fractal and fractional Article Enhancement in Thermal Energy and Solute Particles Using Hybrid Nanoparticles by Engaging Activation Energy and Chemical Reaction over a Parabolic Surface via Finite Element Approach Yu-Ming Chu 1,2 , Umar Nazir 3 , Muhammad Sohail 3, * , Mahmoud M. Selim 4,5 and Jung-Rye Lee 6, *   Citation: Chu, Y.-M.; Nazir, U.; Sohail, M.; Selim, M.M.; Lee, J.-R. Enhancement in Thermal Energy and Solute Particles Using Hybrid Nanoparticles by Engaging Activation Energy and Chemical Reaction over a Parabolic Surface via Finite Element Approach. Fractal Fract. 2021, 5, 119. https://doi.org/ 10.3390/fractalfract5030119 Academic Editors: Lanre Akinyemi, Mostafa M. A. Khater, Mehmet Senol and Hadi Rezazadeh Received: 15 August 2021 Accepted: 10 September 2021 Published: 13 September 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Institute for Advanced Study Honoring Chen Jian Gong, Hangzhou Normal University, Hangzhou 311121, China; chuyuming@zjhu.edu.cn 2 Department of Mathematics, Huzhou University, Huzhou 313000, China 3 Department of Applied Mathematics and Statistics, Institute of Space Technology, P.O. Box 2750, Islamabad 44000, Pakistan; nazir_u2563@yahoo.com 4 Department of Mathematics, Al-Aflaj College of Science and Humanities Studies, Prince Sattam Bin Abdulaziz University, Al-Aflaj 710-11912, Saudi Arabia; m.selim@psau.edu.sa 5 Department of Mathematics, Suez Faculty of Science, Suez University, Suez 34891, Egypt 6 Department of Data Science, Daejin University, Pocheon-si 11159, Korea * Correspondence: muhammad_sohail111@yahoo.com (M.S.); jrlee@daejin.ac.kr (J.-R.L.) Abstract: Several mechanisms in industrial use have significant applications in thermal transporta- tion. The inclusion of hybrid nanoparticles in different mixtures has been studied extensively by researchers due to their wide applications. This report discusses the flow of Powell–Eyring fluid mixed with hybrid nanoparticles over a melting parabolic stretched surface. Flow rheology expres- sions have been derived under boundary layer theory. Afterwards, similarity transformation has been applied to convert PDEs into associated ODEs. These transformed ODEs have been solved the using finite element procedure (FEP) in the symbolic computational package MAPLE 18.0. The applicability and effectiveness of FEM are presented by addressing grid independent analysis. The reliability of FEM is presented by computing the surface drag force and heat transportation coefficient. The used methodology is highly effective and it can be easily implemented in MAPLE 18.0 for other highly nonlinear problems. It is observed that the thermal profile varies directly with the magnetic parameter, and the opposite trend is recorded for the Prandtl number. Keywords: mathematical modeling; ordinary and partial differential equations; parametric investigation; finite element technique; grid independent investigation; thermal enhancement 1. Introduction Due to rapid developments, the modeling of real phenomena is a hot topic of research due to its numerous applications and physical significance. Engineers, physicists and modeling experts have proposed and developed different relations by seeing the materials characteristics. Eyring–Powell is an important material whose constitute relation is τ ∗ = µ ∗ ∇V + 1 γ ∗ sinh −1  ∇V C ∗  ,      ∇V C ∗     ≪ 1. Several important studies on this model have been reported. For instance, Islam et al. [1] studied Powell–Eyring fluid over an inclined slider. They used the lubricant approach to derive a constitutive equation for the considered phenomenon. They used the homotopy perturbation scheme (HPS) to compute the solution. The impacts of numerous involved parameters on the velocity field are sketched out, and their behavior is explained via the underlying physics principles. Patel and Timol [2] computed the numerical solution Fractal Fract. 2021, 5, 119. https://doi.org/10.3390/fractalfract5030119 https://www.mdpi.com/journal/fractalfract