Multiscale simulation of viscoelastic flows : applications to kinetic theory models of polymer melts and liquid crystalline polymers

Abstract

Knowing and understanding the dynamics and molecular configurations of polymer molecules is important for efficient process design and novel product development. Much research has been focused on combining molecular simulations and traditional fluid mechanics computations to simulate the behavior of polymeric liquids in a fabrication process. These simulation approaches require solution of the coupled set of the equations of change, the governing equations from kinetic theory, and the flux expressions that map molecular configurations to macroscopic quantities. Most complex flow simulations so far make use of a mixed finite element method to calculate the velocity field, with stress tensor evaluated by using a stochastic simulation method. This so-called CONNFFESSIT approach suffers from both a large memory requirement and stochastic noise. This thesis focuses on the development and application of a fully deterministic numerical approach for computing viscoelastic flows with constitutive descriptions based directly on diffusion equations from kinetic theory. The numerical approach is based on an operator splitting time integration method that decouples the calculation of microstructure by solution of a hyperbolic diffusion equation from the velocity and pressure field evolution, which is obtained by solution of a generalized Stokes problem. The generalized Stokes problem is written in the DEVSS-G formulation, where a direct interpolation of the components of the velocity gradient tensor is introduced. The efficiency and robustness of this numerical method is demonstrated through calculating the viscoelastic flows of a modified Doi model for liquid crystalline polymer and a number of reptation models for polymer melts in different flow geometries.

(cont.) Simulations of the original Doi model with the Maier-Saupe potential in a pressure-driven channel flow by Nayak showed disclination formation due to the shear-rate-dependent frequencies of the tumbling/wagging states of the Doi model in a shear or mixed shear flow. The lack of an instrinsic length scale in the model leads to an infinitesimal structure refinement that eventually causes numerical instabilities. In this thesis, the effect of concentration variation is incorporated to develop a modified Doi model for introducing an intrinsic length scale through translational diffusion. This changes the mathematical characteristics of the spatial variation of the underlying diffusion equation from that of a hyperbolic equation to that of an elliptic equation. The resulting elliptic diffusion equation is then solved by using a local discontinuous Galerkin method, where an auxillary variable is introduced to rewrite the elliptic diffusion equation into a pair of formal, hyperbolic equations, which in turn is solved by the standard discontinuous Galerkin method. Unlike the original Doi model, a steady state is reached for a variety of De. Although there is structure variation across the channel width, the director profiles point uniformly along the flow direction. The lack of disclination formation may be rectified by introducing Frank elasticity into the modified Doi model ...