(401f) New Finite Element Formulations for Viscoelastic Fluid Flows

Complex fluid flow simulations are important in several industrial and biological applications, e.g., polymer processing, ink-jet printing, and human as well as artificial organs, and they pose several numerical challenges. These flows are governed by the conservation of mass and momentum, and the constitutive equation, which in our case is written in terms of the conformation tensor. In this work, two different new formulations to simulate these flows are presented and validated in benchmark problems. The first formulation is suited for large-scale computations, whereas the second one alleviates the long-standing high Weissenberg number problem associated with the viscoelastic fluid flows.

In the first formulation, the four-field Galerkin/Least-Squares (GLS4) stabilized finite element method is presented. This method yields linear systems that can be solved easily with iterative solvers (e.g., the Generalized Minimum Residual method), and also allows the use of equal-order interpolation functions that can be conveniently and efficiently implemented on modern distributed-memory cache-based clusters. The governing equations are converted into a set of first-order partial differential equations by introducing the velocity gradient as an additional unknown. Thereby four unknown fields---pressure, velocity, conformation, and velocity gradient, are solved using linear shape functions. It is shown that the mesh-convergence of the proposed method is comparable to the state-of-the-art DEVSS-TG/SUPG method and yields accurate results at lower computational cost.

The second formulation---the log-conformation formulation--- replaces the conformation tensor unknown by its logarithm. This guarantees the positive-definiteness of the tensor given by its physical nature, and it is able to capture sharp elastic stress boundary layers. The implementation presented in literature thus far requires changing the evolution equation for the conformation tensor into an equation for its logarithm, and are based on loosely coupled solution procedures; here a simpler yet very effective approach to implement the log-conformation formulation in a DEVSS-type code for a generalized constitutive model is presented and the equations are solved in a coupled way by Newton's method.