 Kaisa_2012_3_photo by Veikko Somerpuro

### Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Mon 2.9.2019
10:15 - 12:00
Tue 3.9.2019
12:15 - 14:00
Wed 4.9.2019
10:15 - 12:00
Mon 9.9.2019
10:15 - 12:00
Tue 10.9.2019
12:15 - 14:00
Wed 11.9.2019
10:15 - 12:00
Mon 16.9.2019
10:15 - 12:00
Tue 17.9.2019
12:15 - 14:00
Wed 18.9.2019
10:15 - 12:00
Mon 23.9.2019
10:15 - 12:00
Tue 24.9.2019
12:15 - 14:00
Wed 25.9.2019
10:15 - 12:00
Mon 30.9.2019
10:15 - 12:00
Tue 1.10.2019
12:15 - 14:00
Wed 2.10.2019
10:15 - 12:00
Mon 7.10.2019
10:15 - 12:00
Tue 8.10.2019
12:15 - 14:00
Wed 9.10.2019
10:15 - 12:00
Mon 14.10.2019
10:15 - 12:00
Tue 15.10.2019
12:15 - 14:00
Wed 16.10.2019
10:15 - 12:00

### Material

Lecture notes and a summary

## Other

#### Exercise set 1 Exercise set 1.pdf Exercise 1 model solutions.pdf

#### Exercise set 2 Exercise set 2.pdf Exercise 2 model solutions.pdf

#### Exercise set 3 Exercise set 3.pdf Exercise 3 model solutions.pdf

#### Exercise set 4 Exercise set 4.pdf Exercise 4 model solutions.pdf

#### Exercise set 5 (slightly modified) Exercise set 5.pdf Exercise 5 model solutions.pdf

#### Exercise set 6 (corrected) Exercise set 6.pdf Exercise 6 model solutions.pdf

#### Exercise set 7 (Ex. 7.6 modified) Exercise set 7.pdf Exercise 7 model solutions.pdf

### Description

MAST students

 Vector analysis I (MAT21003) and Vector analysis II (MAT21020). Alternatively, Matemaattiset apuneuvot I-III (FYS1010, FYS 1011, FYS1012). No prior knowledge on fluid dynamics is assumed. The student is, however, assumed to be familiar with high school physics.
 Introduction to Differential Geometry (MAST31017) gives some proofs that are skipped in Vector analysis I-II and this course. It also aids the understanding of many of the concepts of fluid dynamics.

The course gives an introduction to the Navier-Stokes equations and fluid dynamics in general.

Varying

The course gives a mathematical introduction to fluid dynamics which describes the movement of fluids (liquids, gases and plasmas). Fluid dynamics is used in countless disciplines of physics and engineering and is also a major area of applied mathematics. Our main model of interest if the set of Navier-Stokes equations which describes, e.g., the flow of water. The course material is intended to also be accessible to Master level students who major in physics, meteorology, astronomy, atmospheric sciences and related fields.

We will start the course from the following three basic axioms:

1) Newton’s second law (force equals the rate of change of momentum).

2) The continuum assumption (matter in the fluid is continuously distributed).

3) Mass and energy are neither created nor destroyed.

Different extra assumptions then lead to different models of fluid dynamics. The set of axioms 1)-3) is believed to be very accurate for many macroscopic phenomena (even though, e.g., 2) is highly inaccurate at an atomic scale). We will spend some time on an important basic model, the Euler equations. Towards the end of the course, we discuss some limitations of the Euler equations and formulate a more realistic model, the Navier-Stokes equations.

In the course we will encounter many concepts of vector analysis, such as divergence, curl and the Stokes and Gauss formulas, in a natural, physically motivated context. A lot of emphasis is put on developing a geometric intuition for the objects studied.

We will follow the lecture notes of the course. They, in turn, follow loosely the textbook A. Chorin and J. E. Marsden: A Mathematical Introduction to Fluid Mechanics. Other good mathematically oriented introductions include e.g. the following freely available lecture notes:

Popular textbooks geared towards physicists include e.g.:

• L. D. Landau and E. M. Lifschitz: Fluid Mechanics. Second Edition (1987)
• D. J. Acheson: Elementary Fluid Dynamics (1990)

A collection of useful instructional videos of the National Committee for Fluid Mechanics Films is found at http://web.mit.edu/hml/ncfmf.html

 Lectures and exercise classes