A new numerical algorithm for viscoelastic fluid flows : The grid-by-grid inversion method

We present a new algorithm for solving viscoelastic flows with a general constitutive equation. In our approach the hyperbolic constitutive equation is split such that the term for the convective transport of stress tensor is treated as a source. This allows the stress tensor at each grid point to be expressed mainly in terms of the velocity gradient tensor at the same point. Then, the set of six stress tensor components is found after inverting a six by six matrix at each grid point. Thus we call this algorithm the grid-by-grid inversion method. The convective transport of stress tensor in the constitutive equation, which has been treated as a source, is updated iteratively. The present algorithm can be combined with finite volume method, finite element method or the spectral methods. To corroborate the accuracy and robustness of the present algorithm we consider viscoelastic flow past a cylinder placed at the center between two plates, which has served as a benchmark problem. Also considered is the investigation of the pattern and strength of the secondary flows in the viscoelastic flows through a rectangular pipe. It is found that the present method yields accurate results even for large relaxation times.