Nimi Op Opiskelumuoto Aika Paikkakunta Järjestäjä
Navier-Stokes equations I 5 Cr Luentokurssi 2.9.2019 - 16.10.2019


MAST students

Edeltävät opinnot tai edeltävä osaaminen

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.


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




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.

Oppimista tukevat aktiviteetit ja opetusmenetelmät

Lectures and exercise classes


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

Arviointimenetelmät ja -kriteerit

Exam and exercises, course will be graded with grades 1-5

Suositeltavat valinnaiset opinnot

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.