processes Article Effective Similarity Variables for the Computations of MHD Flow of Williamson Nanofluid over a Non-Linear Stretching Surface Kamran Ahmed 1 , Luthais B. McCash 2 , Tanvir Akbar 1 and Sohail Nadeem 3, *   Citation: Ahmed, K.; McCash, L.B.; Akbar, T.; Nadeem, S. Effective Similarity Variables for the Computations of MHD Flow of Williamson Nanofluid over a Non-Linear Stretching Surface. Processes 2022, 10, 1119. https:// doi.org/10.3390/pr10061119 Academic Editors: Byeong-Ui Moon and Tae-Hyeong Kim Received: 19 May 2021 Accepted: 3 July 2021 Published: 2 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 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 Department of Mathematics, Islamabad Campus, COMSATS University Islamabad, Park Road, Islamabad 45550, Pakistan; kam_ahm@ymail.com (K.A.); tanvir.akbar@comsats.edu.pk (T.A.) 2 School of Mathematics & Acturial Science, University of Leicester, Leicester LE1 7RH, UK; lmccash@fb-fa.com 3 Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan * Correspondence: sohail@qau.edu.pk Abstract: The present study concerns investigating the two-dimensional Magnetohydrodynamics (MHD) boundary layer flow of Williamson nanofluid over a non-linear stretching sheet. The focus of this study is based on the global influence of the non-Newtonian Williamson fluid parameter (λ) rather than the local one that exists in the literature for linear and non-linear stretching cases. The mathematical model of the problem is based on the law of conservation of mass, momentum, and energy. The derived partial differential equations are transformed into ordinary differential equations by applying an appropriate similarity transformation. The subsequent equations are solved numerically by using the Shooting method. The physical quantities Skin friction coefficient, as well as the Sherwood and Nusselt numbers are computed locally. To validate the implemented shooting method, a comparison is made with the results obtained by Matlab function bvp4c, and good agreement is found. The Prandtl number, Pr, has an increasing impact of 25.14% on the wall temperature gradient. The impact of various physical parameters are presented through graphs and tables. Keywords: similarity transformation; non-linear stretching sheet; Williamson nanofluid; shooting method; bvp4c 1. Introduction The two main categories of fluid mechanics are Newtonian and non-Newtonian fluid. The relationship among strain rate is described by deriving the constitutive equation, especially for those fluids that do not maintain the Newtonian law of viscosity. Several researchers have provided mathematical models to determine the rheological properties of such fluids. The models include the power-law, Williamson fluid, Ellis, cross, and Carreau models. Williamson [1] provided the Williamson model for pseudoplastic ma- terials, which is an experimentally verified model. The characteristic of the Williamson fluid model involves choosing minimum (μ 0 ) and maximum (μ ∞ ) viscosity at the same time. In real fluid, minimum as well as maximum viscosity is needed for the mathe- matical model. Pseudoplastic fluids are commonly used in industry as melts of high molecular weight polymer solution, photographic film, and extrusion of polymer sheets [2]. Carmer et al. [3] investigated polymer solution using the Williamson fluid model. Lyubi- mov and Perminov [4] deliberated the flow of Williamson fluid over an inclined wall, with aspects of the gravitational field. Nadeem et al. [5] investigated the numerical solution of the peristaltic flow of Williamson fluid by radially varying MHD in an endoscope. Noreen Sher Akbar et al. [6] used the Carreau model and Ismail et al. [7] used the power-law model to investigate flow of blood in arteries. Ahmed et al. [8] numerically scrutinized the impact of Williamson fluid flow over an exponential stretching surface. Ramzan et al. [9] Processes 2022, 10, 1119. https://doi.org/10.3390/pr10061119 https://www.mdpi.com/journal/processes