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On Using a Lie Group to Represent Increments of Axial Motion in Suspension Flows

On Using a Lie Group to Represent Increments of Axial Motion in Suspension Flows

Eugene Eckstein

Department of Biomedical Engineering, University of Memphis, TN 38152
October 8, 2008, 4:00pm, Manning Hall 201
Refreshments served at 3:30pm, Manning Hall 222

The equation relating increments of displacement over time for the Ornstein-Uhlenbeck process (physicist’s Brownian motion) and persistent random walk was found to fit increments of axial position as observed in a moving reference frame when the sign of the time increment was reversed. The equation then has an exponential growth term and all studies must be done with a targeted distance increment or time of observation.

The main topic is a means of describing this type of data by assuming that the constants of integration for motion as observed in four dimensions (position, axial momentum, balance of kinetic energy and internal energy, and free energy per mass). These constants are to be selected from the Lie group for rotations, SO(3), which can be considered as quaternions showing the change of position on the unit 3-sphere. Gilmore ( treats related ideas in chapter 16 of his book, many chapters of which are available at the web site.

Primarily, he notes that symmetries in the Lie group provide a means of bringing physical principles found by Galileo and Einstein to the organization of such data by means of a differential equation. L.F. Richardson, an editor for G.I. Taylor’s 1921 paper on persistent random walk (Proc. London Math. Soc. Ser. 2, 20, 196), noted that these walks described the separation of two particles.

The unit three-sphere, which surrounds the origin for the moving reference frame, would seem to provide a sound means to understand this remark in respect to the objects in suspension flows. The vector from the origin to any path on the manifold pierces the three sphere and so, the intersection point provides an implicit record of events. These ideas appear to have direct application to far more than suspension flows.

Student Collaborators: J. Lavine, M. Leggas, B. Ma, V. Bhal
Faculty Collaborators: J. Goldstein, J. Marchetta, M. Kiani, D. Goetz
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