Topics in Analysis: Fluid Mechanics


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Figure 12, “Perspective view of Echelon Waves”, from On Ship Waves ([Th87]).

Contact info

Email

miles.wheeler@univie.ac.at

Website

https://mat.univie.ac.at/~wheeler/

Office

3.120, Oskar-Morgenstern-Platz 1

Office hours

By appointment

Class time

Mondays 13:15–14:45 and 15:00–15:45

Class location

Seminarraum 12, Oskar-Morgenstern-Platz 1, 2.Stock

Course description

Fluid mechanics has long been intertwined with both pure and applied mathematics, from the early history of complex analysis to more recent developments in areas such as harmonic analysis, integrable systems, dynamical systems, and the calculus of variations. After deriving the Euler and Navier–Stokes equations from first principles, we will cover several topics of mathematical interest, motivated in part by the interests of the class. No prior knowledge of fluid mechanics is assumed.

Prerequisites: Real and complex analysis, ordinary differential equations. Familiarity with functional analysis and/or partial differential equations may be helpful but is not required. Absolutely no prior knowledge of fluid mechanics (or physics more generally) is assumed.

Schedule and lecture notes

The dates below link to (rough, handwritten) notes for the lectures. For the presentations there are links to the references.

Week

Date

Topics

1

01.10

Introduction, Lagrangian and Eulerian points of view

2

08.10

Conservation of mass, incompressible Euler equations, vorticity

3

15.10

Circulation, steady potential flow in two dimensions

4

22.10

Classical solutions for 2D Euler [Ka67]

5

29.10

Classical solutions for 2D Euler continued

6

05.11

The Navier–Stokes equations in three dimensions

7

12.11

Existence of Leray–Hopf weak solutions (Part II of [Ch06])

8

19.11

Existence of Leray–Hopf weak solutions continued

9

26.11

Boundary layers and singular perturbation theory ([CM79])

10

03.12

Water waves ([St57])

11

10.12

Week 1 of presentations: People who volunteered to go first

12

07.01

Week 2 of presentations: Vorticity and rotation

13

14.01

Week 3 of presentations: Fluids and geometry

14

21.01

Week 4 of presentations: Famous calculations

15

28.01

Week 5 of presentations: Other effects

References

For the lectures

  • Our derivation of the incompressible Euler and Navier–Stokes equations is standard, and versions of it can be found in almost any good textbook on fluid mechanics. I have been interpolating between [CM79] and [Ba00], with a little bit of [Ar62], but unfortunately none of these is freely available online through the university library.

  • The freely available lecture notes [MW16] provide brief derivations of the incompressible Euler and Navier-Stokes equations, and also include a proof of the existence of Leray-Hopf weak solutions in 3D for bounded domains.

  • We used [Ka67] as a reference for the existence of classical solutions to the Euler equations in 2D. For weaker solutions, see for instance [CP12]; this is also a potential presentation topic.

  • For the existence of weak solutions to the Navier–Stokes equations we followed “Part II” of [Ch06], which is available online through the university library (you should be able to save all 70 pages of Part II to a pdf in one go).

  • The freely available textbook [Te18] contains most if not all of the functional analysis background we will need. The Schauder fixed-point theorem (Theorem 18.7) is proved using (the beautiful and extremely useful) Leray–Schauder degree theory. Those uninterested in the degree theory may want to look at Schauder’s shorter original proof, or at the still shorter proofs in other textbooks (e.g. Theorem 11.1 in [GT98]).

  • Our brief discussion of boundary layers followed [CM79]. Much more information (from a mathematical point of view) can be found in the online lecture notes [Ng15].

  • Our brief discussion of water waves was somewhat similar to “Part I” of [St57].

Ar62

R. Aris. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover Publications, New York, 1962.

Ba00

G. Batchelor. An Introduction to Fluid Dynamics (Cambridge Mathematical Library). Cambridge: Cambridge University Press, 2000.

Ch06(1,2)

Jean-Yves Chemin, Mathematical geophysics: an introduction to rotating fluids and the Navier-Stokes equations. Oxford University Press, 2006.

CM79(1,2,3,4)

A. J. Chorin and J. E. Marsden, A mathematical introduction to fluid mechanics. Springer-Verlag, New York-Heidelberg, 1979.

CP12

C. Marchioro and M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids. Springer, 2012.

GT98

David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Springer, 2015.

Ka67(1,2)

Tosio Kato, On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Rational Mech. Anal. 25: 188–200, 1967.

MW16

S. Mishra and F. Weber, Topics in Mathematical and Computational Fluid Dynamics, https://www2.math.ethz.ch/education/bachelor/lectures/fs2016/math/macfd.html

Ng15

Toan Nguyen. Lectures notes for Math 597F, https://nttoan81.wordpress.com/category/math-597f-topics-on-boundary-layers/, 2015.

St57(1,2,3)

J. J. Stoker, Water Waves: The Mathematical Theory with Applications. Interscience Publishers, Inc., New York, 1957.

Te18(1,2)

Gerald Teschl, Topics in Real and Functional Analysis, Graduate Studies in Mathematics, Volume XXX, Amer. Math. Soc., Providence, (to appear).

For the presentations

ADS09

Marino Arroyo and Antonio DeSimone. Relaxation dynamics of fluid membranes. Physical Review E 79.3, 031915 (2009).

Ar79

Hassan Aref. Motion of three vortices. The Physics of Fluids 22, 393 (1979).

Ar10

Hassan Aref. Self-similar motion of three point vortices. Physics of Fluids 22, 057104 (2010).

Be85

T. Brooke Benjamin. Note on added mass and drift. Journal of Fluid Mechanics 169, 251-256 (1985).

Br09

Alberto Bressan. Lecture Notes on Hyperbolic Conservation Laws. (2009).

BKM84

J. T. Beale, T. Kato, and A. Majda. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys., 94:1, 61-66 (1984).

CJ15

Constantin, A., Johnson, R.S.: The dynamics of waves interacting with the Equatorial Undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015)

EM69

Ebin, David G.; Marsden, Jerrold E. Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid. Bull. Amer. Math. Soc. 75:5, 962-967 (1969).

EM70

David G. Ebin and Jerrold Marsden. Groups of Diffeomorphisms and the Motion of an Incompressible Fluid. Annals of Mathematics Second Series 92:1, 102-163 (1970).

Ev10

Lawrence C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Volume 19, Amer. Math. Soc., Providence (2010).

Fi16

Richard Fitzpatrick. Plasma Physics. (2016)

Ha18

SV Haziot. Explicit two-dimensional solutions for the ocean flow in arctic gyres. Monatshefte für Mathematik, 2018

HM18

Susanna V. Haziot and Kateryna Marynets. Applying the Stereographic Projection to Modeling of the Flow of the Antarctic Circumpolar Current. Oceanography 31:3, 68-75 (2018)

Ir14

Fridtjov Irgens. Rheology and Non-Newtonian Fluids. Springer International Publishing Switzerland, 2014.

KC08

Kundu, Pijush K., and Ira M. Cohen. Fluid Mechanics. 4th ed. Amsterdam ; Boston: Academic Press, 2008.

Ja13

Jardetzky, W. S. Theories of figures of celestial bodies. Courier Corporation, 2013. Harvard

JOR18

Thomas Jankuhn, Maxim A. Olshanskii, Arnold Reusken. Incompressible fluid problems on embedded surfaces: Modeling and variational formulations. arXiv preprint, (2018).

MR05

J. Málek and K. R. Rajagopal. Mathematical issues concerning the Navier-Stokes equations and some of their generalizations. (2019)

Ne14

Paul K Newton. Point vortex dynamics in the post-Aref era. Fluid Dynamics Research 46:3 (2014).

No75

E. A. Novikov. Dynamics and statistics of a system of vortices. Zh. Eksp. Teor. Fiz 68, 1868–1882 (1975).

Sa67

P. G. Saffman. The self-propulsion of a deformable body in a perfect fluid. Journal of Fluid Dynamics 28:2, 385-389 (1967).

Sa14

Anders Sandberg. Torus-Earth. Andart (blog), 2014.

Sc60

L.E. Scriven. Dynamics of a fluid interface equation of motion for Newtonian surface fluids. Chemical Engineering Science 12.2 (1960): 98-108.

ST18

Maryam Samavaki, Jukka Tuomela. Navier-Stokes equations on Riemannian manifolds. arXiv preprint (2018).

Sy49

J. L. Synge. On the motion of three vortices. Canadian J. Math. 1, (1949). 257–270

Ta10

Terence Tau. The Euler-Arnold equation. terrytao.wordpress.com (2014)

Th87(1,2)

William Thomson. On Ship Waves. Proceedings of the Institution of Mechanical Engineers 38:1, 409-434 (1887).

Ta14

Terence Tao. Noether’s theorem and the conservation laws for the Euler equations. terrytao.wordpress.com (2010)

Ur53

F. Ursell. Mass transport in gravity waves. Mathematical Proceedings of the Cambridge Philosophical Society, 49:1, 145-150 (1953).

Wo74

C-Y. Wong. Toroidal figures of equilibrium. The Astrophysical Journal 190, 675-694 (1974).

Yi84

Chia-Shun Yih. New derivations of Darwin’s theorem. Journal of Fluid Mechanics 152, 163-172 (1985)

Presentation topics

The assessment for the course is based on in-class presentations. See the schedule above for the 13 topics that were ultimately picked by students, as well as links to the references they used. Below is a list of some other suggested topics which were not chosen: