Simulation of nonisothermal and time-dependent viscoelastic flows

Abstract

The need to predict the interactions between the microstructure of polymeric fluids and the macroscopic flow field in polymer processing operations has lead to the development of many numerical methods for the simulation of viscoelastic flows. The two main obstacles to the use of these tools in the quantitative analysis of polymer processing operations are the need for accurate constitutive equations and the computational efficiency of the numerical methods. The goal of the first part of this thesis is to develop constitutive equations appropriate for modeling nonisothermal viscoelastic flows and to examine the influence of temperature variations on the stress in the fluid. The pseudotime method developed by Crochet and Naghdi (1978) and by Sarti (1977) is used to obtain thermodynamically consistent constitutive equations for the internal energy, the stress, and the heat flux vector. Fourier"s Jaw is assumed for the thermal conductivity, and the internal energy is chosen to depend only on the temperature of the fluid. The non isothermal analog to the Giesekus constitutive equation for the polymer contribution to the stress is then de1ived with the pseudotime approach. The constitutive equations are combined with conservation equations for mass. momentum, and energy to model the nonisothermal viscoelastic extrudate swell problem. For contraction-shaped dies, the final thickness of the extrudate is shown to depend on both the thermal and the strain histories of the fluid emerging, at the die exit. In this thesis second part of this thesis, efficient time integration algorithms are developed within the framework of examining the linear stability of isothermal viscoelastic two dimensional steady flows in complex geometries to three-dimensional perturbations. In this approach, a steady base state is found by discretizing in two dimensions with the discrete elastic-viscous split stress gradient (DEVSS-G) finite element formulation of the momentum-continuity pair combined with either the streamwise upwind Petrov-Galerkin (SUPG) or discontinuous Galerkin (DG) finite element discretization of the hyperbolic constitutive equation for the polymer contribution to the stress tensor. The resulting set of nonlinear algebraic equations is solved by using the Newton-Raphson method. The stability of these states is determinined by time integration of the evolution equations for infinitessimal perturbations governed by the equations of motion and the constitutive equation for the stress linearized about the base state. Two time integration methods are investigated: the {}-method operator splitting scheme and a fourth-order Runge-Kutta method. Both schemes reduce to a solution to a modified Stokes problem and an evaluation of the time-dependent constitutive equation. The overall efficiency of both methods is extremely high, as is the potential for implementation on parallel computers. An algorithm also is presented for calculating eigenvalues with the largest real parts that combines time integration of the linearized equations with a Krylov subspace method to accelerate the calculation of the eigenvalues. Although this method does not dramatically reduce the computational cost over time integration alone. it does provide a more complete analysis of the eigenspectrum. The linear stability of the flow of an Oldroyd-B model is analyzed with both the time integration methods and the hybrid time integration/Krylov algorithm for flow through several model geometries. First, the circular Couette problem is examined to benchmark the methods, and the stability results presented here are in good agreement with those obtained by other methods of analysis. The second model geometry is flow around an isolated cylinder confined in a channel. This investigation is motivated by the observation that the flow of a Boger fluid in this geometry undergoes a transition from a two-dimensional flow to a three-dimensional flow at a critical value of the Weissenberg number (Mc Kinley, 1993 ). Contrary to experimental observations, the flow with the Oldroyd-B model is predicted to be stable for all values of the Weissenberg number investigated. We</=0. 75. The flow of the Oldroyd-B model around a closely spaced linear array of cylinders is predicted to undergo a transition from a two-dimensional steady state to a three-dimensional state at a value of the Weissenberg number that is in good agreement with the experimental observations of Liu ( 1997) for a Boger fluid. An energy analysis of the interactions between the long-time perturbations and the base state flow indicates that the mechanism for this instability is similar to the mechanism proposed by Joo and Shaqfeh ( 1994) for the non-axisymmetric transition observed in the viscoelastic circular Couette flow.