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Fluid Dynamics 3 (MATH 33200)

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Administrative Information

  1. Unit number and title: MATH 33200 Fluid Dynamics 3
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 11/12
  5. First Given in this form: 1996
  6. Unit Organiser: Jens Eggers
  7. Lecturer: Prof. J. Eggers
  8. Teaching block: 1
  9. Prerequisites: Knowledge of vector calculus and complex functions covered in the second year unit MATH 20900 Calculus 2; also MATH 20402 Applied Partial Differential Equations 2 and first year MATH 11009 Mechanics 1 .

Unit aims

The course aims to provides the student with the basic mathematical background and tools to model fluid motion and calculate the flow of an ideal fluid in a variety of situations. The course will develop a physical understanding of the important aspects that govern fluid flows that can be observed in a variety of situations in everyday life.

General Description of the Unit

This unit introduces many of the fundamental aspects of fluid dynamics, developing the mathematical theory behind ideal (inviscid) fluid flows. The theory is applied to a variety of situations that allow the calculation of the fluid flow and its properties.

The unit demonstrates how mathematics can be used to model complex physical phenomena and illustrates how an applied mathematician uses and develops approximations which capture the essential features of realistic phenomena that are observable in the world around us. Examples include: the lift on a aircraft wing, the flow down a bathtub plughole, bubbles rising in a liquid, hydraulic jumps and bores in a fluid flowing within a channel. Some demonstrations of various flows are included.

Relation to Other Units

The ideas of this unit are developed further in the Level M unit Advanced Fluid Dynamics.

Teaching Methods

Lectures including illustrations and some demonstrations of fluid flows. Worksheets and examples classes follow up some applications of the material covered in the lectures. Regular homework assignments are set and marked.

Learning Objectives

After taking this unit, students should:

  1. be familiar with and able to manipulate the mathematics of a continuum model of fluid flow. This includes how to describe the kinematics of the motion, the notion of fluid pressure and the equations expressing the conservation of mass and momentum within the flow.
  2. be able to solve a variety of fundamental fluid flow problems using a variety of techniques introduced during the course. These include the theory of flow hydraulics and surface water waves as well as applications of potential theory and some complex-variable techniques.
  3. be aware of the wide range of applications of fluid mechanics to many practical situations in industry and the environment.
  4. be able to appreciate how to model other physical systems.

Assessment Methods

The assessment mark for Fluid Dynamics 3 is calculated from a 2 and half-hour written examination in April consisting consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted be used in this examination.

Award of Credit Points

Credit points are gained by:

  • either passing the exam,
  • or getting an examination mark of 30 or over and also handing in satisfactory attempts at 50% of the homework assignments given during the unit.

Transferable Skills

The student will learn some of the skills involved in mathematical modelling: namely, transforming a real physical problem into a mathematically tractable form and then being able to interpret and communicate the results of the calculation. The unit will also develop and give practice of various analytical and problem-solving techniques.

Texts

(followed by their shelf numbers in the Queen's Building and Physics Libraries)

  •  Paterson, A First course in Fluid Dynamics, Cambridge University Press QA911 PAT, 47.00 PAT
  • Acheson, Elementary Fluid Dynamics, Oxford University Press TA357 ACH
    very good, brief text, also does viscous fluids
  • Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press QA911 BAT, 47.00 BAT, not very well written, but contains lots of gems

Syllabus

(The number of lectures for each portion of the course is provided only as a rough guide)

Flow kinematics - The idea of a continuum. Pathlines, streamlines and streaklines. Material derivative. Mass conservation. Kinematic boundary condition. Particle acceleration. Revision of vector calculus and introduction of tensor notation. Streamfunctions for an incompressible fluid. Relative motion near a point, rate of strain and vorticity. (6 lectures)

Flow dynamics of an incompressible, inviscid fluid - Euler equation, Momentum integral theorem. Dynamic boundary condition. Energy equation, Steady Bernoulli's theorem. Hydrostatics and pressure, Archimedes' principle. Simple applications including the hydraulics of channel flow, Bores and hydraulic jumps, flows through diverging and converging nozzles, Jets impacting a wall. The vorticity equation, Kelvin's circulation theorem and the persistence of irrotational flow. (8 lectures)

Irrotational flows - Velocity potential. Unsteady Bernoulli's theorem. Sources and images. Kinetic energy. Motion of a sphere. D'Alembert's paradox. Added mass.   Complex potentials, the method of images and the use of conformal mappings. Impact theory. Free streamline theory. (10 lectures)

Rotational flows - Vortex kinematics and the motion of point vortices. Karman vortex street. Blasius theorem. Brief discussion of lifting wings and flight of aeroplanes. (6 lectures)

Gravity waves - Free surface motion. Dispersion relation. Group velocity. Refraction. Standing waves.(2 lectures)

 

Course Home Page

http://www.maths.bris.ac.uk/~majge/fluids3.html