characteristics, necessitating specialized computing techniques for precise and economical solutions. This study presents an innovative computational approach to tackle these difficulties. The main focus is applying the Fractal Runge-Kutta Method to model the time-dependent magnetohydrodynamic (MHD) Newtonian fluid with rescaled viscosity flow on Riga plates. An efficient computational scheme is proposed for handling fractal time-dependent problems in flow phenomena. The scheme is comprised of three stages and constructed using three different time levels. The stability of the scheme is shown by employing the Fourier series analysis to solve scalar problems. The scheme’s convergence is guaranteed for a time fractal partial differential equations system. The scheme is applied to the dimensionless fractal heat and mass transfer model of incompressible, unsteady, laminar, Newtonian fluid with rescaled viscosity flow over the flat and oscillatory Riga plates under the effects of space- and temperature-dependent heat sources. The first-order back differences discretize the continuity equation. The results show that skin friction local Nusselt number declines by raising the coefficient of the temperature-dependent term of heat source and Eckert number. The numerical simulations provide valuable insights into fluid dynamics, explicitly highlighting the influence of the temperature-dependent coefficient of the heat source and the Eckert number on skin friction and local Nusselt number.

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