A variational multiscale framework for non-Newtonian fluid models

This dissertation presents stabilized mixed finite element formulations for non-Newtonian models for complex fluids. One class of non-Newtonian fluids is shear-rate dependent fluids that include nonlinear viscosity functions of the second invariant of the rate-of-deformation tensor, thereby resulting in shear-thinning or shear-thickening effects. Another is viscoelastic fluid models that embody elastic stress term as well as viscous stress term, and therefore reveal memory effects via elasticity in the fluid motion. These two model classes can be combined into shear-rate dependent viscoelastic fluid models that are mathematically sophisticated and reflect complicated fluid motions.;Since the complexity of the mathematical constructs in non-Newtonian fluid models deteriorates numerical stability of discretized formulations, advanced numerical methods with enhanced stability properties are required for efficient numerical implementations. In this dissertation, the Variational Multiscale (VMS) framework is employed to derive stabilized mixed formulations for advection-diffusion, shear-rate dependent, viscoelastic, and shear-rate dependent viscoelastic fluid models. The VMS framework leads to a two-level description of the primary variables, leading to coarse-scale and fine-scale problems. Consistent linearization of the fine-scale problem with respect to the fine-scale fields and the use of bubble functions to expand the fine-scale trial and test functions lead to analytical models for the fine-scale. These nonlinear fine-scale models are variationally embedded in the nonlinear coarse-scale stabilized formulations for the various non-Newtonian fluid models. Advanced computational algorithms that are based on quadratic convergence properties of consistent tangent tensors are derived for efficient nonlinear solution of the system of equations.;The new methods are implemented for equal-order linear and quadratic finite elements in two and three-dimensional space (triangular, quadrilateral, tetrahedral and hexahedral elements). The methods are verified via benchmark problems and then extended to human artery models to highlight the significantly distinctive non-Newtonian fluid response in human blood-flow simulations.