Numerical solution of an extended White–Metzner model for eccentric Taylor–Couette flow N. Germann a,⇑ , M. Dressler b , E.J. Windhab a a Laboratory of Food Process Engineering, Swiss Federal Institute of Technology Zurich, 8092 Zurich, Switzerland b Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, USA article info Article history: Received 30 December 2010 Received in revised form 3 June 2011 Accepted 6 July 2011 Available online 20 July 2011 Keywords: Newton–Krylov methods ILU preconditioning Viscoelastic fluids Eccentric cylinders abstract In this study, we have developed a new numerical approach to solve differential-type vis- coelastic fluid models for a commonly used benchmark problem, namely, the steady Tay- lor—Couette flow between eccentric cylinders. The proposed numerical approach is special in that the nonlinear system of discretized algebraic flow equations is solved iteratively using a Newton–Krylov method along with an inverse-based incomplete lower-upper pre- conditioner. The numerical approach has been validated by solving the benchmark prob- lem for the upper-convected Maxwell model at a large Deborah number. Excellent agreement with the numerical data reported in the literature has been found. In addition, a parameter study was performed for an extended White–Metzner model. A large eccen- tricity ratio was chosen for the cylinder system in order to allow flow recirculation to occur. We detected several interesting phenomena caused by the large eccentricity ratio of the cylinder system and by the viscoelastic nature of the fluid. Encouraged by the results of this study, we intend to investigate other polymeric fluids having a more complex microstructure in an eccentric annular flow field. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction The Taylor–Couette flow between eccentric cylinders is commonly used as a benchmark problem for testing numerical methods and fluid models. Unlike other benchmark problems such as the flow through a contraction or the flow around a sphere in a tube, this problem has a simple flow geometry whose boundaries are smooth and contain no singularities. How- ever, this flow is still complex in that it involves simple shear, planar extension, and rigid body rotation. Another advantage of using the eccentric cylinder system as a flow geometry is that exact analytical solutions exist for many fluid models in the limit of a vanishing eccentricity ratio, and thus, it can be used to verify the consistency of numerical codes. In addition, the eccentric cylinder system has a capability for highly distributive and dispersive mixing, and therefore, it is of interest in the industrial processing of complex fluids. Over the last three decades, many studies have focused on obtaining numerical solutions for viscoelastic fluid models of the Taylor–Couette flow between eccentric cylinders. Beris et al. [1] were the first to report successful calculations for the upper-convected Maxwell (UCM) model at large Deborah numbers. The Deborah number is a dimensionless number that is defined as the ratio of the characteristic timescale of the fluid material to that of the flow process. A spectral/finite-element method was used for the spatial discretization of the benchmark problem. The resulting nonlinear system of spatially dis- cretized algebraic flow equations was solved using a specialized Newton code that employs an out-of-core direct frontal 0021-9991/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2011.07.007 ⇑ Corresponding author. E-mail address: natalie.germann@ilw.agrl.ethz.ch (N. Germann). Journal of Computational Physics 230 (2011) 7853–7866 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp