# Advanced Fluid Dynamics

## Unit aims

Understanding the principles governing fluid flow and the mathematical models used to investigate them.

## Unit description

The behaviour of ordinary fluids like oil, water, or air can be understood on the basis of a single equation, due to Navier and Stokes. The description of fluid motion thus amounts to finding solutions to the Navier-Stokes equation, a mathematical problem of almost infinite variability and often staggering complexity. (A look at a weather map should convince you of that.) Solutions to physically relevant problems generally involve some approximation, motivated by physical insight, and based on the identification of the key parameters that determine the solution.

Close to an equilibrium state, the problem can be solved by linearising the equation around it. Far away from such a state flows are often characterised by widely differing length scales. This seemingly complex structure can be used to one's advantage by investigating the solution under a *change *of scales.

Unavoidably, fluid mechanics has broken up into a great number of subfields. However, this course will try to give a more unified view by emphasizing mathematical structures that reappear in different guises in almost all those sub-specialities.

## Relation to other units

This unit is a continuation of the Level 3 Fluid Dynamics unit and an investiagtion of more advanced topics. This unit is self-contained and it is not necessary to have previously attended Level 3 Fluid Dynamics. However familarity with the key themes and ideas of Level 3 Fluid Dynamics would be advantageous.

## Learning objectives

After taking this unit, students should:

- know the basic equations and the underlying concepts
- realise the importance of the Reynolds number and other non-dimensional parameters
- know how to set up the appropriate mathematical equations for a given flow problem
- appreciate the general concepts of stability and scaling

## Transferable skills

Ability to transfer physical questions into well-defined mathematical problems. Understanding the critical parameters of a problem and developing intuition for the behaviour of a system as a function of these parameters.

## Syllabus

1. The governing equation for viscous fluid flow: Conservation of mass; stress and rate of strain tensors; Navier-Stokes equation; dissipation; boundary conditions; vorticity.

2. Simple fluid flows: Solution to the Navier-Stokes equations for simple geometries; steady flows along pipes and between parallel plates; oscillating flows; impulsively started flows; flows with circular streamlines.

3. Dynamical similarity and the Reynolds Number: Dimensionless governing equations and the definition of the Reynolds number; interpretation; simple examples.

4. Flows with negligible inertia: Stokes' equations; corner flows; settling particles; lubrication flows; spreading droplets.

5. Flows with large Reynolds Numbers: Singular perturbations of the Navier-Stokes equations; boundary layer flows; similarity solutions; production of vorticity; separation; wakes.

6. Instabilities: Linear instability theory of shear flows and vortex sheets.

## Reading and References

- L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon 1959
- D.J. Acheson, Elementary Fluid Dynamics, Oxford.
- T.E. Faber, Fluid dynamics for physicists, Cambridge, 1995
- G.K. Batchelor, An introduction to fluid mechanics, Cambridge 1965

**Unit code:** MATHM0600

**Level of study:** M/7

**Credit points:** 20

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

**Lecturer:** Professor Jens Eggers

## Pre-requisites

MATH11009 Mechanics 1, MATH20901 Multivariable Calculus, MATH20402 Applied Partial Differential Equations 2

## Co-requisites

None

## Methods of teaching

Lectures spread over 12 weeks. Regular homework assignments are set.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

- 100% from a 3 hour exam in January

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.