Annual Report 2008.pdf - SAMSI

Dali Zhang
University of Calgary
Department of Mathematics and Statistics
dzhang@math.utah.edu
“Inverse Electromagnetic Problems for Microstructured Media”
Hong Zhou
Naval Postgraduate School
Department of Applied Mathematics
hzhou@nps.edu
“Mathematical Studies of Nematic Liquid Crystal Polymers”
Complex fluids, or materials that share both liquid-like and solid-like properties, are theoretically
and computationally challenging. An important family of complex fluids is macromolecular
materials (e.g. nematic liquid crystal polymers) and nanocomposites which are technologically
useful materials in our life, from bullet-proof vests, airbags to ultra-fast optoelectronic visual
display devices, artificial muscles for robots, and sensors. The macroscopic properties of these
polymer materials depend on both their structure and their liquid phase processing. The
dynamics of nematic liquid crystal polymers has been widely modeled by the kinetic Doi-Hess
theory in which the orientational probability density function evolves according to a nonlinear
Smoluchowski (Fokker-Planck) equation. The Smoluchowski equation provides a rich variety of
mathematical problems and we have investigated it theoretically for pure equilibria, extensional
flow-induced equilibria, equilibria of dipolar ensembles, and effect of weak shear. Our studies
provide solid mathematical foundations for further numerical and theoretical investigations.
Luoding Zhu
Indiana University-Purdue University Indianapolis
Department of Mathematical Sciences
lzhu@math.iupui.edu
“Scaling Laws for Drag of a Compliant Body in an Incompressible Viscous Flow”
Motivated by an important discovery on the drag scaling law (the four-thirds power law) of a
flexible fiber in a flowing soap film by Alben, Shelley and Zhang (Nature 420, 479 (2002)) at
high Reynolds numbers (2; 000 < Re < 40; 000), we investigate drag scaling laws at moderate Re
for a compliant fiber tethered at the midpoint submerged in an incompressible viscous flow using
the Immersed Boundary (IB) method. Our work shows that the scalings of drag with respective
to oncoming flow speed vary with Re and the range of a dimensionless parameter _ that
measures the relative importance of fluid kinetic energy and body elastic potential energy. In
particular, the exponents of the power laws gradually decrease from approximately two to
approximately four-thirds as Re decreases from 10 to 800 for _ in a certain range.