Lecturer: Julie Staunton

Weighting: 7.5 CATS

Aims:

The module should explain why PDEs (with associated boundary conditions) are an appropriate model for fluids. You should learn how physical ideas and limiting cases can help analyse these PDEs which, in general, cannot be solved. These include the role of the Reynolds number, laminar viscous flow, the boundary layer concept and irrotational flow. The module also prepares you for future applied mathematics modules.

Objectives:

At the end of the module you should be able to

• Recognise and write down the equations of motion for incompressible fluids (the Navier- Stokes equations) and understand the origin and physical meaning of the various terms including the boundary conditions
• Derive Poiseuille's formula and understand the conditions for it to be a valid description of fluid flow
• Use dimensional analysis to analyse fluid flows. In particular, you should appreciate the relevance of the Reynolds number.
• Simplify the equations of motion in the case of incompressible irrotational flow and solve them for simple cases including vortices
• Explain the boundary layer concept

Syllabus:

Introduction

Fluids as materials which do not support shear. Idea of a Newtonian fluid. "Plausibility of from assumption of a relaxation time for stress.

Equations of Motion

Hydrostatics: forces due to pressure and gravity. Hydrodynamics: acceleration, continuity and incompressibility. Euler equation.

Streamlined Flow

Streamlines: Integrating Euler for steady flow along a streamline to give Bernoulli. Derivation of Bernoulli via conservation of energy. Applications of Bernoulli: flux through a hole, Pitot-static tube, aerofoil, waves on shallow water.

Hydrodynamics of Viscous Flow

Forces due to viscosity, Navier-Stokes equation. Derivation of Poiseuille's formula for laminar flow between plates.

Turbulence

Laminar flow only one possibility. Turbulent slugs. Need for dimensionless number, Re, Pressure gradient as a function of Re. 2 Regimes: Physical interpretation of Re as Inertial forces/Viscous forces. Poiseuille works when Re small.

Irrotational Flow

Definition of vorticity and circulation. Importance of irrotational flow, Kelvin's circulation theorem.

Examples of irrotational flow: uniform flow, flow past a cylinder. Derivation of lift on thin aerofoil, as example for Magnus Effect.

Waves on deep water; time-dependent flow potential and time-dependent Bernoulli give dispersion relation. Group velocity of gravity waves. Mention of Kelvin wedge.

Circulation around a cylinder. The vortex. Circulation constant round vortex line, need to close or end on surfaces. Advection of unlike vortices. The vortex ring. Circling of like vortices.

Vortices at edges of wings.

Real Flows

Idea of boundary layer; Boundary layer separation and drag crisis.

Commitment: about 18 Lectures

Assessment: 1 hour examination

Recommended Texts: DJ Tritton, Physical Fluid Dynamics, OUP; TE Faber Fluid Dynamics for Physicists, CUP

Leads from: PX148 Classical Mechanics and relativity; MA250 PDE or PX260 Mathematical Methods for Physicists I

Leads to: PX350 Weather and the Environment, MA3D1 Fluid Dynamics