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Preface
! c, X! w1 y8 mWaves dynamics have fascinated us through out the history of civilization.
7 e; L& K! g; k) qOcean waves like tsunamis, tidal swells, ship bow waves, breaking beach waves
) j4 b+ \( x0 M9 tetc. have been studied for centuries. They have even inspired new fields of: A1 J4 z8 a6 Y% f) X
mathematics. There is, however, a different type of waves that have only been/ L) M! |8 B" W! V( q8 o# B4 _
scrutinized in the last 50 years. Yet their dynamics are even richer than deep-# w0 p2 S! W+ [8 v b9 r. U
water (ocean) waves. In fact, the system that exhibits these waves is a simple, n: r$ C) W3 @8 K
prototype hydrodynamic instability that demonstrates many features of other
* T9 g7 u4 ~; t( m6 y% p. \flow instabilities. As a result, it can become a testing ground for many new
4 }: d a. E1 \6 ^- j7 Thydrodynamic concepts/theories.! l1 l2 I. t4 o: z4 U. Y) v( q
We are referring to interfacial wave dynamics on a thin film that flows down
4 G! P1 ]- ~2 ~4 B/ Xan inclined plane, the subject of this book. These waves appear on the wind-, z5 h; m9 v; }. v' u
shields of our cars, in the evaporator tubes of our refrigerators, within indus-& n7 O- A$ C4 K4 L. h# m
trial/commercial cooling towers and mass-transfer units, during coating pro-. x, ?/ z! [! I* b* B3 q' M [
cesses etc. However, we shall not focus on the engineering aspects of wave
) a; K6 M$ ~ fdynamics but rather on a mathematical theory for its intriguing spatio- tem-
2 M! m! Q9 {2 |* l' @0 jporal dynamics. This description represents a significant extension of classical, D6 T+ L1 G6 }+ J3 `8 ~
Orr-Sommerfeld type linear hydrodynamic theory and offers, for the first time,
7 H6 y5 @3 s# y+ squantitative delineation of the complex wave dynamics.$ |* e) G4 ^" r' a, h9 V
Our ability to quantitatively describe complex wave dynamics on thin films is: Q/ _" E3 N2 I! q
due to a fundamental physical fact - the waves are localized as solitary waves and
' ?- `4 c( T# z/ R, Y1 Nshocks for most,but not all, conditions. Such localized "coherent structures" are
; V7 F- F, j' h) P( \7 [, dalso observed in many other extended-domain dynamics but its mathematical
8 g# n. d, z; ^, p2 o3 x" _analysis is most developed for thin films. Mass conservation and viscous effects f" M$ w& _$ O" t
render the dynamics of these structures very different from energy-conserving, O5 P! X0 T& e' [7 t: _0 V8 A
deep-water solitons. Consequently, a new nonlinear mathematical approach, dif-+ t: u. h3 \# M3 I- B7 Q- }0 U3 L
ferent from inverse scattering and transformation theories, must be formulated5 g& G7 F) C' [; u3 v# z
Our effort is also simplified by certain physical symmetries of the coherent2 R4 |/ [3 x' O4 [4 K& F# k
structures that allow their dynamics to be describe by a few discrete dynamic4 o5 Z8 i3 A6 z& Y& W, X& K( O
zero modes. As such, the complex spatio-temporal dynamics can be captured by
0 q; ]& O9 z+ O: O S Tlow-dimensional dynamical systems. We present these new tools and concepts# I+ q- k2 {" B2 S; m. Q+ Q
for thin-film wave dynamics, as well as some classical ones, in this book.
: y8 ^3 J4 s+ O5 w( R$ cOur new approach hence combines concepts from Dynamical Systems The-, k! v. X: w) D! M4 z3 F
ory, Soliton Spectral Theory and Stochastic Methods with advanced compu-: t& G& w! t/ y9 A+ B8 {/ i5 y' ~
tational methods to explore a classical hydrodynamic instability. It contains* N) {/ c# o) n3 E$ r5 a
results from a six-year collaboration at Notre Dame between the authors, after
1 {( U: f3 d; p' N/ {" `& l) ?7 Ovi% \, l/ E' p$ \% g+ L( @; E1 T$ K
working independently on the same subject for an earlier six years. During# m, I& s* C' L( l! n
both periods, our colleagues, collaborators and students contributed to the re-
. T4 t- F! x, n/ }, W$ L% e( nsults we report here. They include Y.Ben, M.Cheng, E. Kalaidin, S. Kalliadasis,; `9 u6 d7 f6 T9 \. P, b
D. Kopelevich, M. J. McCready, M. Sangalli, S. Saprikin, V. Shkadov and Y.Ye.
1 ^- w! W+ M: I Z7 JIt is our hope that this monograph will trigger similar approaches to other
, `, R- V5 s8 i3 F5 H; gflow instabilities.& N% [1 i$ x7 I& z8 l( D
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