(6ct) Enabling Microscopic Simulators to Perform System-Level Analysis of Viscoelastic Flows

State-of-the-art simulations for viscoelastic flow are currently limited in their ability to perform accurate analysis of stability and flow transitions in viscoelastic fluids of industrial importance. This is primarily due to the need to couple directly the macroscopic conservation equations with a microscopic kinetic theory model in order to accurately describe such flows. While this "hybrid" approach makes it possible to study coarse-grained models of polymeric liquids, there is no simple path either to obtaining long-term dynamics for a set of flow parameters or to providing accurate analyses of stability and flow transitions due to the unavailability of closed equations to which numerical techniques may be applied.

To address this problem, we utilize the coarse time-stepper based approach to stability and bifurcation analysis, which has shown potential to enable models at a 'fine' (microscopic/stochastic) level of description to be used to perform modeling tasks at a 'coarse' (macroscopic/systems) level [1]. The motivation for such an analysis is a framework for obtaining macroscopic time information from appropriately initialized calls to a microscopic simulator for only short times. The process of casting either a Fokker-Planck equation or an equivalent stochastic differential equation from polymer kinetic theory as a time-stepper allows for obtaining stationary states and performing stability/bifurcation analysis of the unavailable closed form macroscopic system. We present several examples that focus on application of this method at equilibrium and for homogeneous and complex flows.

Equilibrium Bifurcation Diagram of the Doi Model with the Onsager Excluded Volume Potential [2]

We present the equilibrium Doi model with the Onsager excluded volume potential as a system of dynamical equations for the basis functions chosen for discretization in orientation space and without any closure approximations. While a time-stepper based description only allows access to the stable steady states of the problem, with an iterative, matrix-free Newton solver and an equation for pseudo-arc length continuation, we were able to obtain the bifurcation diagram for the system at equilibrium, capturing both the stable and unstable states of the isotropic and nematic branches. We also show that most of the eigenvalues of the linearized system lie in a tight cluster about zero, with only a few eigenvalues leaving this cluster near a fold bifurcation.

Dilute Solution of Bead-spring Chains in Homogeneous Shear and Uniaxial Elongational Flows

The stochastic differential equation for a free draining bead spring chain with 5 FENE springs undergoing steady homogeneous shear or uniaxial elongation flow was used to construct a time-stepper for the stress contribution of each chain segment to total fluid stress. For a given set of chain segment stresses, the time-stepper evolved a representative ensemble of chain configurations over a fixed time horizon, before returning the evolved ensemble averaged chain segment stresses. This time-stepper was used in conjunction with an iterative, matrix-free Newton solver to obtain stationary states for the unclosed model that are in excellent agreement with the steady states computed from a dynamic simulation. The eigenvalues of the system at steady state for a range of Weissenberg numbers were shown to lie in a tight cluster, which allowed the use of an inexact Newton method to obtain stationary states from a stochastic model.

Pressure-Driven Flow of Non-Interacting Rigid Dumbbells in a Planar Channel [2]

In this work, a micro-macro simulation of a non-homogeneous flow was recast as a black-box integrator that took a given distribution of dumbbell orientations for the flow domain and returned an evolved distribution after integration over some specified time horizon. The integrator also computed consistent flow fields at each intermediate time-step. Having formulated the black box code, we were then able to make calls to it from a Newton-GMRES solver for the purpose of obtaining the steady state of the underlying nonlinear system, given an initial guess of isotropy and without any modification to the micro-macro simulation algorithm.

[1] I. G. Kevrekidis, C. W. Gear, and G. Hummer. Equation-free: The computer-aided analysis of comptex multiscale systems. AICHE Journal, 50(7):1346?1355, 2004.

[2] Z. Anwar, and Robert C. Armstrong. Using Newton-GMRES for viscoelastic flow time-steppers. J. Non-Newtonian Fluid Mech., Submitted.