Commun. Comput. Phys., 6 (2009), pp. 367-395. Numerical Solution of the Upper-Convected Maxwell Model for Three-Dimensional Free Surface Flows Murilo F. Tome 1, Renato A. P. Silva 1, Cassio M. Oishi 1, Sean McKee 2*1 Departamento de Matematica Aplicada e Estatistica, Instituto de Ciencias Matematicas e de Computacao, Universidade de Sao Paulo, Sao Carlos, Brazil. 2 Department of Mathematics, University of Strathclyde, Glasgow, UK. Received 18 April 2008; Accepted (in revised version) 22 September 2008 Available online 15 December 2008 Abstract This work presents a finite difference technique for simulating three-dimensional free surface flows governed by the Upper-Convected Maxwell (UCM) constitutive equation. A Marker-and-Cell approach is employed to represent the fluid free surface and formulations for calculating the non-Newtonian stress tensor on solid boundaries are developed. The complete free surface stress conditions are employed. The momentum equation is solved by an implicit technique while the UCM constitutive equation is integrated by the explicit Euler method. The resulting equations are solved by the finite difference method on a 3D-staggered grid. By using an exact solution for fully developed flow inside a pipe, validation and convergence results are provided. Numerical results include the simulation of the transient extrudate swell and the comparison between jet buckling of UCM and Newtonian fluids. AMS subject classifications: 76A10, 76D05, 76M20 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Viscoelastic flow, Upper-Convected Maxwell, finite difference, free surface, implicit techniques, Marker-and-Cell. *Corresponding author. Email: murilo@icmc.usp.br (M. F. Tome), silva.renato@gmail.com (R. A. P. Silva), oishi@icmc.usp.br (C. M. Oishi), smck@maths.strath.ac.uk (S. McKee)