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Preface1 N0 w9 I9 g5 _
Waves dynamics have fascinated us through out the history of civilization.
4 k. @1 o; h- f" c3 L# \* c# UOcean waves like tsunamis, tidal swells, ship bow waves, breaking beach waves; y2 d5 F% Z. f+ j! M8 x
etc. have been studied for centuries. They have even inspired new fields of
. P8 K9 K$ s9 d W4 h% Y" ~; @mathematics. There is, however, a different type of waves that have only been
% _- \, ?$ d7 j6 H, R3 escrutinized in the last 50 years. Yet their dynamics are even richer than deep-
; _ a2 G$ j) Kwater (ocean) waves. In fact, the system that exhibits these waves is a simple
z7 ^% d+ m. N) I) Gprototype hydrodynamic instability that demonstrates many features of other! O5 W/ Q! h u" \7 q7 \* \
flow instabilities. As a result, it can become a testing ground for many new5 i: `5 ?% Z1 c, p( v6 T8 }" [ G1 z
hydrodynamic concepts/theories.4 |2 q) A: z) ^( x) y* O
We are referring to interfacial wave dynamics on a thin film that flows down
* C: w8 {. f% g2 |an inclined plane, the subject of this book. These waves appear on the wind-8 c+ y$ `- K4 \0 S5 A2 ?
shields of our cars, in the evaporator tubes of our refrigerators, within indus-
7 R7 t4 z, R! c" Q1 D) \- Vtrial/commercial cooling towers and mass-transfer units, during coating pro-; t/ @( Z: h* r5 U& d* u1 J6 i
cesses etc. However, we shall not focus on the engineering aspects of wave
# g* S/ ~- p3 d- {- A+ v6 Gdynamics but rather on a mathematical theory for its intriguing spatio- tem-; r7 Y% O& z) F# h
poral dynamics. This description represents a significant extension of classical4 H* b/ n" H: M ?/ O9 u
Orr-Sommerfeld type linear hydrodynamic theory and offers, for the first time,
8 I$ L$ v4 I6 b- n: ~7 ^/ K6 kquantitative delineation of the complex wave dynamics.
% f/ @5 v2 l1 SOur ability to quantitatively describe complex wave dynamics on thin films is0 z* n W* S+ G; q; P2 o, ~% o9 a
due to a fundamental physical fact - the waves are localized as solitary waves and
$ h5 q _: T( _7 Lshocks for most,but not all, conditions. Such localized "coherent structures" are8 l& Q% y! S4 W! W% h6 o( E
also observed in many other extended-domain dynamics but its mathematical
& g4 g( O) k' i$ ^9 p1 x1 Ranalysis is most developed for thin films. Mass conservation and viscous effects
) A! K8 A! D* ^+ Orender the dynamics of these structures very different from energy-conserving
3 b" P2 P% h- T2 p, H; b, X3 _deep-water solitons. Consequently, a new nonlinear mathematical approach, dif-3 _. O6 }4 p$ m4 Z% ^: y6 S+ U
ferent from inverse scattering and transformation theories, must be formulated
- { R, e' O) W! w4 ]0 aOur effort is also simplified by certain physical symmetries of the coherent
5 l5 U6 A% b4 G) v' P8 Gstructures that allow their dynamics to be describe by a few discrete dynamic" k" d3 s3 H! c" o% Q5 r
zero modes. As such, the complex spatio-temporal dynamics can be captured by
9 Q3 \0 r7 v: |' mlow-dimensional dynamical systems. We present these new tools and concepts0 s4 G- f! D; ^9 i# L
for thin-film wave dynamics, as well as some classical ones, in this book.
1 [; \, ?1 t' N5 C# aOur new approach hence combines concepts from Dynamical Systems The-
5 v" O' x+ P* Q. S( m, gory, Soliton Spectral Theory and Stochastic Methods with advanced compu-
8 y4 `8 [. J; y5 }tational methods to explore a classical hydrodynamic instability. It contains
s3 u0 }6 Q0 y* q& H$ {: `4 r7 Kresults from a six-year collaboration at Notre Dame between the authors, after3 ]+ `, ~" K: s& k( _
vi$ w( X9 D( F* O% q; k
working independently on the same subject for an earlier six years. During
! V' K- i6 `7 G0 p* eboth periods, our colleagues, collaborators and students contributed to the re-) u+ G! }. U3 y
sults we report here. They include Y.Ben, M.Cheng, E. Kalaidin, S. Kalliadasis,
8 C2 f5 {5 [5 [/ v {' c' oD. Kopelevich, M. J. McCready, M. Sangalli, S. Saprikin, V. Shkadov and Y.Ye.
, X+ p# d% V& v* f- DIt is our hope that this monograph will trigger similar approaches to other
4 L" b6 z5 W6 Q) k# k1 j1 uflow instabilities.
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