Finite element methods for viscoelastic fluid flow simulations: Formulations and applications
Coronado, Oscar M.
Doctor of Philosophy
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, momentum, and conformation equations. In this thesis, two different new formulations to simulate these flows are presented and validated with benchmark problems. This thesis introduces the four-field Galerkin/Least-Squares ( GLS4) stabilized finite element method, which is suited for large-scale computations, because it yields linear systems that can be solved easily with iterative solvers, and use equal-order interpolation functions that increase implementation efficiency on distributed-memory 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 computed using linear interpolation functions. The mesh-convergence of GLS4 is comparable to the state-of-the-art DEVSS-TG/SUPG method and yields accurate results at lower computational cost. The log-conformation formulation, which alleviates the long-standing high Weissenberg number problem associated with the viscoelastic fluid flows, replaces the conformation tensor unknown by its logarithm (Fattal and Kupferman 2004). This guarantees the positive-definiteness of the tensor, and helps in capturing sharp elastic stress boundary layers. Previous implementations are based on loosely coupled solution procedures; here a simpler yet very effective approach to implement the log-conformation formulation in a fully-coupled DEVSS-type code is presented. As an application example, the dynamics of a liquid drop, immersed in a liquid medium under shear flow, is studied. The interface is tracked while preserving the volume of the drop by using the isochoric domain deformation method, where the mesh is treated as an incompressible elastic pseudo-solid (Xie et al. 2007). All governing equations are solved in a coupled fashion using the DEVSS-TG/SUPG finite element method. The critical conditions after which the drop will continue to deform until breakup and the influence of inertia and viscoelasticity on the drop deformation and on the critical conditions are predicted first using a 2-D formulation, which is then extended to 3-D.