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225 pages

Language:

English

The study of fluid mechanics is fundamental to modern applied mathematics, with applications to oceans, the atmosphere, flow in pipes, aircraft, blood flow and very much more.

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The study of fluid mechanics is fundamental to modern applied mathematics, with applications to oceans, the atmosphere, flow in pipes, aircraft, blood flow and very much more. This text provides an introduction to the mathematical approach to this subject and to many of its main ideas, based on material typically found in most university courses. So, firstly, the methods and fundamental results, for a general fluid, are presented, and, secondly, the lift generation of aerofoils is analysed by using complex potentials. Numerous worked examples are provided throughout, as are many set exercises.

- Introduction and Basics
- The continuum hypothesis
- Streamlines and particle paths
- The material (or convective) derivative
- The equation of mass conservation
- Pressure and hydrostatic equilibrium
- Euler’s equation of motion (1755)
- Exercises 1

- Equations: Properties and Solutions
- The vorticity vector and irrotational flow
- Helmholtz’s equation (the ‘vorticity’ equation)
- Bernoulli’s equation (or theorem)
- The pressure equation
- Vorticity and circulation
- The stream function
- Kinetic energy and a uniqueness theorem
- Exercises 2

- Viscous Fluids
- The Navier-Stokes equation
- Simple exact solutions
- The Reynolds number
- The (2D) boundary-layer equations
- The flat-plate boundary layer
- Exercises 3

- Two dimensional, incompressible, irrotational flow
- Laplace’s equation
- The complex potential
- Simple (steady) two-dimensional flows
- The method of images
- The circle theorem (Milne-Thomson, 1940)
- Uniform flow past a circle
- Uniform flow past a spinning circle (circular cylinder)
- Forces on objects (Blasius’ theorem, 1910)
- Conformal transformations
- The transformation of flows
- Exercises 4

- Aerofoil Theory
- Transformation of circles
- The flat-plate aerofoil
- The flat-plate aerofoil with circulation
- The general Joukowski aerofoil in a flow
- Exercises 5

- Appendixes
- Appendix 1: Biographical Notes
- Appendix 2: Check-list of basic equations
- Appendix 3: Derivation of Euler’s equation (which describes an inviscid fluid)
- Appendix 4: Kelvin’s circulation theorem (1869)
- Appendix 5: Some Joukowski aerofoils
- Appendix 6: Lift on a flat-plate aerofoil
- Appendix 7: MAPLE program for plotting Joukowski aerofoils

- Answers
- Index

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