This course is aimed for postgraduate students with their focus on fluid mechanics. The course is an introduction to the kinetic theory of gases. The continuum mechanical equations of fluid dynamics are derived from a kinetic theory approach where the gas is considered a large system of discrete particles, the molecules. The equations of fluid dynamics are shown to result in the continuum limit, but the kinetic theory of gases applies also for micro and nano flows and for rarefied gases. Some basic problems of fluid mechanics are considered in cases when ordinary continuum theory of the gas does not apply.

The course is usually given every second year, depending on the number of expected participants.

After completing this course the student should be able to:

- Give the kinetic theory definitions of the macroscopic continuum properties/variables of a gas.
- State the requirements on a fluid flow for the continuum assumption to be a reasonable approximation.
- Describe the concepts of cross-section and mean free path in a gas and derive an expression for the mean free path.
- Use the mean free path concept to derive an approximate expression for viscosity and heat conductivity in a gas in terms of kinetic variables.
- State the Boltzmann equation and, make an interpretation of the different terms involved.
- State the Maxwellian distribution and when it is valid.
- Give examples of some typical kinetic effects not described by the Navier-Stokes equations.
- Give the main principles of a Direct Simulation Monte-Carlo Simulation (DSMC).
- Describe in broad outline the Chapman-Enskog method to derive the Navier-Stokes equations from the Boltzmann equation at small Knudsen numbers, in particular how viscosity and heat conductivity can be found from the molecular interactions.

The student will be able to describe the connection between the continuum mechanical Navier-Stokes equations for a gas and the kinetic theory description of a gas in thermal non-equilibrium. Also, the student will be able to describe some effects typical to gases at Knudsen numbers of order one or larger, a limit not covered by the Navier-Stokes equations.

About 10 hour lectures.

Project work in groups of 2 students.

Seminars with student project presentations with 2 students per 45 minutes.

Admitted to PhD-program

An advanced course in fluid mechanics on undergraduate level is recommended.

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Course literature

**Gombosi, T.I.**

Gaskinetic Theory, Cambridge University Press, 1994

**Dahlkild, A.A. and Söderholm, L.H.**

Lecture notes in kinetic gas theory, 2011

P, F

- INL1 - Assignment, 1.0 credits, Grading scale: P, F
- PRO1 - Project work, 3.0 credits, Grading scale: P, F
- TEN1 - Oral exam, 3.0 credits, Grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

INL1 Assignment1,0 hp (P, F)

PRO1 Project work 3,0 ho (P, F)

TEN1 Oral exam 3,0 hp P, F

Lists of typical questions at examination are available for the oral exam.

The following items have to be approved in order to obtain a pass on the course:

- Project work and 4-page report on a DSMC-simulation
- Oral examination on kinetic theory of gases

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Third cycle

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Anders Dahlkild

- All members of a group are responsible for the group's work.
- In any assessment, every student shall honestly disclose any help received and sources used.
- In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.