Synopsis for B6a: Viscous Flow

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2A45)

Recommended Prerequisites:

The Part A (second-year) course `Fluid Dynamics and Waves'. Though two half-units are intended to stand alone, they will complement each other. This course combines well with B5 Differential Equations and Applications. The introductory Michaelmas Term course B568a is a prerequisite for both parts of the course, and the material in that course will be assumed to be known.

Overview

Viscous fluids are important in so many facets of everyday life that everyone has some intuition about the diverse flow phenomena that occur in practise. This course is distinctive in that is shows how quite advanced mathematical ideas such asymptotics and partial differential equation theory can be used to analyse the underlying differential equations and hence give scientific understanding about flows of practical importance, such as air flow round wings, oil flow in a journal bearing and the flow of a large raindrop on a windscreen.

Learning Outcomes

Students will have developed an appreciation of diverse viscous flow phenomena and they will have a demonstrable knowledge of the mathematical theory necessary to analyse such phenomena.

Synopsis

Euler's identity and Reynolds' transport theorem. The continuity equation and incompressibility condition. Cauchy's stress theorem and properties of the stress tensor. Cauchy's momentum equation. The incompressible Navier-Stokes equations. Vorticity. Energy. Exact solutions for unidirectional flows; Couette flow, Poiseuille flow, Rayleigh layer, Stokes layer. Dimensional analysis, Reynolds number.Derivation of equations for high and low Reynolds number flows.

Thermal boundary layer on a semi-infinite flat plate. Derivation of Prandtl's boundary-layer equations and similarity solutions for flow past a semi-infinite flat plate. Discussion of separation and application to the theory of flight.

Slow flow past a circular cylinder and a sphere. Non-uniformity of the two dimensional approximation; Oseen's equation. Lubrication theory: bearings, squeeze films, thin films; Hele–Shaw cell and the Saffman-Taylor instability.

1. D. J. Acheson, Elementary Fluid Dynamics (Oxford University Press, 1990), Chapters 2, 6, 7, 8.
2. H. Ockendon and J. R. Ockendon, Viscous Flow (Cambridge Texts in Applied Mathematics, 1995).

   Further reading

3. G.K. Batchelor, An Introduction to Fluid Dynamics (CUP, 2000). ISBN 0521663962.
4. C.C. Lin & L.A. Segel, Mathematics Applied to Deterministic Problems in Natural Sciences(Society of Industrial and Applied Mathematics, 1998). ISBN 0898712297.
5. L.A Segel, Mathematics Applied to Continuum Mechancis(Society for Industrial and Applied Mathematics, 2007). ISBN 0898716209.