NUMERICAL MODELING OF PHAN-THIEN – TANNER VISCOELASTIC FLUID FLOW THROUGH A SQUARE CROSS–SECTION DUCT: HEAT TRANSFER ENHANCEMENT DUE TO SHEAR–THINNING EFFECTS Fouad Hagani Univ Lyon, CNRS, INSA-Lyon, Universit ´ e Claude Bernard Lyon 1, CETHIL UMR5008, F-69621, Villeurbanne, France Email: fouad.hagani@insa-lyon.fr M’hamed Boutaous Univ Lyon, CNRS, INSA-Lyon, Universit ´ e Claude Bernard Lyon 1, CETHIL UMR5008, F-69621, Villeurbanne, France Email: mhamed.boutaous@insa-lyon.fr Ronnie Knikker Univ Lyon, CNRS, INSA-Lyon, Universit ´ e Claude Bernard Lyon 1, CETHIL UMR5008, F-69621, Villeurbanne, France Email: ronnie.knikker@insa-lyon.fr Shihe Xin Univ Lyon, CNRS, INSA-Lyon, Universit ´ e Claude Bernard Lyon 1, CETHIL UMR5008, F-69621, Villeurbanne, France Email: shihe.xin@insa-lyon.fr Dennis Siginer Department of Mechanical Engineering University of Santiago of Chile and Bostwana International University of Science and Technology Email: dennis.siginer@usach.cl ; siginerd@biust.ac.bw ABSTRACT Non-isothermal laminar flow of a viscoelastic fluid through a square cross–section duct is analyzed. Viscoelastic stresses are described by the Phan-Thien –Tanner model and the solvent shear stress is given by the linear Newtonian constitutive rela- tionship. The solution of the set of governing equations spawns coupling between equations of elliptic-hyperbolic type. Our nu- merical approach is based on the finite-differences method. To treat the hyperbolic part, the system of equations are rewritten in a quasilinear form. The resulting pure advection terms are discretized using high-order upwind schemes when the hyper- bolicity condition is satisfied. The incompressibility condition is obtained by the semi-implicit projection method. Finally we investigate the evolution of velocity, shear stress, viscosity and heat transfer over a wide range of Weissenberg numbers. INTRODUCTION Numerical instabilities are often encountered when model- ing non-Newtonian viscoelastic fluid flows. Among these insta- bilities the loss of order of discretization, parasitic oscillations, weak convergences, etc. These instabilities appear in particular in the case of flows with a high Weissenberg numbers. These effects could be related to the coupling between the elliptical equation of momentum and the hyperbolic constitutive equa- tion [1,2]. Several approaches and numerical discretizations have been proposed in the literature, based on finite-volumes [3–5], finite-elements [6–8], and finite-differences [9–11]. The diago- nalization of the hyperbolic part described in [9–11], in a quasi– linear form, seems to be promising. In non-circular tube flow due to the existence and anisotropy of the second-difference of normal stresses [3,12–14], secondary flows with a very small magnitude compared to the main flow take place. It has been shown experimentallly [15], analyti- Proceedings of the ASME 2018 International Mechanical Engineering Congress and Exposition IMECE2018 November 9-15, 2018, Pittsburgh, PA, USA IMECE2018-87568 1 Copyright © 2018 ASME