Math 801: Topics in Applied Mathematics
Mathematical Aspects of Mixing (Spring 2013)

Lecture Room:	B131 Van Vleck
Lecture Time:	1:20–2:10 MWF
Lecturer:	Jean-Luc Thiffeault
Office:	503 Van Vleck
Phone:	(608)263-4089
Email:
Office Hours:	M 10:45–11:45, Th 1:20–2:20 / catch me after class / by appointment

Syllabus

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 mixing.
• The dynamical systems viewpoint: chaotic dynamics and Lyapunov exponents.
• 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:

• J. Kelliher, Oseledec's Multiplicative Ergodic Theorem (Lecture notes).

Generalized Lyapunov exponents:

• J. Vanneste, Estimating generalized Lyapunov exponents for products of random matrices, Phys. Rev. E 81, 036701 (2010).

Strange eigenmodes and intermittency:

• J. Vanneste, 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, Open-flow mixing: Experimental evidence for strange eigenmodes, Phys. Fluids 21, 023603 (2009).

Homogenization theory:

• A. Crisanti et al., Anisotropic diffusion in fluids with steady periodic velocity fields, J. Phys. A 23, 3307 (1990).

Biomixing:

• Z. Lin, J.-L. Thiffeault, and S. Childress, Stirring by squirmers, Journal of Fluid Mechanics 669, 167–177 (2011).
• J.-L. Thiffeault and S. Childress, Stirring by swimming bodies, Physics Letters A 374 (34), 3487–3490 (2010).
• J. S. Guasto, K. A. Johnson, and J. P. Gollub, Measuring Oscillatory 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).
• C. Darwin, Note on hydrodynamics, Mathematical Proceedings of the Cambridge Philosophical Society 49, 342–354 (1953).

Topological mixing:

• P. L. Boyland, H. Aref, and M. A. Stremler, Topological 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, 417–431 (1988).
• A. Fathi, F. Laudenbach, and V. Poenaru, Thurston's 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, 2012).
• 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, 1975).
• 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, 1989).
• M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology 34 109–140 (1995).

Mix-norms, sources and sinks:

• G. Mathew, I. Mezic, and L. Petzold, A multiscale measure for mixing, Physica D 211, 23–46 (2005).
• C. R. Doering and J.-L. Thiffeault, Multiscale mixing efficiencies for steady sources, Physical Review E 74, 025301(R) (2006).
• G. Mathew, I. Mezic, S. Grivopoulos, U. Vaidya, and L. Petzold, Optimal control of mixing in Stokes fluid flows, J. Fluid Mech. 580 261–281 (2007).
• Z. Lin, J.-L. Thiffeault, and C. R. Doering, Optimal stirring strategies for passive scalar mixing, Journal of Fluid Mechanics 675, 465–476, (2011).
• J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport, Nonlinearity 84 (3), R1–R44 (2012).

Books on differential geometry applied to fluid dynamics and physics:

• R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics (Dover, 1990).
• B. F. Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, 1980).
• R. M. Wald, General Relativity (University of Chicago Press, 1984).
• S. Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity (Wiley, 1972).

Last modified: 19:39 1 April 2021

Background: Coarsening of a binary fluid (Simulations by Lennon Ó Náraigh)

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