Barriers and Transport in Unsteady Flows: A Melnikov Approach

Sanjeeva Balasuriya
SIAM 2017, 276 pages
PRICE (PAPERBACK) £75.00 ISBN 978-1-611974-57-7

The author writes well and it is obvious from the outset that he has a deep extensive knowledge of what he writes about. The style is also readable and the book is peppered with humour and useful asides.  That said, technically it is not at all an easy read.  If you need an introduction to fluid dynamics, then this is not the book for you as it contains no traditional fluid dynamics at all, the problems tackled here concern boundaries and special sub-domains that need to be modelled in detail well after the fundamental larger problem has been set up and partially at least, solved.  If you need an introduction to dynamical systems, then you will not find that here either as the book assumes that this knowledge is there already.  This is securely a research text on a very specialist topic.  The title itself although completely right will hardly attract the general reader and it is also not suitable for the engineer or applied physicist with limited mathematical knowledge equivalent to elementary calculus and algebra, even though they might call themselves fluid dynamicists.  The book is a well written very comprehensive account of a particular approach to modelling special rather idealised kinds of flow.  These flows are assumed smooth in the sense of being encapsulated within the solutions to differential equations subject to boundary conditions, and these collections of solutions lie inside spaces that mathematicians label as local stable manifolds; they obey conditions that govern the existence of stable perturbations from equilibrium.  The Melnikov theory sounds specialist but is reasonably general, being a perturbation for determining whether a dynamic equilibrium is or is not chaotic.  Here though it is being used solely to analyse rather specialist flows.

The opening chapter is very dense in terms of material and provides a brief but thorough run through of how Melnikov theory is applied to all kinds of different flows. This might persuade you that learning how to apply them to your own research area is a good idea, or it might convince you to leave Melnikov theory alone.  The second chapter introduces the intricacies of Melnikov theory.  This is quite detailed but also general, not specific to fluid applications, but it is essential to understand for the specialist applications later.  The style of the chapter is theorem – proof and it is here that the general reader by whom I really mean fluid dynamics researcher will see whether the book is going to be suitable for them. There is no fluid mechanics application in Chapter 2, but there is in the rest of the book.

In one sense Chapter 3 may be regarded as the guts of the book. Here the Melnikov method is applied to flux across boundaries.  These fluxes can be impulsive, periodic or vary so they can be represented by a Fourier series.  For each application, the emphasis is on modelling the flow across barriers.  Chapter 4 is on optimising flow, and the final chapter on controlling flow.  The Melnikov approach is essentially a perturbation method and hence a linearisation, but it makes full use of integral transforms.  The technical aspects of the mathematics are quite advanced throughout the entire text.  The book is firmly addressed to those familiar with final year undergraduate or graduate level mathematics.

One minor quibble; the typeface is rather small and where there are no equations a page of writing is a strain certainly on my old eyes and I would guess many of yours too. This text is really aimed at researchers in fundamental fluid mechanics and can be used as a reference book.  The bibliography is extensive (438 entries) and for this limited market it is an excellent text.

Phil Dyke FIMA

Book review published directly onto IMA website (December 2017)


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