Enhancing numerical stability in simulation of viscoelastic fluid flows at high weissenberg number problem

Now days, simulation of viscoelastic flows at high Weissenberg numbers is one of the most obstacles and important issues for rheologists to observe the rheological properties at sufficiently high weissenberg number. 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. In this research we apply a mathematical transformation, the so-called hyperbolic tangent, on the conformation tensor to bound the eigenvalues and prevent the generation of negative spurious eigenvalues during simulations . The flow of FENE-P fluid through a 2D channel is selected as the test case. Discrete solutions are obtained by spectral/hp element methods which based on the high orders polynomials and have high accuracy for physical instability problems. This enhanced formulation, hyperbolic tangent, prevails the previous numerical failure by bounding the magnitude of eigenvalues in a manner that positive definite is always satisfied. Under this new transformation, the maximum accessible Weissenberg number increases 100% comparing the classical constitutive equation(FENE-P classic).