Nonlinear Dynamics of Surface-Tension-Driven Instabilities
Pierre Colinet, Jean Claude Legros, Manuel G. Velarde
price USD 310.- (yes, that's not a typo)
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.
Surface tension effects and interfacial dynamics are important
sources of nonlinear instability in numerous physical settings.
Anyone who has gazed at the rainbow patterns of oil slicks in a
parking lot puddle will appreciate the subtle beauty that a thin
film can yield. Anyone who has tried to spray paint a metal surface
evenly on a hot summer day will appreciate the dynamics that a thin
layer of fluid can exhibit.
The starting point for this text is the well-known phenomenon of
Rayleigh-Benard convection, in which the dynamics of a layer of
fluid heated from below can bifurcate from dissipative heat
transport to convective heat transport via `rolls' of alternating
cells. The primary source of instability in the transition from
conduction to convection is the Marangoni effect --- the change of
surface tension with temperature. This effect allows tangential
stresses to effect motion of the fluid at the surface.
From this starting point, the authors identify a plethora of
fascinating phenomena with similar types of instabilities. For
example, a thin layer of fluid can break up into convection cells
with piecewise-linear interfaces ranging from square to hexagonal to
more complex forms. Such systems are presumably at work in
geological patterns seen in permafrost distribution and also in
mineral distributions observable in evaporated lake beds. (The text
contains some excellent pictures of this phenomenon.)
The body of the text is an analysis of instabilities in systems
marked by such surface tension effects. Particular attention is paid
to patterns and bifurcations.
The text is certainly suitable for a graduate-level topics course.
Although the material is ostensibly orbiting about fluid-dynamical
and thermo-dynamical phenomena, expertise in these topics is by no
means prerequisite. The authors give an accessible introduction to
the tools needed in the initial chapters of the text. Later chapters
develop linear stability, and [weak] nonlinear instabilities,
including monotonic and oscillatory instabilities. There is a strong
experimental component to the text: this is one of its strengths.
There is also an effort to extend the ideas and applications to
differential equations of a broader class. Later chapters give applications of the methods and
techniques to real and complex Ginzburg-Landau equations, nonlinear
Schroedinger equations, Kuramoto-Sivashinsky equations, and
Korteweg-de Vries equations.
There is not much to complain about in this well-written,
well-illustrated text, beyond the well-inflated list price. Anyone
in applied dynamical systems will find this text written in a
familiar language, which the Preface by I. Prigogine helpfully
explains is ``the modern language of Nonlinear Physics.''