• Korea-Australia Rheology Journal
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• v.16 no.4
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• pp.183-191
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• 2004

High Deborah or Weissenberg number problems in viscoelastic flow modeling have been known formidably difficult even in the inertialess limit. There exists almost no result that shows satisfactory accuracy and proper mesh convergence at the same time. However recently, quite a breakthrough seems to have been made in this field of computational rheology. So called matrix-logarithm (here we name it tensor-logarithm) formulation of the viscoelastic constitutive equations originally written in terms of the conformation tensor has been suggested by Fattal and Kupferman (2004) and its finite element implementation has been first presented by Hulsen (2004). Both the works have reported almost unbounded convergence limit in solving two benchmark problems. This new formulation incorporates proper polynomial interpolations of the log­arithm for the variables that exhibit steep exponential dependence near stagnation points, and it also strictly preserves the positive definiteness of the conformation tensor. In this study, we present an alternative pro­cedure for deriving the tensor-logarithmic representation of the differential constitutive equations and pro­vide a numerical example with the Leonov model in 4:1 planar contraction flows. Dramatic improvement of the computational algorithm with stable convergence has been demonstrated and it seems that there exists appropriate mesh convergence even though this conclusion requires further study. It is thought that this new formalism will work only for a few differential constitutive equations proven globally stable. Thus the math­ematical stability criteria perhaps play an important role on the choice and development of the suitable con­stitutive equations. In this respect, the Leonov viscoelastic model is quite feasible and becomes more essential since it has been proven globally stable and it offers the simplest form in the tensor-logarithmic formulation.

• Korea-Australia Rheology Journal
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• v.17 no.3
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• pp.99-110
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• 2005

This work presents results of finite element analysis of isothermal incompressible creeping viscoelastic flows with the tensor-logarithmic formulation of the Leonov model especially for the planar geometry with singular comers in the domain. In the case of 4:1 contraction flow, for all 5 meshes we have obtained solutions over the Deborah number of 100, even though there exists slight decrease of convergence limit as the mesh becomes finer. From this analysis, singular behavior of the comer vortex has been clearly seen and proper interpolation of variables in terms of the logarithmic transformation is demonstrated. Solutions of 4:1:4 contraction/expansion flow are also presented, where there exists 2 singular comers. 5 different types spatial resolutions are also employed, in which convergent solutions are obtained over the Deborah number of 10. Although the convergence limit is rather low in comparison with the result of the contraction flow, the results presented herein seem to be the only numerical outcome available for this flow type. As the flow rate increases, the upstream vortex increases, but the downstream vortex decreases in their size. In addition, peculiar deflection of the streamlines near the exit comer has been found. When the spatial resolution is fine enough and the Deborah number is high, small lip vortex just before the exit comer has been observed. It seems to occur due to abrupt expansion of the elastic liquid through the constriction exit that accompanies sudden relaxation of elastic deformation.