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Preface' e1 ?) r0 X! c" O
Waves dynamics have fascinated us through out the history of civilization.; Y! M+ S5 G; T& ^5 n8 z! N
Ocean waves like tsunamis, tidal swells, ship bow waves, breaking beach waves
2 J& s) x0 S' T0 u/ jetc. have been studied for centuries. They have even inspired new fields of
7 a% s W- u! C, \2 u6 q8 I/ v/ Cmathematics. There is, however, a different type of waves that have only been6 E, I5 `" D$ V$ M9 @( o. _
scrutinized in the last 50 years. Yet their dynamics are even richer than deep-
0 H1 B; ^; }, D9 R- B; Nwater (ocean) waves. In fact, the system that exhibits these waves is a simple+ E5 q9 G: L$ v* O$ c
prototype hydrodynamic instability that demonstrates many features of other! G- ]7 B8 M- }& W
flow instabilities. As a result, it can become a testing ground for many new$ U) \% }* a4 Z# ]6 t
hydrodynamic concepts/theories.% z! o. O" I7 Q u3 ]! o' e
We are referring to interfacial wave dynamics on a thin film that flows down
, C5 g1 d2 L( h5 I' san inclined plane, the subject of this book. These waves appear on the wind-0 t- T& N) u! \
shields of our cars, in the evaporator tubes of our refrigerators, within indus-7 V# @' V# b2 ?. \( Y" p* h# e3 g
trial/commercial cooling towers and mass-transfer units, during coating pro-, I3 B! v3 |7 I' g0 @4 }6 o
cesses etc. However, we shall not focus on the engineering aspects of wave9 q. ]$ D; H9 y+ h
dynamics but rather on a mathematical theory for its intriguing spatio- tem-, o+ a4 B, h4 ^9 h1 w6 q# g( y
poral dynamics. This description represents a significant extension of classical" A& U4 ?5 ^ {7 w" Q
Orr-Sommerfeld type linear hydrodynamic theory and offers, for the first time,
( j$ {% a5 ]: C: \' c) c) Equantitative delineation of the complex wave dynamics.
. `' c4 U; X `8 K- QOur ability to quantitatively describe complex wave dynamics on thin films is6 \7 X4 M* n9 f
due to a fundamental physical fact - the waves are localized as solitary waves and
. W& m/ P7 S' Bshocks for most,but not all, conditions. Such localized "coherent structures" are1 ^+ x5 ~2 d, I2 y- s/ K( X8 y
also observed in many other extended-domain dynamics but its mathematical4 F) h" }. C2 z+ ^
analysis is most developed for thin films. Mass conservation and viscous effects3 z7 _4 h; {, f4 X/ s+ i5 y
render the dynamics of these structures very different from energy-conserving/ Q( c M' w, x) a" Y6 w/ x
deep-water solitons. Consequently, a new nonlinear mathematical approach, dif-
; z+ y4 v# E% u: N* c$ fferent from inverse scattering and transformation theories, must be formulated+ |3 E( D( N6 N+ o7 ]
Our effort is also simplified by certain physical symmetries of the coherent
5 D3 h" H6 p+ d; @structures that allow their dynamics to be describe by a few discrete dynamic& d: U7 H; _3 {- E
zero modes. As such, the complex spatio-temporal dynamics can be captured by( X% M1 G) O! G, b
low-dimensional dynamical systems. We present these new tools and concepts4 e/ o# p- r0 D5 m, ]3 r2 n- B
for thin-film wave dynamics, as well as some classical ones, in this book.% J, g# ]& L9 Q( ?' ` k6 G
Our new approach hence combines concepts from Dynamical Systems The-
- H: M8 N! R* e1 d0 _) u1 ?% bory, Soliton Spectral Theory and Stochastic Methods with advanced compu-8 v8 k2 W3 x; j H9 n; S3 O
tational methods to explore a classical hydrodynamic instability. It contains
4 R& A$ Y( j# q, d; H! [results from a six-year collaboration at Notre Dame between the authors, after
! J/ E1 m9 V1 i7 S) I; o6 Xvi
5 { u7 H( ~, A6 ^$ pworking independently on the same subject for an earlier six years. During9 s9 F) c/ @5 j" V& x) v! l7 g9 z
both periods, our colleagues, collaborators and students contributed to the re-$ T K/ i+ Q( I) V6 I3 L4 v) W
sults we report here. They include Y.Ben, M.Cheng, E. Kalaidin, S. Kalliadasis,
* e( f' C* q" p L; PD. Kopelevich, M. J. McCready, M. Sangalli, S. Saprikin, V. Shkadov and Y.Ye.& ?1 x8 w9 _$ e' M* X/ P, {
It is our hope that this monograph will trigger similar approaches to other
/ y1 B, H/ @: Z" Z% q6 eflow instabilities.2 r0 g0 O+ p) j( j0 N3 S
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