Rheology Projects:
Property reserving reformulation of constitutive laws for the conformation tensor
It is well known that the conformation tensor should, in principle, remain symmetric positive definite (SPD) as it evolves in time. In fact, this property is crucial for the well-posedness of its evolution equation. In practice this property is violated in many numerical simulations. Most likely, this is caused by the accumulation of spatial discretization errors that arises from numerical integration of the governing equations. This gives rise to spurious negative eigenvalues, causing the conformation tensor to lose its SPD property and Hadamard instabilities to grow. This was an obstacle to early attempts to numerically simulate viscoelastic fluids.
In this study, we address the outstanding problem affecting the numerical simulation of viscoelastic flows at a critical value of the Weissenberg number beyond which no numerical solution can be obtained. This study presents a new approach so-called hyperbolic tangent to preserve both symmetric positive definite of the conformation tensors and also bound the magnitude of eigenvalues. The aim of this study is the development of a mathematical model to preserve both the SPD of the conformation tensor and also to bound the magnitude of the eigenvalues. The hyperbolic tangent formulation of the constitutive equation removes some of the stiffness associated with the standard form of the constitutive equation. We demonstrate that this has the effect of increasing the critical Weissenberg number, thereby delaying the so-called high Weissenberg number problem. The flow of FENE-P fluid through a 2-D channel and a 3-D pipe is selected as a test problem. Discrete solutions are obtained by spectral/hp element methods. The shown figure clearly represents the idea of preserving eigenvalues in a new formulation, named hyperbolic tangent.
