**Eric S.G. Shaqfeh**

Stanford University

### The Fast Flow Dynamics of Polymer Solutions As Calculated Using Implicit
Configurational Time Stepping Schemes

Polymer solution dynamics in flows far from equilibrium involves a host
of interesting nonlinear dynamics in systems containing thousands of degrees
of freedom per molecular chain. These dynamics are critical to understanding
many important applications such as fiber spinning, coating, or drag reduction.
Typically these flows are calculated beginning with a bead-spring model
for the polymer which involves constraining forces which become singular
in the limit as the flow tends to pull the chain beyond some finite extensibility.
Solvers for these configurational dynamics can be of two types: stochastic
solvers or Brownian dynamics which try to reproduce a configurational path
or field and preaveraged solvers which within closure approximations allow
an average moment of the configurational field to be determined directly.
For both solvers, time stepping near the singularity in the constraining
force becomes very important for fast flows.

I will discuss a series of algorithms that have been developed to implicitly
time-step these equations and demonstrate that the savings in computer time
per simulation can be more than an order of magnitude over explicit or predictor-corrector
techniques. I will then demonstrate the usefulness of these techniques via
calculations involving three important examples: fast stress relaxation
of polymers following strong extensional flow, critical fluctuations of
polymers in mixed flows near the shear flow boundary, and polymer-induced
turbulent drag reduction.

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