# Fluid Dynamics

## 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.

## Unit description

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 an aircraft wing, motion of vortices in the atmosphere, bubbles rising in a liquid, liquid jets, and waves in a tank. Some demonstrations of various flows may be included if there is interest.

## Relation to other units

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

## Learning objectives

After taking this unit, students should:

- 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.
- 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.
- be aware of the wide range of applications of fluid mechanics to many practical situations in industry and the environment.
- appreciate how a specific flow fits into the wider context of a physical problem

## 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.

## 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 and streamlines. Material derivative. Mass conservation. Kinematic boundary condition. Particle acceleration. Revision of vector calculus and introduction of tensor notation. Streamfunctions for an incompressible fluid. (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)

## Reading and References

(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.

**Unit code:** MATH33200

**Level of study:** H/6

**Credit points:** 20

**Teaching block (weeks):** 1 (1-12)

**Lecturer:** Dr Richard Porter

## Pre-requisites

MATH11009 Mechanics 1, MATH20901 Multivariable Calculus, MATH20001 Methods of Complex Functions, MATH20402 Applied Partial Differential Equations 2.

## Co-requisites

None

## Methods of teaching

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.

## Method of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 90% Examination
- 10% Assessed Coursework

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.