Topics in Analysis: Fluid Mechanics
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Figure 12, "Perspective view of Echelon Waves", from `On Ship Waves`_ ([Th87]_).
Contact info
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: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
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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
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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*
- Point vortices ([Ar79]_, [Ar10]_, [Sy49]_, [Ne14]_, [No75]_)
- Blowup criteria in 3D Euler ([BKM84]_)
- Gyres ([CJ15]_, [Ha18]_, [HM18]_)
---- ------ -----------------------------------------------------------------
12 07.01 Week 2 of presentations: *Vorticity and rotation*
- Rotating stars ([Ja13]_; also see [Wo74]_, [Sa14]_)
- Linear stability of shear flows ([KC08]_)
---- ------ -----------------------------------------------------------------
13 14.01 Week 3 of presentations: *Fluids and geometry*
- Navier–Stokes on manifolds ([Sc60]_, [EM69]_, [ADS09]_,
[JOR18]_, [ST18]_)
- The Euler-Arnold equation ([EM70]_, [Ta10]_, [Ta14]_)
---- ------ -----------------------------------------------------------------
14 21.01 Week 4 of presentations: *Famous calculations*
- Stokes drift (canceled)
- Darwin drift ([Yi84]_, [Be85]_; also see [Sa67]_, [Ur53]_)
- Kelvin wake pattern ([St57]_; also see [Th87]_)
---- ------ -----------------------------------------------------------------
15 28.01 Week 5 of presentations: *Other effects*
- Compressible Euler and shock waves ([CM79]_, [Br09]_, [Ev10]_)
- Waves in Magnetohydrodynamics ([Fi16]_)
- Non-Newtonian fluids ([MR05]_, [Ir14]_)
==== ====== =================================================================
References
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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]
Jean-Yves Chemin, `Mathematical geophysics: an introduction to rotating
fluids and the Navier-Stokes equations <geophy_>`_. Oxford University Press,
2006.
.. [CM79]
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]
Tosio Kato, `On classical solutions of the two-dimensional nonstationary
Euler equation <Kato67_>`_. *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]
J. J. Stoker, `Water Waves: The Mathematical Theory with Applications
<Stoker_>`_. Interscience Publishers, Inc., New York, 1957.
.. [Te18]
Gerald Teschl, `Topics in Real and Functional Analysis <TeFA_>`_, 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]
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 <Schedule and lecture notes_>`_ 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:
.. Below is a list of suggested topics, along with those that have already been
taken. You can of course pick another topic related to the class which is not
on the list. In either case, before you start work in earnest send me an
email and we can meet and have a quick chat. In particular I can give you any
references I have. Most of the links below (e.g. to Wikipedia) are just to
give you a rough idea of the topic.
* `Hele-Shaw flow`_
* Applications of Leray–Schauder degree theory (e.g. [Te18]_)
* Unique weak solutions to 2D Euler (Yudovich's theorem)
* Contour dynamics methods for vortex patches
* `Arnold stability theorem`_ for shear flows
* Vortex filaments
* Gerstner_ and/or Crapper waves (explicit solutions in Lagrangian coordinates)
* `Weak solutions to the Euler equations`_ and isometric embeddings
.. _On Ship Waves: https://doi.org/10.1243%2FPIME_PROC_1887_038_028_02
.. _Kelvin wake pattern: https://en.wikipedia.org/wiki/Wake#Kelvin_wake_pattern
.. _Gerstner: https://en.wikipedia.org/wiki/Trochoidal_wave
.. _Stokes drift: https://en.wikipedia.org/wiki/Stokes_drift
.. _Euler-Arnold equations: https://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/
.. _Arnold stability theorem: http://www.depts.washington.edu/bdecon/workshop2012/g_stability.pdf#page=7
.. _Weak solutions to the Euler equations: https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/
.. _Non-Newtonian fluids: https://en.wikipedia.org/wiki/Non-Newtonian_fluid
.. _Shock waves: https://en.wikipedia.org/wiki/Shock_wave
.. _Hele-Shaw flow: https://en.wikipedia.org/wiki/Hele-Shaw_flow
.. _Kato67: https://link.springer.com/content/pdf/10.1007%2FBF00251588.pdf
.. _Beale–Kato–Majda: https://projecteuclid.org/download/pdf_1/euclid.cmp/1103941230
.. _TeFA: https://www.mat.univie.ac.at/~gerald/ftp/book-fa/
.. _original proof: http://matwbn.icm.edu.pl/ksiazki/sm/sm2/sm2114.pdf
.. _geophy: http://search-ebscohost-com.uaccess.univie.ac.at/login.aspx?direct=true&scope=site&db=nlebk&AN=201079
.. _Magnetohydrodynamics: https://en.wikipedia.org/wiki/Magnetohydrodynamics
.. _Gyres: https://en.wikipedia.org/wiki/Ocean_gyre
.. _Stoker: https://archive.org/details/waterwavesthemat033435mbp
.. _On the motion of three vortices: https://doi.org/10.4153/CJM-1949-022-2
.. _Motion of three vortices: https://aip.scitation.org/doi/10.1063/1.862605
.. _Point vortex dynamics in the post-Aref era: http://iopscience.iop.org/article/10.1088/0169-5983/46/3/031401/meta
.. _Dynamics and statistics of a system of vortices: http://jetp.ac.ru/cgi-bin/dn/e_041_05_0937.pdf
.. _Self-similar motion of three point vortices: https://doi.org/10.1063/1.3425649
.. _Remarks on the breakdown of smooth solutions for the 3-D Euler equations: https://projecteuclid.org/download/pdf_1/euclid.cmp/1103941230
.. _The dynamics of waves interacting with the Equatorial Undercurrent: https://doi.org/10.1080/03091929.2015.1066785
.. _Explicit two-dimensional solutions for the ocean flow in arctic gyres: https://link.springer.com/article/10.1007/s00605-018-1198-3
.. _Applying the Stereographic Projection to Modeling of the Flow of the Antarctic Circumpolar Current: https://www.jstor.org/stable/26509096
.. _Toroidal figures of equilibrium: https://www.researchgate.net/publication/234394497_Toroidal_figures_of_equilibrium
.. _Torus-Earth: http://www.aleph.se/andart/archives/2014/02/torusearth.html
.. _Andart: http://www.aleph.se/andart/
.. _Dynamics of a fluid interface equation of motion for Newtonian surface fluids: https://doi.org/10.1016/0009-2509(60)87003-0
.. _Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid: https://projecteuclid.org/euclid.bams/1183530815
.. _Relaxation dynamics of fluid membranes: https://doi.org/10.1103/PhysRevE.79.031915
.. _Incompressible fluid problems on embedded surfaces\: Modeling and variational formulations: https://arxiv.org/abs/1702.02989v3
.. _Navier-Stokes equations on Riemannian manifolds: https://arxiv.org/abs/1812.09015
.. _Groups of Diffeomorphisms and the Motion of an Incompressible Fluid: https://www.jstor.org/stable/1970699
.. _Noether's theorem and the conservation laws for the Euler equations: https://terrytao.wordpress.com/2014/03/02/noethers-theorem-and-the-conservation-laws-for-the-euler-equations/
.. _The Euler-Arnold Equation: https://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation
.. _New derivations of Darwin's theorem: https://doi.org/10.1017/S0022112085000623
.. _Note on added mass and drift: https://doi.org/10.1017/S0022112086000617
.. _The self-propulsion of a deformable body in a perfect fluid: https://doi.org/10.1017/S0022112067002149
.. _Mass transport in gravity waves: https://doi.org/10.1017/S0305004100028140
.. _Lecture Notes on Hyperbolic Conservation Laws: https://descartes.math.psu.edu/bressan/PSPDF/cetraro09.pdf
.. _Plasma Physics: http://farside.ph.utexas.edu/teaching/plasma/Plasma/
.. _Partial Differential Equations: https://bookstore.ams.org/gsm-19-r
.. _Mathematical issues concerning the Navier-Stokes equations and some of their generalizations: http://msekce.karlin.mff.cuni.cz/~malek/new/images/Handbook-prehled.pdf
.. _Rheology and Non-Newtonian Fluids: https://doi.org/10.1007/978-3-319-01053-3
.. _01.10: notes/01.10.pdf
.. _08.10: notes/08.10.pdf
.. _15.10: notes/15.10.pdf
.. _22.10: notes/22.10.pdf
.. _29.10: notes/29.10.pdf
.. _05.11: notes/05.11.pdf
.. _12.11: notes/12.11.pdf
.. _19.11: notes/19.11.pdf
.. _26.11: notes/26.11.pdf
.. _03.12: notes/03.12.pdf