# University of Minnesota

Aerospace Engineering and Mechanics

**Spring 2001 Seminar Series**

*Analysis and Control of Mixing*

*Abstract*

**Consider a three-dimensional closed container that contains an
incompressible fluid, say water. Introduce ink at some location inside the
container. Roughly speaking, mixing of ink and water will be better if fluid
particle trajectories are more complicated. Fluid particle trajectories are
governed by a three-dimensional system of time dependent volume-preserving
ordinary differential equations. This puts the subject of fluid mixing within
the context of dynamical systems and control theory of volume-preserving
systems. In this talk I will present some elements of the theory of analysis
and control of mixing. It is, in fact, the theory of measure-preserving systems
with specific objectives, typically coming from ergodic theoretic
considerations. The mixing objectives such as partition and Kolmogorov-Sinai
entropy will be considered, a simple prototypical example of optimal control
solved and applied in the context of a recent experiment. Another possible
objective is maximization of flux between different regions of fluid domain. A
prototypical example in which this objective is used will be presented. It
involves reduced-order modeling using vortex elements. Ergodic theory concepts
are also useful in visualization: I will present an idea for visualization of
non-mixed regions in laminar flows. Experimental visualization results based on
this analysis obtained by Sotiropoulos (Georgia Tech) and collaborators will be
presented. Finally, the above analysis will be tested on design of a micromixer
- a promising area for application of these dynamical systems-based ideas. I
will argue that an actively-controlled design has advantages for mixing in
microchannels over the competing strategies.**

### Friday, April 27, 2001

209 Akerman
Hall

2:30-3:30 p.m.

### Refreshments served after the seminar in
227 Akerman Hall.

Disability accomodations provided upon request.

Contact Kristal Belisle, Senior
Secretary, 625-8000.