Kinetic theory of gases SG3131
7 ECTS credits
The course has a double goal.
One of them is to give a microscopic background to fluid dynamics. The other
goal is to extend fluid dynamics to a much wider field where either the gas is
so thin or the physical dimensions so small that fluid dynamics and continuum
theories are no longer valid.
Basically, one keeps track of the molecules, their positions and velocities. As
the number of molecules is huge, statistical methods are used.
In
ordinary fluid dynamics there are a few fields (density, velocity and
temperature) in ordinary three-dimensional space. As in kinetic theory
the velocities as well as the positions of the molecules are taken into
account, the basic space is six dimensional phase space. A point is given by
three coordinates in space and three velocity components. There is just one
field in phase space, the density f, giving the density of molecules at
a given position and velocity.
The basic equation is the Boltzmann equation. It is an equation for f
the density of molecules in phase space. The Boltzmann equation is an integro-differential equation.
The mean free path is a very important concept. It is the typical length
traveled by a molecule between two collisions with other molecules. In air at
standard conditions, the mean free path is very small, 0.05 microns. Collisions
are very numerous and the molecules slowly diffuse in space. Ordinary fluid
dynamics is then valid. From the kinetic theory one can then derive expressions for
the macroscopic properties of the gas such as viscosity and heat conductivity.
In
a rarefied gas, however, the mean free path can be considerably larger. The
Boltzmann equation is then still valid, but not ordinary fluid dynamics. In the
extreme case, when the mean free path is large compared to the typical macro
scales, the molecules essentially move freely, and just collide with the walls.
This is the limit of free molecular flow.
A
widely used numerical method is the so-called Direct Simulation Monte Carlo
Method. The motion of a large number of representative molecules is
simulated directly in a computer. It is a method which has many advantages and
often gives a good rough picture of a given flow. It will be used in many of
the course projects.
A widely Another valuable development
is the Lattice Boltzmann Equation. It
is a simplified Boltzmann equation which is used to give a robust numerical
scheme for fluid dynamics.
1. Introduction
2. Basics of the molecular description
· The molecular model
· The mean free path
· Macroscopic variables
· Mean free path derivation of transport properties
3. Basics of kinetic theory
· Velocity distribution functions
· The Boltzmann equation
· Conservation equations
· The equilibrium distribution
4. The direct simulation Monte-Carlo method
· Introduction
· Principles of a DSMC-code
· Computational examples of relaxation in a homogeneous gas
5. Small Knudsen number and Chapman-Enskog theory
· Deriving the Euler and Navier-Stokes equations from the Boltzmann equation
· Viscosity and heat conductivity from molecular interactions
6. Interaction with a solid surface
· Boundary conditions
· Kinetic layers
7. Transition regime flows
· Higher order Knudsen number effects
8. Large Knudsen number
· Collisionless flow
Text book
In the course we shall be using as a text book
Gaskinetic theory by Tamas I. Gombosi,
Cambridge University Press, where
you can order it,
or you can order it from the Swedish Bokus.se
Reference book
For reference purposes we will also be using
G.A. Bird,
Molecular Gas Dynamics and the Direct Simulation of Gas Flows.
PC versions of the Monte Carlo
codes can be found here
Relevant reading for lectures
I - II Gombosi Chapter 1-3, 4.2-4.3, 5, 6.1-6.2 Bird Chapter 1-4
III - IV Gombosi Chapter 6-7 Bird Chapter 3, 5, 7-11
Relevant reading for project
presentations
· 1. Bird Chapter 12.1 - 12.2
· 2. Gombosi Chapter 7.2.3
· 3. Bird Chapter 12.11, Gombosi Chapter 8
· 4. Bird Chapter 5.1 - 5.3, 11.1 - 11.4
· 5. Bird Chapter 14.2 - 14.4
· 6. Bird Chapter 14.2, 14.8 or, Chapter 15.1 - 15.3
· 7. Bird Chapter 12.1 - 12.2, 13.1 or, Chapter 11.1 - 11.3, 12.11
· 8. Gombosi Chapter 4.4
· 9. Bird Chapter 14
Here you can find the lecture notes by Anders Dahlkild in pdf-format
And the lecture
notes by Lars Söderholm in pdf
Exercises on calculating fluid fields from the distribution function.
You can also find a preliminary
set of study questions as a
help to prepare for the oral examination
Anders Dahlkild ; you can send a mailto:ad@mech.kth.se
&
Lars Söderholm.; you can send a mailto:lars.soderholm@mech.kth.se
Last updated: February 1, 2012.