# B5.3 Viscous Flow - Material for the year 2020-2021

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The Part A (second-year) courses 'Waves and Fluids' and 'DEs 2' would be desirable. This course combines well with B5.2 Applied Partial Differential Equations. Though the two units are intended to be self-contained, they will complement each other.

16 lectures

### Assessment type:

- Written Examination

Viscous fluids are important in so many facets of everyday life that everyone has some intuition about the diverse flow phenomena that occur in practice. This course is distinctive in that it shows how quite advanced mathematical ideas such as asymptotic expansions 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 raindrop on a windscreen.

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.

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.

- D. J. Acheson,
*Elementary Fluid Dynamics*(Oxford University Press, 1990), Chapters 2, 6, 7, 8. - H. Ockendon and J. R. Ockendon,
*Viscous Flow*(Cambridge Texts in Applied Mathematics, 1995). - R. P. Feynman, R.B. Leighton, M. Sands,
*The Feynman Lectures on Physics, volume II*, (Addison Wesley, 1964) Chapter 41 "The Flow of Wet Water" (http://www.feynmanlectures.caltech.edu/II_41.html)

- M. Van Dyke,
*An Album of Fluid Motion*(Parabolic Press, 1982). ISBN 0915760029. - G.K. Batchelor,
*An Introduction to Fluid Dynamics*(CUP, 2000). ISBN 0521663962. - C.C. Lin & L.A. Segel,
*Mathematics Applied to Deterministic Problems in Natural Sciences*(Society of Industrial and Applied Mathematics, 1998). ISBN 0898712297. - L.A Segel,
*Mathematics Applied to Continuum Mechancis*(Society for Industrial and Applied Mathematics, 2007). ISBN 0898716209.