Singular Finite Elements for the Die-Swell Simulation of Viscoelastic Fluids

Abstract

In previous work, we have attempted to apply the singular finite element method (SFEM) developed for Newtonian flow problems to the die-swell simulation of viscoelastic (Giesekus) fluids, and reported that SFEM, with less memory size, gave more accurate results than those obtained by the ordinary decoupled FEM. In this paper, a revised version of above SFEM, i.e. SFEM vpE12, has been presented, where the velocity (v) and stress (E) nodes, as well as the pressure (p) node, are removed from a position of singularity; it implies that asymptotic behaviors of field variables near the singular point (i.e., die-lip) are assumed to be such that v - r 0.5 E ∼ r -0.5 and p - r -0.5 , where r denotes a radial distance from the singular point. Furthermore, as for the stress substitution for momentum equation, we attempted two methods, i.e., the rearranged substitution (RS) method and the direct substitution (DS) method. It is found that SFEM vpE12 with RS (i.e., SFEM vpE12-RS) extends the upper limit of the Weissenberg number (We) of the die-swell flow simulation up to 150, whereas SFEM vpE12 with DS (i.e., SFEM vpE12-DS) fails to converge beyond 1.8 of We; however, starting from converged results obtained by SFEM vpE12-RS at 130 of We, SFEM vpE12-DS simulation is able to converge at the same We of 130, and gives more smoothed and accurate results than the original SFEM especially in the very vicinity of singularity.