J. Non-Newtonian Fluid Mech. 154 (2008) 179–206 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm A finite difference technique for solving the Oldroyd-B model for 3D-unsteady free surface flows M.F. Tom ´ e a , A. Castelo a , V.G. Ferreira a , S. McKee b,∗ a Departamento de Matem´ atica Aplicada e Estat´ ıstica, ICMC-USP, Av. do Trabalhador S˜ ao-Carlense, CEP 13560-970, 400 S˜ ao Carlos, SP, Brazil b Department of Mathematics, University of Strathclyde, Glasgow, UK article info Article history: Received 8 August 2007 Received in revised form 9 April 2008 Accepted 10 April 2008 Keywords: Oldroyd-B Viscoelastic flow Finite difference Marker-and-cell Free surface abstract This work presents a numerical method for solving three-dimensional (3D) viscoelastic unsteady free surface flows governed by the Oldroyd-B constitutive equation. It is an extension of the two-dimensional (2D) technique introduced by Tom ´ e et al. [M.F. Tom ´ e, N. Mangiavacchi, J.A. Cuminato, A. Castelo, S. McKee, A numerical technique for solving unsteady viscoelastic free surface flows, J. Non-Newt. Fluid Mech. 106 (2002) 61–106]. The governing equations are solved by a finite difference method on a 3D-staggered grid. Marker particles are employed to describe the fluid providing both visualization and the location of the free surface. The numerical technique is validated by using an exact solution of the flow of an Oldroyd-B fluid inside a 3D-pipe. Numerical results include the simulation of the transient extrudate swell and jet buckling. © 2008 Elsevier B.V. All rights reserved. 1. Introduction The numerical treatment of free surface flows is an area that has attracted the attention of many researchers over the last two decades and still presents several challenges: the flow is unsteady, non-isothermal, non-Newtonian and can possess several moving free surfaces. Nonetheless, a number of researchers have developed numerical methods capable of simulating free surface flows that can be applied to the design and manufacture of many industrial processes. Among the numerical techniques employed the finite difference method has been used by various researchers (e.g. [28,33,5,26]). The particular case of viscoelastic free surface flows has been the subject of intense research (e.g. [15,31,4,16,18,19,25,13,12]). However, due to the complexity of these flows, the problems that have been tackled tend to have small free surface deformation, be two-dimensional (2D) or be steady-state problems. An exception would appear to be Bonito et al. [3] who have recently presented a finite element-/finite volume-based technique which appears to be capable of dealing with complex three-dimensional (3D) flows of Oldroyd-B fluids with moving free surfaces (see also [17,32]). Recently, Tom´ e et al. [27] developed a numerical method for solving 2D time-dependent viscoelastic free surface flows of a fluid described by the Oldroyd-B constitutive relationship. Numerical results for extrudate swell and jet buckling for high Weissenberg numbers were obtained. In this work we use the ideas presented by Tom´ e et al. [27] and develop a numerical method for solving the governing equations for the 3D flow of an Oldroyd-B fluid. The technique employs the finite difference method on a 3D staggered grid and the fluid free surface is determined by the marker-and-cell method. The numerical method presented in this paper is validated by simulating the flow of an Oldroyd-B fluid in pipe and the numerical results are compared with its analytic solution. Convergence results are obtained throughout using mesh refinement. The paper concludes with the simulation of two important industrial problems: time-dependent extrudate swell and jet buckling. ∗ Corresponding author. E-mail address: smck@maths.strath.ac.uk (S. McKee). 0377-0257/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2008.04.008