MATH 280 Winter 2012

TR 10-11:20 am, MSB 2112

Almost 200 years old, the Navier-Stokes equations provide a mathematical description of the motion of classical fluids such as air and water. Many fluids are composed of mixtures of different materials which interact on different length scales. For example biological fluids such as mucus and cytoplasm consist of water with an immersed polymer network. Understanding the behavior of complex fluids presents additional mathematical, modeling, and computational challenges not encountered in classical fluid mechanics due to the multi-scale and multicomponent nature of the materials. What type of model to use depends on the spatial scale of interest and the questions being asked.

In this course, we will introduce both continuum and multi-scale models of complex fluids. We will discuss analytical solutions which provide insight into the behavior of these models and the mathematical structure of the equations. We will present different computational techniques for simulating more complex problems. The students will gain a solid grounding in complex fluids, and they will be exposed to recent research papers on the application of these theories to cell motility, micro-scale mixing, active suspensions and gels.

The target audience is students who have taken Mathematical Fluid Dynamics (MAT 221A) or Advanced Fluid Mechanics (ECH 253A) and/or participants in the proposed Complex Fluids RFG 2011-2012. However, the course will be self-contained so as to make it accessible to students who did not take these courses..

Instructor: | Becca Thomases |

** Announcements:**

- Becca's last lecture Tues. 2/28, after which the class will consist of individual presentations
- Individual presentations schedule:
- Thurs. March 1 - Jun Allard
- Tues. March 6 - Ryan Halabi
- Thurs. March 8 - David Stein
- Tues. March 13 and Thurs. March 15 TBA
- Ben Fogelson, Jeff Bronson, and Owen Lewis still need to schedule times. We can schedule 2 people per lecture or else schedule a make-up day. Please see me.

**In class presentations **

- Lecture on Stokes-Oldroyd-B numerics
- Lecture on viscoelastic instabilities
- Lecture on mixing - note: movies unavailable at this time

**Topics for individual/group study** including some possible papers to focus on. When you have picked a topic or a paper please contact me and we will schedule a time for your discussion

- Swimming
- Forces and Torques on Rotating Spirochete Flagella, Jing Yang, Greg Huber, and Charles W. Wolgemuth, PRL 2007
- Force-free swimming of a model helical flagellum in viscoelastic fluids, Bin Liu, Thomas R. Powers, and Kenneth S. Breuer, PNAS 2011

- Suspensions
- Fluctuations, Dynamics, and the Stretch-Coil Transition of Single Actin Filaments in Extensional Flows, V. Kantsler and R. Goldstein, PRL 2012
- Chemotaxis Driven Instability of a Confined Bacterial Suspension, T. Kasyap and D. Koch, PRL 2012
- The shape of an elastic filament in a two-dimensional corner flow, N. Autrusson, L. Guglielmini, S. Lecuyer, R. Rusconi, and H. Stone

- Biological Rheology
- Physics of Actin Networks. 1. Rheology of Semi-Dilute F-Actin, K. Zaner, Biophysical Journal 1995
- Particle-Tracking Microrheology of Living Cells: Principles and Applications,D. Wirtz, Annu. Rev. Biophys. 2009
- Multiple Particle Tracking and Two-Point Microrheology in Cells, J. Crocker and B. Hoffman, Methods in Cell Biology 2007
- Elasticity, dynamics and relaxation in biopolymer networks, K. Kroy, Current Opion in Colloid and interface science, 2006

- Mixing
- Using multiscale norms to quantify mixing and transport (a review article), Nonlinearity, 2011
- Stirring by swimming bodies, J.-L. Thiffeault and S. Childress, Physics Letters A, 2010
- Stirring by squirmers, Z. Lin, J.-L. Thiffeault and S. Childress, Journal Fluid Mech. 2011

- Cells
- A model of cytoplasmically driven microtubule-based motion in the single-celled C- elegans embryo, Tamar Shinar, Miyeko Mana, Fabio Piano, and Michael J. Shelley, PNAS 2011
- Cell Motility Resulting from Spontaneous Polymerization Waves, K. Doubrovinski and K. Kruse, PRL 2011
- Optical Rheology of Biological Cells, F. Wottawah
et al. , PRL 2005 - How Changes in Cell Mechanical Properties Induce Cancerous Behavior, P. Katira
et al. , PRL 2012 - Shape Dynamics and Rheology of Soft Elastic Particles in a Shear Flow, T. Gao
et al. , PRL 2012