  Citation: Bodnár, T.; Sequeira, A. Analysis of the Shear-Thinning Viscosity Behavior of the Johnson–Segalman Viscoelastic Fluids. Fluids 2022, 7, 36. https:// doi.org/10.3390/fluids7010036 Academic Editors: Kannan N. Premnath and Ramesh Agarwal Received: 30 November 2021 Accepted: 12 January 2022 Published: 14 January 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/). fluids Article Analysis of the Shear-Thinning Viscosity Behavior of the Johnson–Segalman Viscoelastic Fluids Tomáš Bodnár 1,2,∗ and Adélia Sequeira 3 1 Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University in Prague, Karlovo námˇ estí 13, 121 35 Prague 2, Czech Republic 2 Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic 3 Department of Mathematics and CEMAT, Instituto Superior Técnico, University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal; adelia.sequeira@tecnico.ulisboa.pt * Correspondence: Tomas.Bodnar@fs.cvut.cz; Tel.: +420-2-2435-7548 Abstract: This paper presents a numerical comparison of viscoelastic shear-thinning fluid flow using a generalized Oldroyd-B model and Johnson–Segalman model under various settings. Results for the standard shear-thinning generalization of Oldroyd-B model are used as a reference for comparison with those obtained for the same flow cases using Johnson–Segalman model that has specific adjustment of convected derivative to assure shear-thinning behavior. The modeling strategy is first briefly described, pointing out the main differences between the generalized Oldroyd-B model (using the Cross model for shear-thinning viscosity) and the Johnson–Segalman model operating in shear-thinning regime. Then, both models are used for blood flow simulation in an idealized stenosed axisymmetric vessel under different flow rates for various model parameters. The simulations are performed using an in-house numerical code based on finite-volume discretization. The obtained results are mutually compared and discussed in detail, focusing on the qualitative assessment of the most distinct flow field differences. It is shown that despite all models sharing the same asymptotic viscosities, the behavior of the Johnson–Segalman model can be (depending on flow regime) quite different from the predictions of the generalized Oldroyd-B model. Keywords: viscoelastic fluid; shear-thinning viscosity; Johnson–Segalman model; generalized Oldroyd-B model 1. Introduction Many fluids of practical interest exhibit a complex behavior that cannot be predicted using mathematical models employing the classical Newtonian rheological laws. Phe- nomena such as shear-thinning/thickening, yield stress, stress relaxation or viscoelastic behavior are quite commonly observed in real fluids, but fail to be properly represented using classical Newtonian fluids models. A wide class of so-called non-Newtonain models was developed and used to capture specific fluid properties and flow behavior. A compre- hensive overview and discussion of complex fluid rheology and corresponding models can be found for example in classical books [1–3] or in papers [4,5]. From the plethora of non-Newtonian fluids properties we will only remind and discuss two, the shear-thinning and viscoelasticity, that are relevant within the scope of this paper. The shear-thinning behavior is typically captured by a specific sub-class of the so called generalized Newtonian models. In classical Newtonian models the stress tensor is directly proportional (by a constant coefficient named viscosity) to the fluid rate of strain tensor (which is nothing but symmetric part of velocity gradient). The generalized Newtonian models follow this concept, but allow the proportionality coefficient (viscosity) to be variable, typically depending on some relevant physical quantities, most importantly the invariants of the rate of strain tensor. Classical representative of this class of generalized Newtonian models is the well known power law viscosity model. The shear-thinning Fluids 2022, 7, 36. https://doi.org/10.3390/fluids7010036 https://www.mdpi.com/journal/fluids