Math 801: Topics in Applied Mathematics
Mathematical Aspects of Mixing (Spring 2013)
B131 Van Vleck|
|Lecture Time: ||
1:20–2:10 MWF |
|Office: ||503 Van Vleck|
|Office Hours: ||M
10:45–11:45, Th 1:20–2:20 / catch me after class / by appointment|
This course will focus on the mathematical theory of mixing in the context of fluid dynamics. Topics will include:
- Diffusion and random walks.
- The ergodic theory viewpoint: ergodicity, weak and strong
- The dynamical systems viewpoint: chaotic dynamics and Lyapunov
- The PDE viewpoint: homogenization theory.
- The probabilistic viewpoint: card shuffling and cut-offs.
- The topological viewpoint: topological entropy and mapping classes.
- Physical examples: stirring viscous fluids, biomixing.
Schedule of Topics
All lectures are available in
a single file
in PDF (14MB) or a
smaller DjVu file
(3.36MB, see DjVu format).
Bibliography and Resources
Filament models, linear flows:
J.-L. Thiffeault, Scalar
Decay in Chaotic Mixing, in Transport and Mixing in Geophysical
Flows, Lecture Notes in Physics 744, 3-35, 2008 (Springer,
Berlin). (Proceedings of the Gran Combin Summer School, Valle
d'Aosta, Italy, 14-24 June 2004.)
Lyapunov exponents and Oseledec's theorem:
Generalized Lyapunov exponents:
Strange eigenmodes and intermittency:
of passive-scalar decay: Strange eigenmodes in random shear
flows, Phys. Fluids 18, 087108 (2006).
P. Haynes and
J. Vanneste, What
controls the decay of passive scalars in smooth flows?,
Phys. Fluids 17, 097103 (2005).
E. Gouillart, O. Dauchot, J.-L. Thiffeault, and S. Roux,
mixing: Experimental evidence for strange eigenmodes,
Phys. Fluids 21, 023603 (2009).
Z. Lin, J.-L. Thiffeault, and S. Childress,
by squirmers, Journal of Fluid Mechanics 669, 167–177
J.-L. Thiffeault and S. Childress,
Stirring by swimming bodies, Physics Letters A 374 (34),
J. S. Guasto, K. A. Johnson, and J. P. Gollub,
Velocity Fields Due to Swimming Algae (2010) [video].
J. Clerk-Maxwell, On
the Displacement in a Case of Fluid Motion, Proceedings of the
London Mathematical Society s1–3, 82–87 (1869).
Note on hydrodynamics,
Mathematical Proceedings of the Cambridge Philosophical
Society 49, 342–354 (1953).
P. L. Boyland, H. Aref, and M. A. Stremler,
fluid mechanics of stirring, Journal of Fluid Mechanics
403, 277–304 (2000).
W. P. Thurston, On
the geometry and dynamics of diffeomorphisms of
surfaces, Bull. Amer. Math. Soc. (N.S.) 19,
A. Fathi, F. Laudenbach, and
work on surfaces (Princeton University Press, 2011).
English translation of the 1979 French original.
B. Farb and
D. Margalit, A
Primer on Mapping Class Groups (Princeton University Press,
J.-L. Thiffeault and M. D. Finn,
Topology, Braids, and Mixing in
Fluids, Philosophical Transactions of the Royal Society A
364, 3251–3266 (2006).
M. D. Finn and J.-L. Thiffeault,
Topological optimization of rod-stirring devices,
SIAM Review 53 (4), 723–743 (2011).
- J. S. Birman, Braids, Links and Mapping Class Groups, Annals of
Mathematical Studies 82 (Princeton University Press,
- A. J. Casson and S. A. Bleiler, Automorphisms of Surfaces after
Nielsen and Thurston, London Mathematical Society Student
Texts 9 (Cambridge University Press, 1988).
- P. L. Boyland, "Isotopy Stability of Dynamics
on Surfaces" (1999).
- P. L. Boyland and J. Franks, Notes
on Dynamics of Surface Homeomorphisms (University of Warwick,
- M. Bestvina and
M. Handel, Train-tracks
for surface homeomorphisms, Topology 34
Mix-norms, sources and sinks:
G. Mathew, I. Mezic, and
L. Petzold, A
multiscale measure for mixing, Physica D 211, 23–46
C. R. Doering and J.-L. Thiffeault,
efficiencies for steady sources, Physical Review E
74, 025301(R) (2006).
G. Mathew, I. Mezic, S. Grivopoulos, U. Vaidya, and L. Petzold,
control of mixing in Stokes fluid flows, J. Fluid
Mech. 580 261–281 (2007).
Z. Lin, J.-L. Thiffeault, and C. R. Doering,
stirring strategies for passive scalar mixing,
Journal of Fluid Mechanics 675, 465–476,
multiscale norms to quantify mixing and transport,
Nonlinearity 84 (3), R1–R44 (2012).
Books on differential geometry applied to fluid dynamics and physics: