Journal of Applied Mathematics
Volume 2012 (2012), Article ID 420189, 15 pages
Research Article

Viscoelastic Flow through an Axisymmetric Contraction Using the Grid-by-Grid Inversion Method

Department of Chemical and Biomolecular Engineering, Sogang University, Seoul 121-742, Republic of Korea

Received 1 December 2011; Revised 5 January 2012; Accepted 15 January 2012

Academic Editor: M. F. El-Amin

Copyright © 2012 H. M. Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The newly developed algorithm called the grid-by-grid inversion method is a very convenient method for converting an existing computer code for Newtonian flow simulations to that for viscoelastic flow simulations. In this method, the hyperbolic constitutive equation is split such that the term for the convective transport of stress tensor is treated as a source which is updated iteratively. This allows the stress tensors at each grid point to be expressed in terms of velocity gradient tensor at the same location, and the set of stress tensor components is found after inverting a small matrix at each grid point. To corroborate the robustness and accuracy of the grid-by-grid inversion method, we apply it to the 4 : 1 axisymmetric contraction problem. This algorithm is found to be robust and yields accurate results as compared with other finite volume methods. Any commercial CFD packages for Newtonian flow simulations can be easily converted to those for viscoelastic fluids exploiting the grid-by-grid inversion method.