polymers Article An Iterative Approach for the Parameter Estimation of Shear-Rate and Temperature-Dependent Rheological Models for Polymeric Liquids Medeu Amangeldi 1 , Yanwei Wang 2,3, * , Asma Perveen 4 , Dichuan Zhang 5 and Dongming Wei 1, *   Citation: Amangeldi, M.; Wang, Y.; Perveen, A.; Zhang, D.; Wei, D. An Iterative Approach for the Parameter Estimation of Shear-Rate and Temperature-Dependent Viscosity Models for Polymeric Liquids. Polymers 2021, 13, 4185. https:// doi.org/10.3390/polym13234185 Academic Editor: Jay McCarty Received: 18 October 2021 Accepted: 5 November 2021 Published: 30 November 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 Department of Mathematics, School of Sciences and Humanities, Nazarbayev University, Nur-Sultan 010000, Kazakhstan; medeu.amangeldi@alumni.nu.edu.kz 2 Department of Chemical & Materials Engineering, School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan 010000, Kazakhstan 3 Laboratory of Computational Materials Science for Energy Applications, Center for Energy and Advanced Materials Science, National Laboratory Astana, Nur-Sultan 010000, Kazakhstan 4 Department of Mechanical & Aerospace Engineering, School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan 010000, Kazakhstan; asma.perveen@nu.edu.kz 5 Department of Civil & Environmental Engineering, School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan 010000, Kazakhstan; dichuan.zhang@nu.edu.kz * Correspondence: yanwei.wang@nu.edu.kz(Y.W.); dongming.wei@nu.edu.kz (D.W.) Abstract: Numerical flow simulations play an important role in polymer processing. One of the essential prerequisites for accurate and precise flow simulations is to obtain accurate materials functions. In the framework of the generalized Newtonian fluid model, one needs to obtain shear viscosity as a function of the rate-of-shear and temperature—as determined by rheometry—and then fitted to a mathematical model. Often, many subjectively perform the fitting without paying attention to the relative quality of the estimated parameters. This paper proposes a unique iterative algorithm for fitting the rate-of-shear and temperature-dependent viscosity model under the time– temperature superposition (TTS) principle. Proof-of-concept demonstrations are shown using the five-parameter Carreau–Yasuda model and experimental data from small-amplitude oscillatory shear (SAOS) measurements. It is shown that the newly proposed iterative algorithm leads to a more accurate representation of the experimental data compared to the traditional approach. We compare their performance in studies of the steady isothermal flow of a Carreau–Yasuda model fluid in a straight, circular tube. The two sets of parameters, one from the traditional approach and the other from the newly proposed iterative approach, show considerable differences in flow simulation. The percentage difference between the two predictions can be as large as 10% or more. Furthermore, even in cases where prior knowledge of the TTS shifting factors is not available, the newly proposed iterative approach can still yield a good fit to the experimental data, resulting in both the shifting factors and parameters for the non-Newtonian fluid model. Keywords: rheology model; polymers; non-Newtonian fluid; time–temperature superposition; curve-fitting; parameter estimation 1. Introduction In polymer rheology, the proper non-Newtonian viscosity models are essential for modeling and flow simulations [1]. Experimental measurements of polymeric liquids (solution and melt) are routinely carried out to obtain the necessary data on their rheological properties. Then, various numerical methods were used to find a suitable set of parameters for the rheology model to facilitate efficient flow simulations [2–8]. In general, there is no universal rule of fitting the rheological data. As stated by Singh et al. (2019) [9], people subjectively choose the fitting approach—thus, leading to non- unique inferences. Moreover, Gallagher et al. (2019) [10] reported the non-identifiability in Polymers 2021, 13, 4185. https://doi.org/10.3390/polym13234185 https://www.mdpi.com/journal/polymers