Pattern Formation at Interfaces

SpringerWienNewYork

CISM COURSES AND LECTURES

Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider - Wien

The Secretary General Bernhard Schrefler - Padua

Executive Editor Paolo Serafini - Udine

The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 513

PATTERN FORMATION AT INTERFACES

EDITED BY PIERRE COLINET UNIVERSITE LIBRE DE BRUXELLES, BRUSSELS, BELGIUM ALEXANDER NEPOMNYASHCHY TECHNION-ISR. INST. OF TECHNOLOGY, HAIFA, ISRAEL

This volume contains 113 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2010 by CISM, Udine Printed in Italy SPIN 12821907

All contributions have been typeset by the authors.

ISBN 978-3-7091-0124-7 SpringerWienNewYork




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Interfacial Patterns and Waves in Liquid Layers and Thin Films



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+ FY2 V(MK( (M(

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FV( Y+( Y
=6F Y=6F N H

M=MC =

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Y( #(: CM (#( T: T C(2 ( #C FV( =FC=M = :6YFY CM FV(   = K M:( Y(( C <2 (F2 FV( + =CCFT = FV( ) (
(CM F 6KV VCV( F(+(

F6 ( FV M FV( +( C+V( T = FV( K M:( WM=FV(  =6Y ( +( CY FV( +V(M=(M=M = F( Y = CM(N Y(( C <2 CVF V=Y( (KV MCY CY V=(#(  = ( K=+CK F(:2 CM#=# CM + (( (MFC  (# +=

FC=M = FV( K=V=2 (FFCM + =+( FC(Y CFV  YY2 FV(   M: Y=6F  H

M=MC (0(KFY2 M: 

#CF FC=M  CMYF
CCFT = )6C:

CYCM =M  YY  Y W F(MF FC#( (+ M FC=M = FVCY +V(M=(M=M  Y 


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4

Pierre Colinet

C#(M Y ( T Y CM <"XX
T  (Y /V=Y=M /V=Y=M2 <"XX2 (#(M FV=6V FV( H

M=MC (0(KF  Y C#(M CFY M ( ==CM FV(  F( = = 4 = H

M=MC CM FV( <"5L'Y

  3  Z(F  FV( =K +C T )= CM K M:(   RCVF  F( Y = CM( CM  YY #C((:  =
=#( 4=6 F(YT = =VM D6YV M: WM(FF( >=Y=C2 H,/ ,M FV(Y( (KF6 (Y2 ( C =YFT
( K=MK( M(: CFV FV( +V(M=(M==T

M: =:(CM = FV(   H

M=MC )=Y (M(

F(: CM C6C:  T( V( F(:  =
(= Y(( C * =  =
=#( ,M FV( = ( K Y(2 V( =M  YF 6K F6 (Y ( ( ; YF =
Y( #(:
T >(M C DG(M : DG(M :2 <L< F FV(
(CMMCM = FV( K(MF6 T V(MK( FV( M ( = H

M=MCDG(M : )=Y2 (#(M FV=6V FV(C K=

(KF CMF( + (F FC=M CM F( Y = FV( H

M=MC (0(KF  Y C#(M 6KV  F( D=K 2 <XE

  3  Z(F  V( =M  + FF( M =
Y( #(:
T YV :=

+VT2 +( +(M :CK6 T F= FV(  (( Y6  K( = C6C:  T( V( F(:  =
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Interfacial Patterns and Waves in Liquid Layers and Thin Films

5

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T Y+( CM2 Y(#(

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Y(: =M FV( Y=K (: :C06Y(CMF(  K( = +V Y(;(: FV(= T V #( + =#(M 6CF( )(C
( M: :(6 F( F= M6( CK T YC6 F( + =
(Y CM#=# CM K=+( CMF(  K( YV +(Y M: :TM CK  (#(MFY Y6KV Y 6+F6 ( M: K= (YK(MK( WF FV( Y ( (#(2  FFCK( D=F? MM (FV=:Y2 + =K((:CM =M FV(
YCY = :CYK (F( =:(Y = FV( K((


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( CMF(  K(Y Y M=M?( = FVCK M(YY =
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=6M: T K=M:CFC=MY 6YF
( (+ (YY(: /V(Y( (KF6 (Y C (K6YC#(T
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((M

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( 6Y(: :=M F= #( T Y  YK (Y2 (#(M CM FV( Y6
CK =(F(

M( ,+= F MFT2 FV( ( (CYFY =   M =T
(F((M FV( + =
(Y = V( F F

MY( CM#=#CM FV( =K +C CFT2 M:  YY F

MY( YY=KC F(: CFV K=M K(MF

FC=M 

:C(MFY (M(

FCM Y=6F=K +C T )=Y WKF6 T2 FV(  FV ( FCK  :(YK C+FC=MY = Y6KV +V(M=(M ( 6CF( YCC 2 + =#C:(: Y=( K=+CK FC=MY Y+(KC;K CFV  YY F

MY( FV =6V M CMF(  K( ( CM= (:2 Y6KV Y Y=6F( KK66 FC=M F FV( CMF(  K(2 :Y= +FC=M CM(FCKY2  4=MY( 6(MFT2 F ( YF CM FV( YC+(YF :(YK C+FC=MY2 CF C
( =  T (6C# (MF F= Y T FV F +6 ( C6C:  T( CY V( F(:  =
(=2 = FV F F=K=+=M(MF CY=FV(   C6C:  T( 6M:( =(Y :(Y= +FC=M FV =6V CFY 6++(  (( Y6  K( = FV( K=+=M(MF VCKV =( Y FV( Y6  K( F(MYC=M V( ( F( K (: Y6   KF MF ,M FV( Y(K=M: YCF6 FC=M K=MYC:( (: CM FV(Y( (KF6 (Y2 C6C:  T(

CY V( F(:  = FV( F=+ = 6M:( =(Y
Y= +FC=M = Y=6
( Y6  KF MF ,M FV(Y( K Y(Y2 =M( FT+CK T =
Y( #(Y Y(=  MC? FC=M = FV( YTYF( CM FV( =  = + =+  FCM  #( F

CMY2 ( ?C6FV T CM M MM6 K=MF CM(

Y CM C (F $KK YC=M T2 YC6F M(=6Y + =+  FC=M =  #(Y CM
=FV

?C6FV  :C (KFC=MY  T Y=
( =
Y( #(:2 Y6KV Y CM C  CVF @6KV  #( F

CMY V #( YF =M M =C(Y CFV Y=CF T  #(Y2 FV=6V FV(C ( :CYYC+ FC=M =M  ( FC( YK ( K M V( (
(
 MK(:
T (M( T CM+6F  =

M 6M:( TCM CMYF
CCFT /V(Y( CMF( (YFCM Y+(KFY V #(
((M YF6:C(: CM :(F CY
T ( :( M: K== ( Y Y(( ( 4V CYF=# M: ( :(2 <XO

6

Pierre Colinet

  3 ! Z(F  ?C6FV T+ =+  FCM  #( F

CMY #CY6 C?(:
T YV : =

+V F(KVMC6( :6 CM
Y= +FC=M = +(MF M( # += CMF= F=6(M( 4=6  F(YT = WM: ( Y C( YKV( RCVF  K=6MF( + =+  FCM  #( F

CMY =
 Y( #(: CM V( F F

MY( (+( C(MF CFV V( FCM  = FV( F=+  F( (V

M: ZCM:(2 <5

R(:MC =# M: ( :(2 *LLL D=FV K Y(Y = V( FCM  =
=#( =
(= 
Y=
CM = :(Y=
CM Y6   KF MF FV( (= ( ( : F= Y+=MF M(=6Y
( CM = FV( M F6

 YT(F C(Y = FV( DG(M :  T( 2 (CFV( CM FV( =  =  #(Y2 = = YF( :T K=M#(KFC=M + FF( MY ,M:((:2 CM K Y( )6 (CFV( = V( F = =  FF(  =KK6 Y +(  +(M:CK6 T F= M CMF(  K(2 Y6  K( F(MYC=M CY (+(KF(: F=
( 6MC= 2 C( M= )= CY (+(KF(: 
 ,M  FV( FCK  F( Y2 FV( YTYF( CY CM# C

MF CFV (Y+(KF F= F

MY FC=MY =M CFY CMF(  K( CM FV( C:( C?(: K Y( =  F(

T CM;MCF(  T( 2 M: (M3=TY CY=F =+T Y (2 C( MT =F FC=M CFV (Y+(KF F= M CY = FV==M  F= FV(  T( ( #(Y FV(  FF( 6M 0(KF(: @T(F T
( CM YY=KC F(: CFV FV( (( (MK( = = :( (: YF 6KF6 (Y CM M=M(6CC
C6 YTYF(Y CY FT+CK  = CMYF
CCFT +V(M=(M Y(( =FV(

( +(Y CM FV( Y ( #=6(2 =M VCKV Y6  K(F(MYC=M: C#(M  M:
6=T MKT: C#(M CMYF
CCFC(Y :CYK6YY(: CM FV(Y( (KF6 (Y



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Z(F 6Y ; YF K=MYC:( FV( Y=K (: R T(CVDG(M : CMYF
CCFT C 2 (F2 YY=KC F(: CFV
6=T MKT: C#(M 

FV( FV M Y6  K(F(MYC=M: C#(M K=M#(KFC=M W  T( = +6 ( C6C: VCKV C
( K=MYC:( (: Y CMK=+ (YY

Interfacial Patterns and Waves in Liquid Layers and Thin Films

7

C
( M: 7(F=MC M CY K=M;M(:
(F((M F= V= C?=MF  + F(Y  CMF CM(:

F K=MF =
( F(+(

F6 (Y

  3  @ (FKV(Y C6YF

FCM FV( (KV MCYY F FV( = CCM = FV( R T(CVDG(M : (F M: FV( H

M=MCDG(M :  CVF CMYF
CCFC(Y ,M
=FV K Y(Y2 FV( F(+(

F6 ( = FV(
=FF= + F( CY " VC( FV F = FV( 6++( Y6  K( (CFV( Y=C: = =+(M F= C  CY " . " FV(  T( CY V( F(:  =
(= WYY6( = FV( =(MF FV F FV( :CYYC+ FC#( (0(KFY = =(K6 F

MY += F = V( F M: =(MF6 ( M(CC
( C( YY=KC F(: CFV #( T =M FC( YK (Y2 Y(( * /V(M2 V(M =K C?(: )6KF6 FC=M = F(+(

F6 ( Y T2 V=F Y+=F CY(Y CM K( F CM #=6( (((MF2 M 6+ : = K( =M FVCY )6C: + FCK( CY CM:6K(:
T
6=T MKT2 Y CFY :(MYCFT CY :(K ( Y(: K=+ (: CFV CFY (M#C =M(MF :C0( (MFC 
6=T MKT = K( WY =(K6 :C06YC=M CY YY6(: F=
( M(CC
(2 FV( )6C: + FCK( C Y(F CMF= =FC=M CM FV( 6+ : :C (KFC=M2 CFV=6F  CKFC=M2 M: C ( CM F FV( Y ( F(+(

 F6 ( :6 CM FVCY =FC=M /V(
6=T MKT = K( CF C (+( C(MK( C FV(M CMK ( Y(2
(K 6Y( FV( + FCK( =#(Y F= :Y K=:( (C=MY2 Y6KV FV F FV( #(=KCFT CY CMK ( Y(:2  /VCY CY FV( = CCM = FV( +C;K FC=M CMYF
CCFT +V(M=(M=M /V( H

M=MCDG(M : (KV MCY  T
( (+ CM(: YCC T C 2

CVF  C )6KF6 FC=M = F(+(

F6 ( Y T2 V=F Y+=F =KK6 Y F FV(  (( Y6  K( = CMF(  K(2 =K  Y6  K( F(MYC=M 

:C(MF CY K ( F(:2 :C (KF(:

:C T  T  = FV( )6KF6 FC=M /V( YY=KC F(: F M(MFC  YF (YY(Y FV(M CM:6K(

:C T :C#( (MF Y6  K( )6C: =FC=M FV( CMF(  K(  T
( Y((M Y

M ( YFCK (


M( VCKV ( (Y F FV( +=CMF V( ( FV( :CYF6
MK( CY K ( F(: 7=2 K=MFCM6CFT = FV( )6C: (6C (Y #( FCK  YK(M:CM )= F= F ( + K(
(= FV( :CYF6
MK( , FV(  T( = )6C: CY V( F(:  =
(=2 FVCY 6+ CYCM )6C:2
(CM V=FF( 2 C  ( FV(  (( Y6  K( F(+(

F6 ( CMK ( Y( F FV( CMCFC  =K FC=M = FV( :CYF6
MK( 4=MY(6(MFT2 FV( Y6  K( F(MYC=M F FV F +=CMF :(K ( Y(Y2 FV6Y CMK ( YCM FV( F M(MFC  YF (YY(Y 4( T2 FV(   :C06YC#CFT C KF  CMYF FV(Y( CMYF
CCFT + =K(YY(Y2

8

Pierre Colinet


T  #= CM FV(  C? FC=M = )6C: + FCK(Y CFV FV(C (M#C =M(MF2 C( : +CM = )6KF6 FC=MY CYK=YCFT C Y= (M(

F( YF (YY(Y2 + (#(MFCM  K =YK=+CK =FC=MY F= :(#(=+ CFVCM FV( )6C: WY (Y6F2 FV( F(+( 

F6 ( :C0( (MK( P" - " " V Y F= (K((: FV (YV=: # 6( = DG(M : CMYF
CCFC(Y F= :(#(=+2 CM = :( = FV( :(YF
CC?CM (KV MCYY F= =#(  K=( FV( :CYYC+ FC#( (0(KFY /VCY C
( 6 MFC;(: CM F( Y = FC( YK (Y

M: :C(MYC=M(YY M6
( Y CM * 7=F( FV F K=MF

T F= FV( H

M=MCDG(M : CMYF
CCFT ( :CM F= V(  =MY Y(( C * V(M FV( CMYF
CCFT FV (YV=: CY (K((:(:2 FV( R T(CV DG(M : CMYF
CCFT FT+CK T TC(:Y = + FF( MY2 FV( YF6:T = VCKV V Y =FC# F(: C+= F MF + = (YY(Y CM FV( FV(= T = + FF( M =  FC=M Y(( =

CMYF MK( 4 =YY M: >=V(M
( 2 < 7=F( FV F
=FV =Y M: V( =MY

( #( T Y(:= +( (KFT (6 2 M: # C=6Y CM:Y = :((KFY V #( CM :((:
((M =
Y( #(:2 M: YF6:C(: (F(MYC#(T = FV(C CMF( (YFCM 6MC#( Y  + =+( FC(Y = CMYF MK(2 V( =M  + FF( MY #( T =F(M :CY+ T CY= F(: +(MF =MV(+F =M :((KFY2 VCKV K M KF Y (:C F= Y CM FV( F

MYCFC=M  = V( =M  F= Y6 ( + FF( MY =
Y( #(: F CMK ( YCM F(+(

F6 ( :C ( (MK( P" 2 #C + =K(YY = M6K( FC=M = CM(Y = +(MF =MY K ( F M: /V(YY2 < 4=MYC:( CM M= FV( =++=YCF( K Y( = C6C:  T( V( F(:  =
=#(2 ; YF CM FV(
6=T MKT: C#(M K Y(2 CF CY 6CF( M F6

 F= (+(KF M =YKC F= T
(V #C= = )6KF6 FC=MY2  CM CM FV( K Y( V( ( FV( (0(KF = :CYYC+ FC=M CY M=F F== YF =M ,M:((:2 :6 CM CFY 6+ : =FC=M2 )6C: + FCK( CMC FC T  ( FV M CFY (M#C =M(MF C M= =#( F= :Y / (C=MY /VCY ( MY FV F FV( 6+ :
6=T MKT = K( CF (+( C(MK(Y C ; YF :(K ( Y(2 FV(M (#(MF6 T KV M( YCM V(M FV( + FCK( =#( YV==FY CFY M(6F

T
6=T

MF +=YCFC=M2 :6( F= CM( FC  /V( + FCK( CY FV(M KK((

F(: :=M :Y2 F= :Y K=:( (C=MY2 M:  T =YKC F( =6M: CFY M(6F

T
6=T MF +=YC FC=M Y(#(

 FC(Y
(= ( :CYYC+ FC=M (#(MF6 T : +Y =6F MT )6C: =FC=M /VCY =YKC F= T (KV MCY CY M=M CM FV( CF(

F6 ( Y D 6MF A CYA

A

 CMF( M   #(Y2 =
Y( #(: CM YCF6 FC=MY = YF
( :(MYCFT YF

FC;K FC=M ( CM FV( F=Y+V( ( = FV( =K( M2 Y(( /6 M( 2 <5 @CC T2 Y6  K(  #(Y :6( F= FV( =K +C CFT CM  T( V( F(:  =


=#( ( Y= (+(KF(:2 M:  T
( (+ CM(: K=MYC:( CM =MK(  CM V=F Y+=F F FV(  (( Y6  K( WY 6Y6  FV( H

M=MC (0(KF : C#(Y Y6  K( )=  T  = FV( Y+=F2 VCKV M=
CMY "/ C6C:  = FV(
6  D(K 6Y( = )6C: CM( FC 2 FV( CMCFC T V=F Y+=F (#(MF6 T
(K=(Y K=:(

FV M CFY M(CV
= V==: =#( YV==F2 FV6Y (#( YCM FV( )= M: ( :CM F= =YKC FC=MY /V(Y(  #(Y CM#=#( =FC=MY (YY(MFC T +

( F= FV(  (( Y6  K(2 M:  T
( K (: =MCF6:CM   #(Y = FVCY ( Y=M 7=F( FV F
=FV CMF( M  M: Y6  K(  #(Y ( 6Y6 T : +(:2 FV=6V

Interfacial Patterns and Waves in Liquid Layers and Thin Films

9

CF V Y
((M YV=M (K(MFT R(:MC =# (F 2 *LLL FV F FV(C CMF(

KFC=M M( (Y=M MK(  T ( : F= +C;K FC=M M: =:(CCM Y(( Y(KFC=M   WKF6 T2 CMF( M   #(Y (KCF(:
T FV( H

M=MC (0(KF ( CM:((: FV=6VF C( YKV( (F 2 *LLL F=
( (Y+=MYC
( = FV( + =+  FCM  #( F

CMY YV=M CM C (F2 C( CM YCF6 FC=M V( ( +(MF M( CY
Y=
(: CM F=6(M( QC#(M FV F +(MF M( CY CVF( FV M F=6(M(2
Y= +FC=M CM:((: (M(

F(Y YF
( :(MYCFT YF

FC;K FC=M + =M( F= F C( CM = CMF( M   #(Y
T (F( M  :CYF6
MK(Y2 36YF Y CM FV( FV(   K Y( :CYK6YY(:
=#( WY +(MF M( Y= =( Y FV( Y6  K( F(MYC=M = FV( CF6 (2 =MCF6:CM  Y6  K(  #(Y  T Y= =KK6 M: (Y=M MFT CMF(

KF CFV F

MY#( Y( CMF( M   #(Y CM T2 FV( ( M=M K +C T

#CFT  #(Y ( T(F M=FV( =YKC  F= T =:(2 VCKV =KK6 Y V(M(#(  (( Y6  K( CY +( F6
(:
T M ( F( M  = K( RC++(Y + =+  FCM =M  (( Y6  K( CM#=#( M (YY(MFC  :(=  FC=M = FV==M  F= FV(  FF( 2 M: ( F

MY#( Y(  #(Y CM FVCY ( Y+(KF ,MF( (YFCMT2 Y6KV K +C T

#CFT  #(Y  T Y=
( (KCF(: CM  T( Y V( F(:  = FV( F=+2 #C (Y=M MF CMF(

KFC=M M: CCM CFV FV( Y6  K( =MCF6:CM   #(Y 36YF :CYK6YY(: R(:MC =# (F 2 *LL< W ( #C( = (+( C(MF  ;M:CMY =M  #(Y (M(

F(:
T V( F =  YY F

MY(

FV =6V M CMF(  K(2 FV(C CMF( + (F FC=M CM F( Y = CMF( M 2 Y6  K( M: K +C T

#CFT  #(Y2 Y ( Y FV(C CM Y CFV Y=CF T  #(Y2 ( C#(M CM FV(
== = 7(+=MT YVKVT (F 2 *LL* 

/

#%( 3%3 1
3#1

V(M  T( = )6C: CY V( F(:  =
(=2 FV( YF F( = +6 ( K=M:6KFC=M + (# CCM F Y  P" CY ; YF (+ K(:
T K(6 K=M#(KFC#( YF 6KF6 ( F

K( F CM P" 2 Y (+ CM(: CM FV( + (#C=6Y Y(KFC=M 4=M#(KFC#( V( F F

MY += F M= ::CM F= FV( (0(KF = V( F K=M:6KFC=M2 FV( =
 V( F F

MY(

FV =6V FV(  T( CY (MV MK(:2 VCKV  T
( Y((M Y M CMK ( Y( = FV(

++ (MF FV(   K=M:6KF MK( = V( F F

MY( K=(KC(MF = FV( C6C:  T( :(;M(:2 (2
T .P" V( ( . CY FV( ( M V( F )6 K =YYCM FV(  T(  @6
Y(6(MF F

MYCFC=MY2 ( F= FC(:(+(M:(MF (C(Y2 (M(

T

(Y6F CM 6 FV( CMK ( Y(Y = FV( V( F F

MY( K=(KC(MF /VCY V Y
((M (F(MYC#(T YF6:C(: = FV( K Y( = R T(CVDG(M : K=M#(KFC=M2 (#(M CM FV(

(C(Y = Y=K (: ( M: YF =M FV(   F6
6(MK( Y(FFCM CM F #( T VCV P" P" Y(( = CMYF MK( Q6T=M (F 2 << ,M Y6  K(F(MYC=M: C#(M YCF6 FC=MY2 6KV (YY CY M=M2 (#(M FV=6V CMF(  KC  F6
6(MK( V Y
((M =
Y( #(: Y ( T Y CM FV( XL'Y2 CM FV( K=MF(F = 4V(CK  MCM(( CM ,M  MT YCF6 FC=MY CM#=#CM CMF(  KC  V( F M:  YY F

MY( = CMYF MK( CM F(KVMC6(Y Y6KV Y C6C:C6C: (F

KFC=M2

10

Pierre Colinet

C6C: Y
Y= +FC=M = :(Y= +FC=M2 :CYFC FC=M2 2 = :( =  MCF6:( KV M(Y = F

MY(

F(Y  = =M( +V Y( F= M=FV( V #(
((M ( Y6 (:

M: K=

( F(: CFV (+C CK  ( FC=MYVC+Y WM (F(MYC#( (#C( = Y6KV +V(M=(M M:

FV( K=+(F(
C
C=

+VT ( + =#C:(: CM FV( FV(YCY = / H=(M + H=(M +2 <" 7=F( Y= FV F CM KV(CK  ( KF= Y2  YY F

MY( FV =6V CMF(  K(Y CY =F(M KK=+ MC(:
T KV(CK  ( KFC=MY2

M: CMF( (YFCM K=6+CMY CFV Y6  K(F(MYC=M: C#(M K=M#(KFC=M K M =KK6

K ( F M: Q

VM2 <O K ( F (F 2 *LL  H

M=MCDG(M : K=M#(KFC=M CY Y= C+= F MF CM + =K(YY(Y CM#=#CM (# +=

FC=M2 Y6KV Y CM FV( : TCM = + CMF ;Y2 CM FV( K= FCM CM:6YF T2

M: = V( F F

MY( :(#CK(Y ,M C X2 FVCM  T( = (FVT K=V= CY (# +=

F(: 6M:( M CM( F  Y MCF =(M )= #(M FV=6V M= (F( M  V( FCM CY C+=Y(: CM FVCY (+( C(MF2 K(6 K=M#(KFC=M :(#(=+Y Y Y==M

Y FV( )= CY C+=Y(:2 :6( F= FV( K==CM = FV(  (( Y6  K( CM:6K(:
T (# +=

FC=M K=MY6+FC=M =  F(MF V( F /V(  KF FV F FV(Y( K=M#(KFC#( + FF( MY ( Y6YF CM(:
T (# +=

FC=M CY 6 FV( K=M; (:
T FV(C Y6::(M :CY ++(

MK( V(M # += (=# 
T FV(  Y )= CY CMF(

6+F(: C (Y " M:  ,M F6 M2 K=M#(KFC#( F

MY+= F = V( F (:6K(Y FV( =#(

 FV(  

(YCYF MK( = FV( C6C:  T( 2 FV( (= ( K=MF C
6FCM F= M CMK ( Y( = FV( (# +=

FC=M

F( ,M ::CFC=M F= FV( CMK ( Y( = FV( CMF(  KC  (KV M( K=(KC(MFY C( +( 6MCF Y6  K(2 H

M=MC )=Y =F(M CM:6K( YCMC;K MF CMF(  KC  :( =  FC=M2 ( :CM F= C+= F MF # C FC=MY = FV( F=F  Y6  K( = (KV M( CFY( ,M FV( K Y( = FVCM C6C: ;Y2 Y+(KC;K CMYF
CCFC(Y Y(( *   T =KK6 2 ( :CM CM Y=( K Y(Y F= : T (C=MY VC( Y6KV : T Y+=FY ( (M( 

T 6M #=


( = CMYF MK(2 CM FVCM; (# +=

F= Y2 FV(T ( : F= +=YYC
( =#( V( FCM M: :(YF 6KFC=M = FV( Y6
YF

F(2 FV(T Y= V #( +=YCFC#( ( (KF YCMK( FV( V( F F

MY( (YCYF MK( YK (Y + =+= FC=M T F= FV( =K  ; FVCK M(YY2 Y ; YF ++ =C FC=M WY K=MY(6(MK(2 FV( =K  V( F )6 K M
(K=( =K T #( T VCV CM FV( F

MYCFC=M (C=M
(F((M FV( ; M: FV( : T Y+=F C( F FV( K=MF KF CM(2 Y  Y 6 MFC;(:
T  @F(+V M CM FV( K=MF(F = V( FF

MY( :(#CK(Y Y6KV Y V( F +C+(Y M:
=C( Y @F(+V M

M: D6YY(2 <* Q(M(

T Y+( CM2 H

M=MC (0(KFY M: YY=KC F(: CMYF
CCFC(Y ( (+(KF(: F=
( :=CM MF F Y  (MFV YK (Y2 V( ( 

#CFT M:
6=T

MKT ( CM(0(KFC#(  Y C
( Y((M  = :C(MYC=M(YY  =6+Y CM FV( M(F Y(KFC=M /V( ( CY FV( (= ( (M((: CMF( (YF CM (K(MF T( Y2 (Y+(KC T CM K=MM(KFC=M CFV FV(

+C:T :(#(=+CM ;(:Y = CK =)6C:CKY M: (#(M M M=F(KVM==C(Y WF Y6KV YK (Y2 Y(=  MC? FC=M +V(M=(M Y6KV Y FV( Y6  K(F(MYC=M: C#(M DG(M : CMYF
CCFT K=6: = CMYF MK(
( 6Y(: Y  T F= :(YCM Y+(KC  FT+(Y = Y=C: Y6  K(Y2
T :(+=YCFC=M = M M=K TYF Y

Interfacial Patterns and Waves in Liquid Layers and Thin Films

11

CM FV( = Y = +=T=M 

TY M: CMY Y(( ( H C : (F 2 *LLL

  3 0 @(6(MK( = CM

(: K (

C (Y = FV(  (( Y6  K( = FVCM C6C:  T( = (FVT K=V=2 = K(: F= (# +=

F( 6M:( )6 = MCF =(M CM:CK F(:
T FV(

= F == F(+(

F6 ( M: F=Y+V( CK + (YY6 (

!

*3%1%311 %3 1 % 3 1#31

@(#(

 FC( YK (Y K M
( :(;M(:  = +VTYCK  + =+( FC(Y = FV( )6C:

M: (=(F CK  KV

KF( CYFCKY Y6KV Y CFY :(+FV ) ( ; YF V #( FV( FV(    FC( YK ( )  < % C( FV( FT+CK  FC( YK ( CF F (Y = F(+(

F6 ( )6KF6 FC=MY F=
( : +(: =#( :CYF MK( ) % CY FV( FV(   :C06YC#CFT  @CC T2 #(=KCFT =



0 %8#; ;=,%#0 $= $# =0=  =*= %  = ># ;#%%= %  %= =%= #==0,= 1  $=:= * 0 #;%8= #0 ># ;##  %#= ;, =; 0 #2 =0;#0 =;; 0=;  = =0= = ,#0 1  1 ' % #, ;; *,%#0 #==0,= =%$==0 %% 0 %    '%8= #;%8= #;#0 ,=.,#=0% 0 =0#0  '    0   '    $8==  0  = %8= ;*,= %=0;#0 0 %8= : #, ;; =; =,%#:= 

12

Pierre Colinet

#= FCKCFT )6KF6 FC=MY :(K T =M #CYK=6Y FC( YK ( " -

) 

*

V( (  CY FV( CM( FCK #CYK=YCFT 7=2 CF CY +=YYC
( F= K=MYF 6KF F= =FV( FC( YK (Y VCKV := M=F :(+(M: =M MT =(K6 :C06YC#CFT (KV MCY /V( ; YF =M( CY FV( 
6=T MKT FC( YK (N )   P" VCKV CY =6VT FV( FC( CF F (Y = )6C: + FCK( F FV(
=FF= = FV(  T( F=
( KK((

F(:
T
6=T MKT 6+ F= CFY 6++( Y6  K( Y(( C 2 C( FV( FT+CK  FC( YK ( = FV( CMYF
CCFT ( :CM F= K=M#(KFC#( + FF( MY ,M  2  CY FV( FV(   (+ MYC=M K=(KC(MF2  CY FV( 

#CFT KK((

FC=M2

M: P" - " " CY FV( F(+(

F6 ( : =+ K =YY FV(  T(  @CC T2 FT+CK  FC( YK ( = FV( FV( =K +C T = H

M=MC CMYF
CCFT K M
( :(;M(: Y !)     P" V( ( ! CY FV( C6C: #=6CK  YY2 M:  - 88" CY FV( K=(KC(MF = # C FC=M = Y6  K( F(MYC=M CFV F(+(

F6 ( +=YCFC#( = 6Y6  C6C:Y 7=F( FV F V(M FV(  T( CY V( F(:  = FV( F=+2   M:  

+ =#C:( = :( =  MCF6:( (YFC F(Y = FV(  (6(MKC(Y = CMF( M  M: Y6  K(  #(Y Y(( *<2 (Y+(KFC#(T 4=+ CM FV( :C0( (MF FC( YK (Y C#(M
T Y < 2 CF  T
( ( +(KF(: FV F CMYF
CCFT C :(#(=+ C FV( FC( = )6C: + FCK( F= F

#( =#( :CYF MK( = = :( ) CY YV= F( FV M FV( FC(Y M(K(YY T = FV( + FC K( (CFV( F=
( Y=(: :=M
T #CYK=YCFT2 = FV(  T (6CC


F(: CFV CFY (M#C =M(MF /V6Y2
6=T MKT: C#(M CMYF
CCFT =6: =KK6 FT+CK T C > -

P" ) "  - < % 

X

VC( Y6  K(F(MYC=M: C#(M CMYF
CCFT C =KK6 V(M 6 -

P" ) "  < !% 

E

/V(Y( (6 FC=MY :(;M(
=FV FV( R T(CV M6
( > M: FV( H

M=MC M6
( 6 2 VCKV ( FV( 6Y6  ( Y6 (Y = FV( ( FC#( C+= F MK( =

Interfacial Patterns and Waves in Liquid Layers and Thin Films

13

:(YF
CC?CM M: YF
CC?CM (0(KFY /T+CK T2 FV( K CFCK  # 6( = FV( R T(CV M6
(
=#( VCKV CMYF
CCFT Y(FY CM CY = = :(
M=MC M6
( CY = = :(
 (( M: +== TK=M:6KFCM2 2 M: 6YF
( :(F( CM(:
T CM( YF
CCFT

M TYCY Y((   7=F( FV F =FV( 6Y6  :C(MYC=M(YY +

(F( Y ( FV( 

M:F M6
(

  - % -  " M: FV( Q C(= M6
( ; - ) % -  "  

 V( (   - ) 2 C( FV( FT+CK  FC( YK ( =
=:T F=
( KK(( 

F(: =#( :CYF MK( ) 6M:( FV( KFC=M = 

#CFT CM T2 CM K Y( FV( C6C:  T( + (Y(MFY M 6++(  (( Y6  K(2 CF CY K= =M F= :(YK C
( V( F F

MY( FV =6V FV(  FF( 6YCM K=MYF MF V( F F

MY( K=(KC(MF @ = K=MYF MF DC=F M6
( 32 - @) CM :C(M YC=M(YY = 2 V( (  CY FV( C6C: FV(   K=M:6KFC#CFT #(M FV=6V FVCY CY K 6:( ++ =C FC=M YCMK( FV( V( F F

MY( K=(KC(MF (M(

T :(+(M:Y =M K=M#(KFC#( =FC=MY CM FV(  Y +V Y(2 M: FV( (= ( =M FV( F( +(

F6 ( = FV( CMF(  K( CFY(2 CF :=(Y + =#C:( FV( K=

(KF F(M:(MKT = FV( CM)6(MK( = V( F F

MY( F FV( 6++( Y6  K( M (T M CMK ( Y( = FV( K CFCK  H

M=MC M6
( CFV CMK ( YCM DC=F M6
(  W = ( C= =6Y :(;MCFC=M = (M(

C?(: DC=F M6
( N C
( + (Y(MF(: CM  

-.3 %1' 3#.%11 % ,3 ' .% " 1

8(F M=FV( CMYF
CCFT =:( K M =KK6 2 Y+(KC;K T = FVCM C6C: ;Y = #CYK=6Y C6C:Y /VCY CY C6YF

F(: CM C E  K=MYC:( FV F M CMCFC T ) F V( F(: C6C: ; F=+ ;6 ( CY +( F6
(:2 (Y6FCM CM M=M6MC=  C6C: :(+FV C::( ;6 ( WF FV( FVCMM( + FY = FV( ;2 FV(  (( Y6  K( CY  ( YCMK( CF CY K=Y( F= FV( V( F(: + F( FV M F FVCK (

(C=MY W Y6  K( F(MYC=M 

:C(MF FV( (= ( CY(Y2 M: (M(

F(Y )= :

CM )6C:  T  = FV( :(+ (YYC=M2 V(MK( :(+(FCM FV( FVCMM( + FY (#(M = ( /VCY  T (Y6F CM FV( =  FC=M = : T Y+=FY 
=FF= ;6 (2 (#(M FV=6V FV( K=

(KF :(YK C+FC=M = FV( ; (#=6FC=M M( 6+F6 ( YV=6: CM#=#( CMF( =(K6 = K(Y
(F((M FV( )6C: M: FV( Y6
YF

F(2 FT+CK T V(M FV( =K  FVCK M(YY :(K ( Y(Y
(=
( KV#CC2 <*O $ =M (F 2 <5 7=F( FV F FV( V= C?=MF  (MFV YK ( = K=M#(KFC#( =FC=MY  M: = Y6   K( :(=  FC=MY (M(

F(:
T FVCY (KV MCY CY (M(

T 6KV  (

FV M FV( )6C: :(+FV2 V(MK( FV( M ( =M #( :(=  FC=M  =:( =

FVCY CMYF
CCFT (KV MCY ,M K=MF

YF2 +=T=M  K=M#(KFC=M K(Y FT+C K  = H

M=MCDG(M : K=M#(KFC=M F6 M =6F F= YK ( + =+= FC=M T F= FV( :(+FV = FV( C6C:  T( Y(( C * CM K Y( FV( CC:
=FF= + F( CY

14

Pierre Colinet

==: FV(   K=M:6KF= Y6KV Y (F  ,M K Y( FV(
=FF= + F( V Y = V( F F

MY( K=(KC(MF2 36YF Y FV( 6++(  (( Y6  K(2 FV( H

M=MC DG(M : K=M#(KFC=M K(Y
(K=(  (YK ( Y (2 VCKV =0( Y  T F= :(YK C
( FV( CM F( Y = =M #( YT+F=FCK FV(= C(Y


 



  3  Z=M #( :(=  FC=M  CMYF
CCFT = V( F(: FVCM C6C: ; /V( =M #( :(=  FC=M  =:( =KK6 Y FT+CK T = 6  6  ; 2 = +6 (T CMY6 FCM 6++(  (( Y6  K(  ==: ++ =C FC=M = C 6C:Y CM K=MF KF CFV  Y +V Y( WY CMF =:6K(: CM FV( + (#C=6Y Y(KFC=M2 FV( Q C(= M6
( ; - ) % ( Y6 (Y FV( YF
CC?CM CM)6(MK( = 

#CFT =M Y6  K( :(=  FC=MY2 M: CY 6Y6 T (F ((T  ( 
M( ,M FV(Y( K Y(Y2 K(6 K=M#(KFC=M (KV MCYY C  TY =KK6 ; YF2 YCMK( FV(C K CFCK  H

M=MC M6
( CY FT+CK T
M=MCDG(M : K=M#(KFC=M VC( =

 F( FVCY K=

(Y+=M:Y F= 6M6Y6 T FVCM C6C: ;Y ) == (F   M>== (F 2 <5 V #( CM:C K F(: FV F FV( K =YY=#(
(F((M FV( F= =:(Y  T CM:((:
( =
Y( #(:2 +=YYC
T (Y6FCM CM K=(CYF(MK( =  (YK ( : T Y+=FY CFV :(+FVYK (: K=M#(KFC=M K(Y Y6KV Y Y (FKV(: CM C E 
=FF= M>== (F  V #( 6 FV( =
Y( #(: M: (+ CM(: FV F FV( =M #( :(=  FC=M  =:( CM :((: (M(

T ( :Y F= ; 6+F6 ( Y(( Y= $ =M (F 2 <52
6F = ( K=+( YCF6 FC=MY  T =KK6 Y6  K( ((# FC=MY = VCV Y+=FYN2 K YK :(Y = YF 6KF6 (Y2  7=F( FV F YCF6 FC=MY CM#=#CM (YY(MFC  Y6  K( :(=   FC=M C M=F
( K=MYC:( (: 6 FV( CM FV(Y( M=F(Y2 Y FV(T C
( :(YK C
(: CM :(F CY CM FV( (KF6 (Y = H D(YF(V= M M: Z CY(M2 CMK6:CM FV( :(

Interfacial Patterns and Waves in Liquid Layers and Thin Films

15

YK C+FC=M = 6+F6 ( = 6F

FVCM ;Y :6( F= =(K6 CMF(

KFC=MY CFV FV( Y6
YF

F(2 : =+(FY2 



$ % $         $

#(M FV=6V FV( :(YK C+FC=M = DG(M : CMYF
CCFC(Y CM  T( Y CFV  (( Y6   K( (M(

T CM#=#(Y )6C: =FC=MY M: F(+(

F6 ( )6KF6 FC=MY CM
=FV C6C: M:  Y +V Y(Y F= T( YTYF(2 CF CY =F(M +=YYC
( F=  ( Y=( YC+CTCM YY6+FC=MY C#(M V( ( F(  CM = :( F= :(YK C
( FV( CM F( Y = C6C: 6 MFCFC(Y =M( =M(YC:(: =:(2 CFV Y6CF
(
=6M: T K=M:C FC=MY YCM FV( FV(   FC( YK (  - ) %2 FV( (MFV YK ( )2 FV( + (YY6 ( YK ( +%) 2 M: T(F 6M:(;M(: F(+(

F6 (

M( -2 FV(
YCK :C(MYC=M(YY (6 FC=MY (M(

T 6Y(: F= :(YK C
( FV(  T: C#(M DG(M : CMYF
CCFC(Y = C6C:  T( (    - L  5           ; < I< -" "& % -   8 B  " P 85 8"   "  B   85 V( (  - # # 2 " 2 M: ( (Y+(KFC#(T FV( :C(MYC=M(YY ;(:Y = #(=KCFT2 F(+(

F6 ( M: + (YY6 (2 < CY FV( 6MCF #(KF= :C (KF(: #(  FCK T 6+ :Y2 "& - " B " *- CY FV( :C(MYC=M(YY (( (MK( ( M F(+(

F6 ( F VCKV FV( +VTYCK  + =+( FC(Y ( (# 6 F(: M=F( FV F FV(C

+=YYC
( # C FC=M CY M((KF(:2 CM FV( F(+(

F6 (

M( - K=MYC:( (:2 M: =FV( :C(MYC=M(YY 6 MFCFC(Y V #(
((M :(;M(: ( C(  6 FC=MY 5 ( Y+(KFC#(T (+ (YY FV( K=MY( # FC=M =  YY = CMK=+ (YYC
( C6C:Y2 = =(MF6 CM FV( + (Y(MK( =
6=T MKT = K(2 M: = (M( T M((KFCM #CYK=6Y V( FCM ,+= F MFT2 FVCY =  = FV( (6 FC=MY  (Y 6Y( = FV( D=6YYCM(Y

++ =C FC=M2 C( -  < /VCY ( MY FV F FV( # C FC=MY = :(MYCFT  T
( M((KF(: (#( TV( (2 6M(YY V(M 6FC+C(:
T  ( 6 MFCFC(Y Y6KV Y FV( Q C(= M6
( ; 2 Y CF CM:((: =KK6 Y = FV(
=:T = K( F(  = FV( 7 #C( @F= (Y (6 FC=MY " WKF6 T2 FV( R T(CV M6
( :(;M(: 6YCM FV( F(+(

F6 (

M( - CY FV( + =:6KF > - -; 2 C( FV( + =:6KF = Y  M:  ( M6
( 2 M: K M FV( (= (
( Y6KC(MF F= F C(

CMYF
CCFT = FV( DG(M :  T(  D=6M: T K=M:CFC=MY =M FV(
=FF= + F( 1 - L ( 6Y6 T F (M Y FV( M=YC+ M: K=MYF MF F(+(

F6 ( K=M:CFC=MY  - " "  - L F 1 - L 
16

Pierre Colinet

V( ( "  - - " 2 VC( FV=Y( F FV( 6++(  (( Y6  K( YV=6: CM + CMKC+(
(  CFF(M F 6M M=M +=YCFC=M2 VCKV YV=6:
( =6M: Y Y=6FC=M = FV( 6 + =
( >=(#( 2 ( C M=F :(  CFV Y6KV :(= 
(  (( Y6  K(Y V( ( FV( ( :( CY ((

(: F= FV(
== = 4=CM(F (F 2 *LL<2 =

(M(

 CMF(  KC  K=M:CFC=MY M: YY6( FV(  (( Y6  K( F=
( =K F(:

F 1 - <2 C( :CY ( :CM FV( K Y( = #( T FVCM C6C: ;Y FV( (0(KF = 

#CFT2 6 MFC;(:
T FV( Q C(= M6
( 2 CY Y6KC(MFT YF =M F= + (#(MF  (( Y6  K( :(=  FC=MY ,M ::CFC=M2 Y (MFC=M(:
=#(2 K=6+CM = C6C: M:  Y +V Y(Y CY (+(KF(: F 1 - < ( M= 6M:( CM( FV(  CM VT+=FV(Y(Y M(K(YY T F=

( KV =M(YC:(: :(YK C+FC=M2 C( F=  CF( FV( K=

(KF M6
( =  (( Y6  K(
=6M: T K=M:CFC=MY2 FV=6V CM F( Y = C6C: 6 MFCFC(Y =M( C YF2 FV( C6C: #(=KCFT = FV==M  F= FV( CMF(  K( CY ?( =2 YCMK( CF CY

YY6(: =FC=M(YY >(MK(2 -L

F 1 - <

<<

@(K=M:2 CM #C( = FV(  KF FV F FV( :TM CK #CYK=YCFT = FV(  Y CY (M(

T 6KV =( FV M FV F = FV( C6C:2 (  T M((KF  Y #CYK=6Y YF (YY(Y CM FV( F M(MFC  =(MF6
 MK( F 1 - < / CM CMF= KK=6MF Y6  K( F(MYC=M 

:C(MFY M: CM :C(MYC=M(YY = 2 FVCY ( :Y 8  B 6 8 " - 8 B 6 8 " - L

F 1 - <

<*

V( ( 6 - -)!% CY FV( H

M=MC M6
(  7=F( FV F M (6C# (MF =  CY =
F CM(:
T :C0( (MFC FCM FV( ; YF = FV(Y(
=6M: T K=M:CFC=MY CFV (Y+(KF F= 2 FV( Y(K=M: CFV (Y+(KF F= =2 ::CM FV( (Y6FY M: 6YCM FV( CMK=+ (YYC
CCFT K=M:CFC=M 5 /VCY TC(:Y 8 - 6 P "

F 1 - <

<

V( ( P - 8 8 B 8 8= CY FV( V= C?=MF  Z + KC M =+(

F=  W YF

CVF=  :  T F= K=Y( FV( + =
( =6:
( F= M((KF FV( FV(    K=M:6KFC#CFT = FV(  Y2 M: FV( (= ( F= YY6( 8 " - L F FV(  (( Y6   K( ?( = V( F F

MY( K=(KC(MF M:( Y=( YY6+FC=MY (MFC=MM(: CM Y(KFC=M *2 (# +=

FC=M K M Y=
( CMK6:(: CM FV( (M( T K=MY( # FC=M

F FV( CMF(  K(2 VCKV TC(:Y 8 " -



F 1 - <

< 

V( ( - B7 - CY M (# +=

FC=M M6
( 2 K=+ CM FV(  F(MF V( F = (# +=

FC=M B F= 7 -2 C( FV( V( F M((:(: F=   6+ FV( C6C: =#( FV(

Interfacial Patterns and Waves in Liquid Layers and Thin Films

17

Y((KF(: F(+(

F6 (

M( - 7=F( FV F  <  CMF =:6K(Y FV( (# +=

FC=M

F(  V( ( YK (:
T !%)2 VCKV CY =F(M  CFF(M Y   -

6 *<> "!

-)  " # "!  !%

- "! 

<X

C( =M( = FV( +=YYC
( = Y = FV( >( F?M6:Y(M CM(FCK   Y(( Y= D6 (
KV (F 2 <""2 CM VCKV  CY FV( KK==: FC=M K=(KC(MF2 6 CY FV( C6C: =(K6  YY2 > CY FV( 6MC#( Y   Y K=MYF MF2 # "  CY FV( (6CC


C6 + (YY6 ( F F(+(

F6 ( " ( FV( 4 6YC6Y4 +(T =M K=(CYF(MK( K6 #(2 M: "! CY FV( :C(MYC=M  Y F6

FC=M + (YY6 ( F FV( =K  # +=

+ (YY6 (  2 C( <E # "!  -  >(MK(2 = +6 ( # += +V Y( V( ( FV( # += + (YY6 (   T
( YY6(: K=MYF MF2 "! CY K=MYF MF M: Y <  M: <X K=
CM( F= C#( C(: FV(   K=M:CFC=M 8 " B 32# " - "!  - L

<5

V( ( FV( (0(KFC#( V( F F

MY( K=(KC(MF C( FV( DC=F M6
( 2 CM :C(M YC=M(YY =  ( :Y  32# - 

6 *<>"!

B)  # "!  

<"

M: FV( CM:( YF M:Y = +6 ( # += +V Y(N 7=F( FV F FV( # 6(Y K  K6 F(: = 32# ( (M(

T #( T VCV + F = #( T FVCM C6C: ;Y =

#( T Y  KK==: FC=M K=(KC(MF 2 Y6KV FV F FV( CMF(  KC  F(+(

F6 (

( CMY (#( TV( ( #( T K=Y( F= "! =K  FV( =:TM CK (6CC
C6
(F((M FV( # += M: FV( C6C:2 M: FV( H

M=MC (0(KF CY #( T (  >(MK(2 FV( + =
( CY M= K=Y(:2 (CFV( = FV( K Y( V( ( FV( C6C: CY M=M#= FC( M: FV( # += +V Y( V Y #( T Y  FV(   K=M:6KFC#CFT 32# V Y F=
( Y(F F= ?( = CM  <52 = V(M FV( C6C: K M (# +=

F( CMF= CFY =M # += CM VCKV K Y( 32# CY #( T  ( 6 MFCFT >=(#( 2 FV( ( ( K Y(Y V( ( =M( =
=FV = FV(Y( YY6+FC=MY  C2 M: = ( (M(



++ = KV CY M((:(: 

9;  ; #;,;;=   #0=% =%   -- %8= ,;;2:= =%$==0 %8= =,%#02 ##%= =#=    0 %8= 8=% %0;*=2 ##%= =#=    ,,; *  = %8  7'-- 0 * $%= 0= %8= 0 # #0 #0% 0 0 ,,%#0 ,=.,#=0%    8#; = %8 #0,=;=; $8=0  =,=;=; 8$=:=

18 !

Pierre Colinet %(%3  3 1 .3  #% #,'

WY FV( YY6+FC=M = # MCYVCM  Y FV(   K=M:6KFC#CFT  T M=F
( 36YFC;(: CM  K Y(Y ( V(C6 CY (6 T K=M:6KFCM Y YCCK=M( =C2 CF Y((Y + ((


( F= (M(

C?( FV( =6V ++ = KV = FV( + (#C=6Y Y(KFC=M2 FV=6V ; YF CM FV( M=M#= FC( K Y( ,M (M(

2 FV( FV(   :C06YC#CFT % =  Y CY
=6F =M( V6M: (: FC(Y  ( FV M FV F = C6C: /V( (= (2 FV( FC( YK ( = FV(   )6KF6 FC=MY CM FV(  Y2 C( ) % V( ( ) CY FV(  Y :(+FV2 CY 6KV Y ( FV M FV( C6C: FV(   FC( YK (  - ) % ,M ::CFC=M2 (  T Y= (+(KF FV( (M( T F

MY+= F :6( F=  Y )=Y F=
( M(CC
( K=+ (: CFV FV(   :C06YC=M Y  G(K(F M6
( YY6+FC=M 2 M: FV( (M( T (6 FC=M CM FV(  Y  T
(  CFF(M CM FV( 6 YCYF( :T ++ =C FC=M P" - L

<

VCKV  T CM + CMKC+(
( Y=#(: 6YCM
=6M: T K=M:CFC=MY " - " # =# 1 <# 5 F 1 - < M: " - " F 1 - < B , V( ( , - ) ) QC#(M Y6KV

=   Y=6FC=M " I" 1 - <%2 (  T FV(M (+ (YY FV( K=MFCM6CFT = V( F )6 Y  8 " I" 1 - <% F 1 - < 8 " -  *L  V( (  -   CY FV(

FC= =  Y M: C6C: FV(   K=M:6KFC#CFC(Y ( V #( M= ( KV(: (M(

C?(: =M(YC:(: :(YK C+FC=M = FV( + =
(2 C( FV( YTYF( = (6 FC=MY 5 CFV
=6M: T K=M:CFC=MY =(#( 2 Y6KV + =
( CY M=M=K 2 YCMK( FV( V( F )6 *L :(+(M:Y =M FV( CMYF MF M(=6Y F(+(

F6 ( -$ " 1 - < F FV(  (( Y6  K(  M: M=F =MT =M CFY =K  # 6( = F6M F(T2 FVCY + =
(  T ( YCT
( V M:(: CM =6 C( Y+ K( !( M=FCM FV( V= C?=MF  #(

(
T 2 FV( V= C?=MF  =6 C( F

MY= Y
T FC:(: 6 MFCFC(Y2 M: ( C?CM FV F  =  <2 "  6YF # T CM( T  = " 1 - < F 1 - < F= " F 1 - < B ,2 FV( V= C?=MF  #(

( =  *L TC(:Y  " "  F 1 - < 8 "  - ,  *< VCKV YV=Y FV F FV( CMK ( Y( = V( F F

MY( :6( F= +=YYC
( K=M#(KFC=M CM FV( C6C: CY :C (KFT CM (: CFV FV( CMK ( Y( = FV( ( M Y6  K( F(+(

 F6 ( 7=2 = =6 C( K=+=M(MFY CFV  #(M6
( 
- L2  < TC(:Y 8  "& - L2 VCKV V Y F=
( Y=#(: CFV
=6M: T K=M:CFC=MY "& - "&


%= %8% * %8#; =:=  =0% % 8  #% #; 0=,=;; % ;;= %8% %8= ; 8;= #; 
 #= #%; = %8  ;% 0% = %6=0 ,8 = %80 %8= #+# = %8  (=:= %8= ,;= * = %=0  # ;= *;% ; >$; #;  ; = , =

Interfacial Patterns and Waves in Liquid Layers and Thin Films

19

F 1 - < M: "& - L F 1 - < B , /V(M2 6YCM FVCY (Y6F CM FV( =6 C(

F

MY=  =  *L2 ( (F 8 "& - 32 "&

F 1 - <

**

 K=FV, CY (M(

C?(:2 */$/&/
/$/$ DC=F V( ( 32  -  M6
( C( :C(MYC=M(YY V( F F

MY( K=(KC(MF2 Y(( (M: = * 7=M =K CFT FV( (= (  MC(YFY CFY( V( ( FV =6V FV(  KF FV F FV( :C0( (MF =6 C( K=+=M(MFY = FV( F(+(

F6 ( ;(: (#=#( CFV :C0( (MF V( F F

MY ( K=(KC(MFY F FV( 6++(  (( Y6  K( !

&3%3 3 %3(1 3  3%  3,
 %

V(M FV( C6C: CY #= FC( M: (# +=

F(Y CMF= M CM( F  Y Y6KV Y C 2 FV(
=6M: T K=M:CFC=M <5 K MM=F
( ++C(: MT= (2
(K 6Y( FV( + FC  + (YY6 ( = # += K MM=F
( YY6(: K=MYF MF CM (M(

 WYY6CM FV(  Y CF6 ( F=
( +( (KF2 ( V #(  - '  B < '  V( ( CY FV( F=F   Y + (YY6 (  YY6(: K=MYF MF V( (2 Y K M
( 36YFC;(: 
-/2  - 66 CY FV(

FC= = =(K6  YY(Y = FV( # += M: = FV( CM( F  Y2 M: ' CY FV( # +=  YY 

KFC=M = VCKV  YY :C06YC=M (6 FC=M YV=6:
( Y=#(: /V(  FF( C V( (
(  CFF(M P' - L

*

C( CM FV( 6 YCYF( :T ++ =C FC=M2 36YF Y = FV( F(+(

F6 ( ;(: CM FV( + (#C=6Y Y(KFC=M FV(Y( YY6+FC=MY ( K=V( (MF YCMK( FV( Z(CY M6
(

B - %2 V( (  CY FV( CY=FV(   :C06YC=M K=(KC(MF2 CY = = :( 6MCFT =  Y(=6Y CF6 (  V(MK( FV( (  FC=M FC(Y CM FV(  Y ( (M(

T 6KV Y ( FV M FV=Y( CM FV( C6C:2 + =#C:(: FV( :(+FV = FV(  Y CY M=F F==  ( ,M = :( F= 6M:( YF M: FV(
YCK =( = M CM( F  Y CFV=6F (MF( CM CMF= :(F C(: :(YK C+FC=MY = FV(  Y +V Y( :TM CKY2 ( C V( ( K=MYC:(

FV F FV(  YY 

KFC=M2 36YF Y FV( F(+(

F6 (2 CY C+=Y(: F FV( 6++(


=6M: T2 C( ' - ' F 1 - < B , *  VC( F FV( CMF(  K(2 CF C
( YY6(: FV F FV( CM( F  Y CY M=F
Y=
(:
T FV( C6C:2 TC(:CM < '  B !%  B8 ' - L

F 1 - <

*X

V( ( ! - ! ! M: %  - % % ( FV(

FC=Y = :(MYCFC(Y M: = FV(   :C 6YC#CFC(Y2 (Y+(KFC#(T W CM2 >( F?M6:Y(M CM(FCK (6 FC=M CY M((:(:

20

Pierre Colinet

= FV( (# +=

FC=M

F( 2 VCKV C V( (
(  CFF(M 6M:( FV( =   -

! " 



'  B < '

*E

V( ( " - " 1 - < CY FV( :C(MYC=M(YY CMF(  KC  F(+(

F6 (2 ' ' 1 - < CY FV(  YY 

KFC=M = # += F FV( CMF(  K(2 ! "  - # " - CY FV( YK (: Y F6

FC=M + (YY6 (2 M: FV( CM(FCK (YCYF MK( F= (# +=

FC=M CY 6 MFC;(:
T  !% *<>" *5 ) 6 C( CM#( Y(T + =+= FC=M  F= FV( 6Y6  KK==: FC=M K=(KC(MF  7=F( FV F CM FV( + (Y(MF F ( F(MF = (# +=

FC=M2 (CFV( = +6 ( # += = =  Y CF6 (2 +=YYC
( =FC=MY = FV( CMF(  K( ( M((KF(:2 C( 1 - @5 < /VCY 6 YCYF( :T YY6+FC=M CY # C: + =#C:(: CFY ( FC#( =FC=M :6 CM FT+CK  (  FC=M FC( ) % CY Y 2 C( 8 @  < CM =6

KV=CK( = (MFV M: FC( YK (Y WKF6 T2 FV( F=F  36+  YY
 MK(

F FV( CMF(  K( ( :Y  - 8 @ - ! 8 @

F 1 - @5

*"

V( ( M: ( #( FCK  #(=KCFC(Y = FV( C6C: M: = FV(  Y CF6 (2

(Y+(KFC#(T W#(

CM CM FV( V= C?=MF  :C (KFC=M M: F CM CMF= KK=6MF FV F   - L
(K 6Y( = Y 5 M: (MK(2 FV( 6 YCYF FCK ++ = KV C
( # C: + =#C:(: FV( ( M (# +=

FC=M

F(  CY Y  (M=6V K=+ (: F= 6MCFT WM=FV( YY6+FC=M CY FV F FV( #( FCK  #(=KCFT = FV( C6C: F FV( CMF(   K( CY M((KF(:2 C(  << CY YY6(:2 (#(M FV=6V ( YV=6: KF6 T V #( 1 - < -   /VCY YY6+FC=M2 VCKV  T
( KV(K (: +=YF(

C= C2 YV=6:
( # C: + =#C:(: FT+CK  #(=KCFC(Y CM:6K(:
T FV( H

M=MC = R T(CV (0(KF ( 6KV VCV( FV M FV( C6C: #(=KCFT CM:6K(:
T (# +=

FC=M!  ,M = :( F= #=C: M=MCM( CFC(Y CM FV(  YY F

MY(
=6M: T K=M:CFC=MY2

M: F= ( CM K=V( (MF CFV FV( YY6+FC=M = Y  (# +=

FC=M

F( 2 CF 

0 *,% #0  ,8#,= * ;, =;   #;  % #, "& =, =% 0= ;= 0 %8= #+# := ,#% #0,=  =: %#0 0 $= ;;= #% % = ; #0 %8= =;=0% 0 ;#; 0 %0

%8= (00# 0=    
 #; 0%8= "& =, =% 0= ;= 0 %8= % #, %8=, #  := ,#% 
  9;   #; * = 0#% ' = %8= %# * %8=, #  0 =: %#0 := ,#%#=; #; * =      $8#,8 ;%#=; + ' ;#0,= %8=  6 : = *  $#  $; = ,8 = %80  '   %= %8% %8= =;0#0  ; 8 ; * 0,2#:=0 ,0:=,%#0

Interfacial Patterns and Waves in Liquid Layers and Thin Films

21

CY K=M#(MC(MF F= YY6( FV( YTYF( F=
( K=Y( F= =
 (6CC
C6 YF F( :(;M(:
T FV( =#(

  Y + (YY6 ( (2 F=Y+V( CK M: FV( (F( M T C+=Y(: # +=  YY 

KFC=M '  WF (6CC
C62  - L M: FV(  YY 

KFC=M CY (6  F= ' (#( TV( ( CM FV(  Y2 VC(
=FV FV( C6C: M: FV(  Y ( F F(+(

F6 ( "  :(F( CM(:
T  *E2 C( ! "



-

'  B < '

*

/V(M2 YY6CM Y  :(#C FC=MY =6M: FVCY (6CC
C6 YF F(2
=6M: T K=M:CFC=MY *X M: *E K M
( CM( C?(:2 ( :CM F=  B8 ' - L < '  B !%

M:  -

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" 

F 1 - <

L

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<

V( (  CY (# 6 F(: F FV( Y F6

FC=M F(+(

F6 ( "! - -"   7=2 Y(+

FCM FV( V= C?=MF  #(

(  = FV( =6 C( K=+=M(MFY CFV  #(M6
( 2 Y CM Y(KFC=M <2 ( V #( F= Y=#( 8 '  - L M: 8  '& - L Y6
3(KF F= K=

(Y+=M:CM
=6M: T K=M:CFC=MY WF( Y=(

(


2 M: F CM CMF= KK=6MF FV F FV( K=+(F( CMF(  KC  (M( T
 MK(   "  M: FV F FV( FV(   + =
( CM FV(  Y K M
(

( :Y 8 " - 8 Y=#(: Y
(= (2 (  CM (F C(:
=6M: T K=M:CFC=M 8 "& B 32"& - L

F 1 - <

*

= FV( =6 C( K=+=M(MFY CFV  #(M6
( 
- L2 V( ( FV( (0(KFC#( DC=F M6
( CY M= C#(M
T  K=FV, B 32 - 32  B 32#  - 

! "   B

 
      
 "#   




M: FV( =6 C( K=+=M(MFY = FV( (# +=

FC=M

F( ( & - 32# "& - , K=FV,

!%  B '& ,< ' 

 

,M ::CFC=M2 FV( V= C?=MF  #(

( = FV( CMF(  KC  (M( T
 MK( C#(Y FV(
=6M: T K=M:CFC=M = " 2 C( 8 "  B 32 "  "  B 32# "  "   - L

F 1 - <

X

22

Pierre Colinet

V( ( 32 - 32  L M: 32# - 32#  L2 VC( FV( #(

(: (# +=

FC=M

F( ( :Y  - 32# "  "   -

!%  B '  '  ,< ' 

E

/= Y6 C?(2 FVCY Y(KFC=M V Y :(=MYF

F(: FV F2 CM Y=( #CKCMCFT = =
 (6CC
C6 YF F( :(;M(:
T FV( F=F   Y + (YY6 ( M: FV( # +=

 YY 

KFC=M ' F Y=( :CYF MK(  = FV( CMF(  K(2 K=M#(KFC#( (# +=

 FC=M = +6 ( C6C: CMF= M CM( F  Y  T
( :(YK C
(:
T FV( YTYF( = (6 FC=MY 52 F=(FV( CFV
=6M: T K=M:CFC=MY 
FC= , /VCY + =
( CY M=F =MT M=MCM( 2
6F Y= M=M=K 2 Y K=6:
( ( C?(:
T ++TCM M CM#( Y( =6 C( F

MY=  F= Y *2 ( : CM F= K=M#=6FC=M + =:6KF >=(#( 2 CF CY =YF K=M#(MC(MF F=  CF(  M: Y=#( CF CM =6 C( Y+ K( !!

)3 3 3%#3 131

!6( F= CFY YT(F C(Y2 FV( 6 YTYF( = (6 FC=MY M:
=6M: T K=M :CFC=MY  TY :CFY V= C?=MF T V==(M(=6Y Y=6FC=M $M =M FC( YK (2 FVCY Y=6FC=M (#=#(Y
(K 6Y( = FV( :(K ( Y( = FV(  T( FVCK M(YY CM FC( >=(#( 2 CM FV( 6 YCYF( :T ++ =C FC=M :(YK C
(: CM FV( + (#C=6Y Y(KFC=M2 FV(  (( Y6  K(  T
( ;(: F 1 - <2 M: YF( :T Y=6FC=M (CYFY2 M (T  # - L #

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V( ( FV( :C(MYC=M(YY F(+(

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F6 ( YK ( --

32 " "  B 32# " "!  < B 32 B 32#

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Y K M
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:(;MCFC=M = FV( =
 (6CC
C6 YF F( 7=2 6YCM  E M: F(

Y=( (


2 FV( (# +=

FC=M

F( CM FV( M=M(6CC
C6 (( (MK( YF F( 5  T
(  CFF(M Y  # -

" " 

32

# B < B 32 



Interfacial Patterns and Waves in Liquid Layers and Thin Films V( ( " -

  "  B 32 " ,"  B "  < B 32 , B

23

 L

CY FV( CMF(  KC  F(+(

F6 ( CM ;KFC#( +6 (T K=M:6KFCM YF F(2 V( ( (# +=

FC=M CY + (#(MF(: C(
T YY6CM # MCYVCM KK==: FC=M K=( ;KC(MF ( :CM F= 32# - L ,F C
( 6Y(6 CM FV( ==CM F= :(;M( > -  #

> -

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> -

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M: F= (K YF   CM FV( =  > B > B >  # -

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VCKV K( T YV=Y Y(( Y= > 6F M: 4=CM(F2 *LLX FV F FV( (# +=

FC=M )6 CY CCF(:
T FV (( (YCYF MK(Y CM Y( C(YN2 M (T FV( CMF(  KC   C M(FCK (YCYF MK( > 2 FV( (YCYF MK( > F= :C06YC=M CM FV(  Y CF6 (2 M: FV( (YCYF MK( > F= V( F K=M:6KFC=M F= :Y FV( CMF(  K(  =
=FV C6C:

M:  Y +V Y(Y $FV( 6Y(6 (+ (YYC=MY = FV( (( (MK( (# +=

FC=M

F( K M
( =6M:  = Y <2 E M:  *  > B >  # -

" 

 

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' # '   X  B < ' 

> B >  # > B >  #

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 = VCKV # C=6Y CCFCM K Y(Y K M
( C:(MFC;(:  C V(M >  > =

>  > 2 K=+ CM Y  * M:   ( :CT YV=Y FV F " # - " 2 C( FV( CMF(  KC  F(+(

F6 ( CY M=F 0(KF(:
T (# +=

FC=M M: ( KV(Y CFY # 6( CM FV( +6 (T K=M:6KFCM YF F(O CC V(M >  > = >  > 2 Y  * M:   ( : F= ! "  " # "   - ' # '  B <

'  2 VCKV ( MY FV F =K  FV( =:TM CK (6CC
C6 + (# CY F FV( CMF(  K(2 KK= :CM F=  <O CCC V(M >  > = >  > 2 Y  *

M:  X ( :CT TC(: ' # - ' 2 M: FV(  Y K=+=YCFC=M CY 6MC=  4( T2 F= = FV(Y( FV (( YCF6 FC=MY  T
( ( C?(: YC6F M(=6YT =

CMYF MK(2 CM FV( K Y( >  > M: >  > 2 FV( (# +=

FC=M

F(  #  ! "  " "   CY /"$&/  2 (#(M FV=6V FVCY K Y( YV=6: =MT 

:=0 %88  8;= %0;#%#0 #; 0%  =,%#0 %8#; %= #; ;= 8== % =0 %8% %8= ##%%#0 ;%= #; #06= % 6#0=%#, ==,%; #= %8= #0%=*,= #; 0% % =+# ##

24

Pierre Colinet

++T F= (F ((T FVCM C6C: M:  Y +V Y(Y M=F( FV F  ! "   CY (6  F= 32#  Y((  <" = FV( +6 ( # += K Y(  FV=6V 32# CY V( ( #( T Y  V(M > :=CM F(Y2 (# +=

FC=M CY +/&-$&/2 C( (MFC (T :(F( CM(:
T K=M:6KFC=M F= :Y FV( CMF(  K(2 VCKV ( CMY

F FV( Y F6

FC=M F(+(

F6 ( "   CM T2 V(M > :=CM F(Y2 =M( V Y FV( --&-$&/ (C(2 V( ( (# +=

FC=M CY : C#(M
T FV(  YY 

KFC=M :C0( (MK( ' ' 2 CM VCKV ' CY FV( (6CC
C6  YY 

KFC=M K=

(Y+=M:CM F= FV( CMF(  K( F(+(

F6 ( " CM FV( +6 (T K=M:6KFCM YF F( 7=F( FV F FV(Y( K=MYC:(

FC=MY ( # C: = FV( (( (MK( YF F(2
6F := M=F M(K(YY CT ++T F= )6KF6 FC=MY =6M: FV(  FF(  !

2%3 1' %'11

/V( YF
CCFT = FV( YF( :T (( (MK( YF F( 5 CY YF6:C(:
T ::CM CM;MCF(YC  +( F6
FC=M ;(:Y F=  # C
(Y2 C(  -  # B  #

" - " # B "  #

-

#

B



 E

M: CM( C?CM FV(
YCK (6 FC=MY CFV (Y+(KF F= FV( + C(: 6 MFCFC(Y /V(M2 =CFFCM FV( + C(Y2 ( K M =
F CM YC+( YTYF(
T ; YF = CM FV( :C#( (MK( =  " M: 6YCM  52 ( :CM F= P - > 8 "  /VCY

=Y (CCM FCM  Y ( Y  M:   = FV( 6 YTYF(2
T ++TCM FV( Z + KC M =+(

F= F=  "2 M: 6YCM FV( ( FC=M 36YF =6M: ( FV(M =
F CM 8 P B > P " -   P  5 85 8" P" B  " 85 V( (  YF M:Y = FV( :C(MYC=M(YY 1:( C# FC#( FV =6V=6F CM FV( == CM2 VC( P - 8 B 8 CY FV( V= C?=MF  Z + KC M =+(

F=  /VCY YTYF(  M: YY=KC F(:
=6M: T K=M:CFC=MY :CYK6YY(: CM + (#C=6Y Y(KFC=MY K M
( =6 C( F

MY= (: CM FV( V= C?=MF  + M(2 M: :CFY Y=6FC=MY = FV( =      & 1 (+I5%   C1 "& /V( K=

(Y+=M:CM (C(M# 6( + =
( =  FV(M ( :Y      > C -     

XL

  C B  - C

X<

CFV
=6M: T K=M:CFC=MY (6C# (MF F= Y 
F 1 - L

X*

Interfacial Patterns and Waves in Liquid Layers and Thin Films  -   B  6 C - C B 32C - L

F 1 - <

25 X

!6( F= YT(F C(Y = FV( = CCM  + =
( CM# C MK( CFV (Y+(KF F= =F  FC=MY M: F

MY FC=MY CM FV( V= C?=MF  + M(2 FV( (C(M# 6( + =
( =MT :(+(M:Y =M FV( =:66Y  -  B    = FV(  #(#(KF=  -  #   = FV( =6 C( =:( K=MYC:( (: /V(
=#( Y+(KF

 + =
(  T
( Y=#(: ( KFT
T Y6+( +=YCFC=M = Y=6FC=MY = FV( =  (+I 1%2 CFV 
 C#(M
T FV( ==FY = FV( KV

KF( CYFCK (6 FC=M          B  > - L

X 

/V6Y2 FV( 1:(+(M:(MKC(Y = F(+(

F6 ( M: #(=KCFT +( F6
FC=MY ( : (K6:CM :((M(

F( K Y(Y Y C-






7 (+I 1% #

-



7  B    (+I 1%

XX




M: FV( YC CM:(+(M:(MF CMF(

FC=M K=MYF MFY V #( F=
( :(F( CM(:  = FV(
=6M: T K=M:CFC=MY /VCY ( :Y F= CM( V==(M(=6Y YTYF( = FV( =  A $ - L2 V( ( $ CY FV( #(KF= = 6M M=MY 7 M: A CY E  E  F C /V(M2 FV( K=+ FC
CCFT ( FC=M = FV( YTYF( VCKV 6

MF((Y FV( (CYF(MK( = M=MF C#C  Y=6FC=MY CY :(FA - L2 VCKV ( :Y F= FV( :CY+( YC=M ( FC=M  I# #  # 32# 6 # > % - L

XE

QC#(M   M: 322 FVCY ( FC=M  T CM + CMKC+(
( Y=#(: = FV( :CY +( YC=M ( FC=M  - > # 6 #  >=(#( 2 FVCY CY F(KVMCK T C+=YYC
( FV( ( Y=M = FVCY
(CM FV( K=+CK F(: :(+(M:(MKT = FV( ==FY  6+=M 2 M: FV( :(F( CM MF XE CM (M(

 V Y F=
( K K6 F(: M6( CK T2

M: K=
CM(: CFV M CF(

FC#( (FV=: = ==F ;M:CM ( 7(F=M'Y (FV=: /V( ( ( V=(#( + FCK6 K Y(Y2 = VCKV M TFCK  (Y6FY K M
( =
F CM(: = C#(M # 6(Y = FV( K=MF = +

(F( Y > M: 6 M: ;(: # 6(Y =   M: 322 FV( (( (MK( YF F( C
( 6MYF
( C FV( ( (CYFY F ( YF # 6( =  = VCKV   L2 VC( CF C
( YF
( F= CM;MCF(YC  +( F6
FC=MY C  . L =   /V( FV (YV=: = CMYF
CCFT CY FV(M C#(M
T  - L , ( -  - L2 =M( Y+( Y = M(6F

 YF
CCFT =M=F=MCK =:(Y2 VC( C (
- L F K CFCK CFT2 =M( V Y =#( YF
CCFT =YKC F= T =:(Y2 =  #(Y

26

Pierre Colinet

  3  Z(F  M(6F

 YF
CCFT K6 #(Y = FV( H

M=MCDG(M : + =
( > - L = # C=6Y "$-$ DC=F M6
( Y 32 RCVF  =K6Y = CMC = M(6F

 YF
CCFT K6 #(Y = FV( R T(CVH

M=MCDG(M : + =
(2 =

32 - L

%%#  31 ,M K Y(  - L M(6F

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CCFT2 FV( + =
( XLX K M
( Y=#(: M TFCK T = > - L2 Y :=M(
T ( Y=M ( Y=M2 <X" 4( T2 =  - L2 FV( + =
( CY CM:(+(M:(MF = FV( 

M:F M6
(  2

M: FV( K=+ FC
CCFT ( FC=M K M
( Y=#(: = 6 2 ( :CM F= 6 -

<E K=YVI% B 32 YCMVI%* YCMVI*%   K=YVI% B  YCMVI% YCMVI%

X5

/VCY ( FC=MYVC+ CY (+ (Y(MF(: CM C 5(F = # C=6Y DC=F M6
( Y2 V( ( F (M Y K=MYF MF ,F CY Y((M FV F FV( M(6F

 YF
CCFT K6 #(Y + (Y(MF

CMC6 6  F ;MCF(  -  2 + =#C:CM FV( K CFCK  K=M:CFC=MY V(M 6 - CMK ( Y(Y2 FV( FV (YV=: CY ; YF ( KV(: F 6  2 V( (   KV M(Y  = M( FC#( F= +=YCFC#( W
=#( 6  2 ;MCF(
M: =  #(M6
( Y V #( +=YCFC#(  =FV

F( WM CMK ( Y( = FV( DC=F M6
( ( :Y F= M CMK ( Y( = 6  2 FV( (= ( YF
CC?CM FV( (( (MK( YF F( = >
- L2 FV( + =
( XLX CY FV( =M( Y=#(:
T 7C(: CM <E  ,FY Y=6FC=M CY (+ (Y(MF(: 

+VCK T CM C 5 CVF2 = 32 - L M= (# +=

FC=M2 M(CC
(  Y FV(   K=M:6KFC#CFT /V( =K6Y = FV (YV=: K=M:CFC=MY ++( Y K=Y( F= YF

CVF CM( C#(M
T >  6  B -< >  6 

X"

M: =CM F= :(F( CM( FV( K CFCK  K=M:CFC=MY +=CMF  = C#(M # 6( = FV( :TM CK D=M: M6
( 30 - > 6 - !)  #(M FV=6V FV(Y( CM( YF
CCFT (Y6FY ( ( = CCM T =
F CM(: CM FV( M=M#= FC( K Y(2 FV( :CYK6YYC=M = FV( + (#C=6Y Y(KFC=MY YV=(: FV F

Interfacial Patterns and Waves in Liquid Layers and Thin Films

27

= Y6KC(MFT Y  (# +=

FC=M )6(Y2 FV( YF
CCFT = #= FC( C6C:  T( K M
( YF6:C(: =M FV(
YCY = FV( Y ( Y+(KF

 + =
(2 6YCM (M(

C?(: DC=F M6
( 32 C#(M
T   7=F( FV F CM FVCY 

(2 Y6KV M=M#= FC( CCF =  T K=

(Y+=M:Y F= # MCYVCM KK==: FC=M K=(KC(MF 2 =   CM FV( CMF(  KC  CM(FCK   *E ,M Y6KV  K=FV,2 /"$&/ (C(2   ( :Y F= 32 - 32  -  C( V( F F

MY( F FV(  (( Y6  K( CY CCF(: F= ( V( F K=M:6KFC=M  L < H= (=#( 2 KK= :CM F=  "2 CM FV(  Y +V Y( FT+CK T2 

M=M?( = F(+(

F6 ( :C0( (MK( - K M =MT
( K ( F(: V(M (F( M T K=MF =(: F(+(

F6 (Y F FV(
=FF= " M: F FV( F=+ " ( :C0( (MF V(M FV( CM(FCK (YCYF MK( F= (# +=

FC=M > -  CY ;MCF(2 (# +=

FC=M K M (M(

F( ;MCF( -2 V(MK( +=YYC
( CMYF
CCFT2 (#(M CM FV(
Y(MK( = (F( M  : C#CM = K( " "  H= (=#( 2 FV( (0(KFC#( DC=F M6
( CM K ( Y(Y :6( F= FV( ::CFC=M  K==CM (0(KF (M(

F(:
T (# +=

FC=M2 VCKV 6CK T
(K=(Y  ( FV M +6 ( V( F K=M:6KFC=M CM FV(  Y WKF6 T2 CM FV( K Y( = C6C: (# +=

FCM CMF= CFY
/ *
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(  T F6 M =6F F=
( (F ((T  ( :(+(M:CM =M FV( KK= =: FC=M K=(KC(MF2 CF K M
( = FV( = :(
(Y+=M:CM K CFCK  H

M=MC M6
( CY FV(M Y= #( T VCV2 FV( (( (MK( YF F( CY FV(M (M(

T YF
( F= FV( H

M=MC (0(KF /VCY CY ( YCT 6M:( YF M:
( YCMK( CM Y6KV +/&-$&/ (C( V( ( >  > M: > - L2 FV( CMF(  K( YV=6: ( CM K=Y( F= FV( Y F6

FC=M F(+(

F6 ( K=

(Y+=M:CM F= FV( # += + (YY6 ( VCKV2 + F = Y  VT: =:TM CK )6KF6 FC=MY2 YV=6:
( =6VT K=MYF MF /VCY ( MY FV F Y6  K( F(MYC=M 

:C(MFY C (M(

T ( CM 6CF( Y 2 + (#(MFCM FV( =K +C T )=Y >=(#( 2 (#(M FV=6V M=F K=MYC:( (: V( (2 CF K M
( K=M3(KF6 (: FV F FV( YTYF( K=6: FV(M
( VCVT 6MYF
( F=
6=T MKT: C#(M K=M#(KFC=M 7=2 V(M ::CM M CM( F  Y Y6KV Y MCF =(M CM FV( # += +V Y(2 FV( (0(KFC#( DC=F M6
( : =+Y :=M 6CF(

+C:T FV( Y(K=M: F(  CM FV( :( M=CM F= =  2 + =+= FC=M  F= > 2 6CK T (K((:Y > -  /V( CMF(  K( ( CMY M( =K  FV( =:TM CK (6CC
C62
6F )6KF6 FC=MY = FV(
 # += + (YY6 (2 CM (: F= K=MK(MF

FC=M )6KF6 FC=MY2 ( M=

=(: 4( T2 FV( CMK ( Y( = > CY :6( F= FV( ::CFC=MM  :C06YC=M


C( YF M MF ; = CM( F  Y FV=6V VCKV FV( # += 6YF :C06Y(2
6F FVCY (0(KF CY Y=(V= K=6MF( 
 MK(:
T FV(  KF FV F FV( + FC  + (Y Y6 ( C+=Y(: F Y=( :CYF MK(  = FV( CMF(  K( Y= :(K ( Y(Y2 FV( (= ( CMK ( YCM FV( : C#CM = K( = (# +=

FC=M FV( CVFV M:YC:( =   * K M CMK ( Y(2 YCMK( KK= :CM F=  *2 "  :(K ( Y(Y V(M ' :(K ( Y(Y WKK= :CMT2 FV( M(F (0(KF CY CM (M(

 :(YF
CC? FC=M = FV( (# +=

FC#(

(( (MK( YF F(2 Y C6YF

F(: CM C "

28

Pierre Colinet

  3 + ZCM( YF
CCFT (Y6FY = (FV M= (FVT K=V= (# +=

FCM CMF= MCF =(M2 = " - " - *" 2 M KK==: FC=M K=(KC(MF  - L <2

M: M(CC
(
6=T MKT (0(KF Z(F 9 - 6 6  Y 6MKFC=M = FV( # +=  YY 

KFC=M ' 2 = :(+FVY ) - ) - X  = C6C: M:  Y +V Y(Y2 M: - - L <# L *# L X  RCVF 9 - 6 6  Y 6MKFC=M = ) CFV ) - ) = F= F(+(

F6 ( : =+Y - - L <# L X  M: Y(#(

 # +=

 YY 

KFC=MY '  /V( CMYF
CCFT FV (YV=: 9 - < CY CM:CK F(:
T : YV(: CM(

,M FVCY ;6 (2 FV( Y6+( K CFCK CFTN 9 - 6 6  CY :(;M(: Y FV(

 FC=
(F((M FV( KF6  # 6( = FV( H

M=MC M6
( + =+= FC=M  F= FV( F(+(

F6 ( : =+ -
(F((M FV(
=FF= + F( M: FV( CMF(  K(2 M: CFY K CFCK  # 6( =
F CM(:
T CMCC?CM X5 CFV (Y+(KF F=  ,M = :( F=

( CM K=Y( F= =
 FV( =:TM CK YF F( =  K=M:CFC=MY = C "2 FV( F=F  + (YY6 ( V Y
((M :36YF(: CM ( KV K Y(2 Y6KV Y F=  CMF CM K=M YF MF F(+(

F6 ( : =+ -2 C( K=MYF MF : C#CM = K( = FV( CMYF
CCFT = FV( K6 #(Y K=

(Y+=M:CM F= - - L <  =M FV( CVF +=F2 FV( K=

( Y+=M:CM # 6(Y ( - L L"E# L <*# L <# L X# < < F2 (Y+(KFC#(T =

' - L # L 5# L X# L # L <2 M: =YF CM:(+(M:(MFT = ) 7=F( FV F =

' - <2 9  < =  ) YF
( (( (MK( YF F( WKF6 T2 FV( # 6(Y = 9 CM C "(F = ' - < M: - - L <# L *# L X  ( 9 - L LL"# L L<5# L L *2

(Y+(KFC#(T 7=F( ;M T FV F2 CM FV(Y( K K6 FC=MY2 FV(  Y CF6 ( + =+ ( FC(Y :(MYCFT M: FV(   :C06YC#CFT ( (YFC F(: 6YCM +( (KF  Y  Y

M: CCM 6(Y :(YK C
(: CM DC : (F 2 <EL -1# '  31 WY (MFC=M(: CM Y(KFC=M *<2 =YKC F= T =:(Y ( M=F (+(KF(: V(M
=FV > M: 6 ( +=YCFC#( V( FCM  = FV(
=FF=2
6F

FV( CM FV( =++=YCF( K Y( > # 6 . L V( FCM  = FV( F=+ /V( K Y( = M( FC#( :TM CK D=M: M6
( Y 30 - > 6 CY

FV( (=FCK

Interfacial Patterns and Waves in Liquid Layers and Thin Films

29

M: M=F K=MYC:( (: V( ( F(  ,M = :( F= :(=MYF

F( FV F FV( K=6+CM
(F((M CMF( M  M: Y6  K( =MCF6:CM   #(Y K M ( : F= CMYF
CCFT2 Y YV=M ; YF CM R(:MC =# (F 2 *LLL2 =M( V Y F= Y(( = Y=6FC=MY = FV( K=+ FC
CCFT ( FC=M XE CFV  - 2(2 V( ( ( CY FV(  (   (6(MKT ,M (M(

2 Y6KV K K6 FC=M =  CM  YF
CCFT K=M:CFC=MY = =YKC F= T =:(Y YV=6:
( +( = (: M6( CK T2
(K 6Y( = FV( K=+CK F(: :(+(M :(MKT =  XE 6+=M  /VCY K M
( :=M( CM Y(#(

  TY2 (
T ; YF Y=#CM  XE CFV (Y+(KF F= 6 2 ( :CM F= 6MKFC=M 6 # > #  VCKV CY (M(

T K=+( C 
- L /V(M2 FV( # 6(Y = (# >  Y FCYT CM I6  - 2 (# > # % - L ( M6( CK T F

K (: M=F( FV F (# >  K M
( 6FC# 6(: > #CM K MK((: FV( C CM T + F2 FV( (  + F :C

(KFT + =#C:(Y FV( H

M=MC M6
( = VCKV  #( =:( CY  CM T YF
( F FV(Y( # 6(Y =  M: >  H= (=#( 2 F CM CMF= KK=6MF FV F > 6 - 302 FV( (6 FC=M > - 30 6 2(# > # > #  6YF
( Y=#(: CFV (Y+(KF F= >  /V(  CM  YF
CCFT K6 #(Y ( (+ (Y(MF(: CM C (F Y 6MKFC=M = 6 2 = 32 - L M= (# +=

FC=M2 M(CC
(  Y FV(   K=M:6KFC#CFT ,M C 2 FV( =K6Y = K CFCK  K=M:CFC=MY CY (+ ( Y(MF(: CM FV( + M( > # 6  7=F( FV F K CFCK  # 6(Y = FV( R T(CV M: H

M=MC M6
( Y ++( F=
( 6KV VCV( FV M = =M=F=MCK =:(Y2 FV=6V ( C?
( CM + CMKC+(2 = = #CYK=YCFT C6C:Y Y6KV Y  F( M: :(+FVY CM FV( K(MFC(F(

M( WY YV=M CM R(:MC =# (F 2 *LLL2 M TFCK  (Y6FY K M Y=
( =
 F CM(:2 CM FV( CCF >  M: 6  YFC CFV 30 - > 6

;MCF( 6YCM  FKV(: YTF=FCK (+ MYC=MY $MT + F = FV( FV(= T CY + (Y(MF(: V( ( C YF2 CF C
( 6Y(6 F= KV M( FV( FC( YK (  = FV( FV(   =M(  - ) % F= FV(
6=T MF =M(  -  )-  M=F( FV F - . L V( ( /VCY =6MFY F= 6FC+TCM
=FV  M: 
T   -   >   CM Y XLX !(;MCM * -   >   2 FVCY YTYF( K M
( ( CFF(M



*     B  C -   

X

*     C B  - C

EL

 -  - C - L

E<







 - *    30

F 1 - L

C - C B 32C - L

F 1 - <

E*

WY :CYYC+ FC#( FC( YK (Y ( 6KV =M( FV M FV(
6=T MKT  M: FV( = K +C CFT FC( YK (Y = >   <# 30 - <2 CF CY M=F Y6 + CYCM FV F 

%= %8% %8= = =0=0, * %8= #; =;#0 = %#0  0 %8= =%=; ;,8 ;
 0 ' #; #%%= 8== * , #% * 0%%#0

30

Pierre Colinet

  3  $MY(F =  #(Y CM C6C:  T( V( F(:  =
=#(2 =   - E

M: 32 - L Z(F  CM  YF
CCFT K6 #(Y = Y(#(

 # 6(Y = FV( :TM CK D=M: M6
( 30 - > 6  /V( : YV(: CM( CY FV( =K6Y = CMC K CFCK  K=M:CFC=MY RCVF YF
CCFT :C 

 CM FV( 7C(:'Y + M(N /V( K CFCK   #(M6
( # C(Y  = L F=  V(M 30 CMK ( Y(Y

VCV(YF = :( :CYYC+ FC#( F( Y CM FV(
6 (6 FC=MY X M: EL ( 6FC+C(:
T *  < >=(#( 2 M=F 
=6M: T K=M:CFC=MY F 1 - L M: 1 - < K M
( Y FCY;(: C FV(Y( F( Y ( YC+T : =++(: =6F CM FV( CCF * L2 M: FV( K=

(KF (Y=6FC=M = FV( + =
( (6C (Y FV( 6Y( = YF (FKV(: #( FCK  K== :CM F(Y CM FVCM
=6M: T  T( Y =K F(: M( FV(
=6M: C(Y ,M FV(
=FF=
=6M: T  T( 2 ( :(;M( 1& - 1*2 Y6KV FV F  - * 8 WF FV( ( :CM = :( 2 FV( (6 FC=MY F=
( Y=#(: ( FV( (= ( 8  & - 8  &

E

& -  C&   8 C& B 

E 

V( ( FC:(Y ( 6Y(: F= :CYFCM6CYV # C
(Y CM FVCY
=FF=
=6M: T  T(  7=F( FV F C:(MFCK  (6 FC=MY V #( F=
( Y=#(: =   M: C CM FV( 6++(


=6M: T  T( 2 CFV FV( YF (FKV(: K== :CM F( V( ( :(;M(:
T 1 - < 1* CFV FV(Y( :(;MCFC=MY2 FV(
=6M: T K=M:CFC=MY (  & - 8 & - C& - L

F 1& - L

  30 C - 8C *32C - L   - 8 

EX

F 1 - L

EE

M: YT+F=FCK  FKVCM K=M:CFC=MY
(F((M FV(
=6M: T  T( Y M: FV(
6 ( : & 
  C
  & 
  
 # C E5   
  
#

 
  C


C

E"

Interfacial Patterns and Waves in Liquid Layers and Thin Films

31

V( (  M: C ( FV( +=F(MFC  )= Y=6FC=MY =
F CM(:
T Y=#CM Y X M: EL = * - L /VCY :C (KFT ( :Y F= C - K=YVI1% B
YCMVI1% M:  - C CFV  - 
  FKVCM K=M:CFC=MY E5 CY C -
YCMVI1% #

 -
 YCMVI1%

E

VC(  & - C& - L (#( TV( ( CM FV(
=FF=
=6M: T  T(  ,M FV( F=+
=6M: T  T( 2 YCC (Y=6FC=M TC(:Y     -
 YCMVI% < (+I   1% 5L

 ( :T Y FCYTCM   
 - L M: FV( ; YF = Y E" /V(M2 Y=#CM = C

M: F CM CMF= KK=6MF FV F FV( V( F F

MY(
=6M: T K=M:CFC=M (:6K(Y  
 - L F FV( =(YF = :( 2 ( =
F CM F= 8C   (+I   1% (+I    1% B C -
YCMVI% < B   <    <   

5<

CM T2 CF ( CMY F= (+ (YY FV(  YF H

M=MC
=6M: T K=M:CFC=M2 VCKV K M ( YCT
( (

M(: Y   
YCMVI%  B -L 5* 30< B     / CM CMF= KK=6MF FV F  - < B    2 M=MF C#C  Y=6FC=MY

- L ( +=YYC
( =MT = :CYYC+ FC=M(YY  #( =:(Y  - 2(2 V( ( (CFV(

  ( - ( # : - <# *# 5  B : < = V(M

 ( - (" -  30< B    

5 

/V( ; YF = FV(Y(  #( =:(Y CY FV( D 6MF A

CYA

A

 CMF( M   #(2 VCKV2 CM ::CFC=M F= FV(  #(M6
( 2 CY +

(F C?(:
T FV( M6
( = #= FCK(Y

32

Pierre Colinet

: CM FV( #( FCK  :C (KFC=M # C YCMI:<1% /V( Y(K=M: =:( CY (M6CM(T

( F(: F= FV( H

M=MC (0(KF2 M: K=

(Y+=M:Y F= FV( Y6  K( =MCF6 :CM   #(2 FV( (KV MCY = VCKV  Y :(YK C
(: CM Y(KFC=M *< 7=F( FV F K=MF

T F= FV( CMF( M   #(Y = VCKV   - L F FV( =(YF = :(

CM FV( 6++(
=6M: T  T( 2 Y6  K(  #(Y ( KV

KF( C?(:
T  
- L2 V(MK(
T M CMF(MY( V= C?=MF  #(=KCFT ;(: &  - 2 * 82  C( * FC(Y  ( FV M =FV( 6 MFCFC(Y CM FV( CCF * L D=FV FV(Y(  #( =:(Y ( M(CFV( : +(: M= +C;(: =M FV(  YF FC( YK ( K=MYC:( (:2 M: =MT M(F = :( K=

(KFC=MY F= FV( K=+(  =FV

F(  K M ( : F= CM=  FC=M =M FV(C =MF( 
(V #C=6  $CFFCM :(F CY = K=MKCY(M(YY2 ( V #( F= (+ M:  - 2( B * 2 Y ( Y F= K=MYC:( * K=

(KFC=MY F= #(=KCFT

M: F(+(

F6 ( ;(:Y CM ( KV YT+F=FCK (C=M @((KFCM (CFV( (  =

(" = FV( =(YF= :(  (6(MKT ( 2 ( ;M T =
F CM FV( K=

(Y+=M:CM  = ( KV  #(2 + =K((:CM YCC T 
6F = ( F(:C=6YT Y
=#( ,M F( (YFCMT2
=FV # 6(Y =  CM:((: +=YY(YY M=M?( = (  + F2 M: ( =6M: F= :C#( ( V(M 30 - 30 #" -

 B : < < B   

5X

CM:CK FCM FV F FV( YTF=FCK M TYCY  CY V(M FV(  (6(MKC(Y =
=FV  #(Y ( (6  CM( (Y=M MK( WM C+ =#(: YTF=FCKY CY KF6 T M((:(: CM FV( #CKCMCFT = FVCY +=CMF R(:MC =# (F 2 *LLL H= ( (C
(

(Y6FY =
F CM(:  =  ( FV(  CM  YF
CCFT FV (YV=:Y I % - L2 VCKV TC(: < B *   30 - 30  B : <  5E   B    = FV( CMF( M   #(2 M: 30 - 30" -

 B  < B   

55

= FV( Y6  K(  #(2 V( ( FV( 
 ( FV(  (  ==FY = FV( F

M YK(M:(MF  (6 FC=M  K=FI% B 32< B     - L = Y  322 =M( V Y 30" -

 B : <* < *32 B B 32  = 32  < < B    < B    

5"

M: = ( (M(

T2 : <*<  :< /V(Y( YT+F=FCK (Y6FY ( (+ (Y(MF(: CM C
Interfacial Patterns and Waves in Liquid Layers and Thin Films

33

  3 $ WYT+F=FCK (Y6FY CM FV( CCF > # 6 2 = 32 - L M:   - E  CM  YF
CCFT FV (YV=:Y = Y6  K( Y  M: CMF( M  CMF   #(Y2 F=(FV( CFV =KC = (Y=M MK(Y  (Y 2 = =YF : M( =6Y =:(Y : - <# *#  /V( (( (MK( YF F( CY 6MYF
( F= =M(  #( =:( CM FV( V FKV(: (C=MY2 F= F= =:(Y CM FV( K =YY(:V FKV(: (C=MY2 

/V(
=#( (Y6FY M: FV( ==CM :CYK6YYC=MY 6YF
( 6M:( YF==: CM FV( YT+F=FCK CCF > 2 6 2 30 - < 4=MYC:( CM 30 Y  (( +

(F( + = (YYC#(T CMK ( Y(:  = L +6 ( H

M=MC (0(KF2 CMYF
C CFT ; YF =KK6 Y F= Y6  K(  #(Y F FV( CMC  # 6( = 30"  WKK= :CM F=  55 M: C
32 - L2 FVCY C#(Y 30 - <  < B     - L 5<X =   - E2 CM  (( (MF CFV 6T M6( CK  (Y6FY ( CM C (F2 FV( CVF YT+F=F( = FV(  CM  YF
CCFT K6 #( ++( Y F  - L2 + (KCY(T F FVCY # 6( = 30 /V(M2 K=MYC:( CM ;(: # 6(Y =  M: : - < CM C
MYCFC=M =KK6 Y
(F((M FV( Y6  K(  #( =:( M: FV( CMF( M   #( =:(  M: #CK(#( Y 2 +V(M=(M=M K (: /&$  /V( (2 VCV(

:( (( = +C;K FC=M CY + (:CKF(: = FV( 6MYF
( C(: =:(2 VC( FV( YF
( C(: =:( CY = ( : +(: V=(#( FV( (  + FY =
=FV =:(Y YFC ( CM Y  K=+ (: CFV FV(  (6(MKT /VCY =:( CCM (Y6FY CM FV(  KF FV F
=#( 30 #" 2 FV( =:( CFV +=YCFC#(  =FV

F( F6 MY =6F F=
( FV( CMF( M   #( =:( ,MK ( YCM 30  CM YFC F ;(:  M: : - <2 FV( +C;K FC=M

F( = FV( CMF( M   #( =:( :(K ( Y(Y2 M:

34

Pierre Colinet

+ YY(Y FV =6V ?( = F 30 : - < /VCY :=(Y M=F ( M FV F FV( (( (MK( YF F( CY YF
(
=#( FVCY # 6( V=(#( 2
(K 6Y( M=FV(  #( =:( CFV : - * V Y
(K=( +C;(: F 30" : - * . 30 : - < Y(( C  M: 6 - > 30 WM=FV( = ( +VTYCK  (+ (Y(MF FC=M CY =
F CM(: F C#(M D=M: M6
( 30 WF C#(M :2 FV( (Y=6FC=M = 30 - 30 2 30 - 30 #" M: 30 - 30" CFV (Y+(KF F= FV( V= C?=MF   #(M6
(  ( :Y F=2 Y T2  2  #" M: "

(Y+(KFC#(T = : - <2 Y=6FC=M "  L (CYFY C 30   < B    2 VC(   L (CYFY C 30  < < B *     B     ,M FVCY K Y(2 CF CY

( :CT YV=M FV F "   #"   2 Y Y= Y((M CM C
(Y+=M: F= FV( (F M: CVF YT+F=F(Y =  CM  YF
CCFT K6 #(Y = C (F CCFY = FV( YF
CCFT
=6M: T = 6 2 :(;MCM FV( CMF( #  = 6MYF
(  #(M6
( Y F ;(: : 7=F( FV F FV( (0(KF = 32 V(MK(2 = +=YYC
( (# +=

FC=MK=M:(MY FC=M + =K(YY(Y F FV( CMF(  K( F6 MY =6F F= 0(KF FV( Y6  K(  #( =MT F FVCY = :(  Y = 6 Y 5X55 YV=2 FV(  =FV

F( = VCKV CY  TY :(K ( Y CM CFV 32 WKF6 T2 (#(M FV( CMF( #  = 6MYF
(  #(M6
( Y I 2 " % YV CM Y2 " (FFCM K=Y( F=  #" V(M FV( (0(KFC#( DC=F M6
( 32 CM K ( Y(Y /= K=MK6:( FVCY Y(KFC=M2 CF CY = FV (+V YC?CM FV( (M(

CFT = FV( =:(CCM + =K(YY :(YK C
(:
=#(2
T +=CMFCM =6F M=FV( ( +( V( ( #( T YCC YT+F=FCK ( F6 (Y ( =
Y( #(: CFV Y=( Y6
F( :C ( (MK(Y2 V=(#(  WY (MFC=MM(: CM Y(KFC=M *<2 FV( K +C T

#CFT  #( CY T(F M=FV( VCV (6(MKT  #( =:( VCKV K M (Y=M MFT CMF(

KF M: C CFV FV( Y6  K(  #( =:( :(YK C
(: CM FVCY Y(KFC=M ,M FV( 

# CF FC=M  FC( YK (   - )  2 FV( =(YF = :( :CY+( YC=M ( FC=M = C:(  K +C T

#CFT  #(Y CM:((: ( :Y ( - < B 3   F MVI% V( ( 3 - !)  CY FV( -" D=M: M6
(  V( ( :(M=F(Y FV( Y6  K( F(MYC=M /VCY CY CM:((: FV( (Y6F =6M: CM FV( CCF     " CFV 3 - < Y(( = CMYF MK( R(:MC =# (F 2 *LL< YCM FV( FV(   FC( YK (  =  FV ((  #(Y CMF( M 2 Y6  K(2

M: K +C T

#CFT2 FV( K=

(Y+=M:CM :CY+( YC=M ( FC=MY ( : ( -

    >   B : <

    6  < B    -  ; < B 3   F MVI% (" -

(

: - <# *#

5 "L "<

Interfacial Patterns and Waves in Liquid Layers and Thin Films

35

V( ( ; - ) % CY FV( Q C(= M6
(  /V( +=YYC
CCFT = (Y=M MK(
(F((M (CFV( F= = FV(Y(  #( =:(Y K M
( M T?(: CFV FV( V(+ = :C 

 = ( #( Y6Y   = CMYF MK(2 CMY+(KFC=M = FV( Y=+(Y = FV( :CY+( YC=M ( FC=MY F FV( = CCM ( :CT YV=Y FV F ( - (" F Y=( M=M?( =  + =#C:(: 30  : < < B    2 VCKV F6 MY =6F F=
( KVC(#
( = 6Y6  C6C:Y Y6KV Y  F( CFV :(+FVY CM FV( K(MFC(F(

M(2 F ( YF = : - <  Y :CYK6YY(: ( C(  @CC T2 (Y=M MF CMF(

KFC=M
(F((M =MCF6:CM  Y6  K(  #(Y M: F

MY#( Y( K +C T

#CFT  #(Y C K( F CMT =KK6 F Y=( ;MCF(  + = #C:(: ; . 6 < B     #(M FV=6V (Y=M MK( K M CM + CMKC+( =K K6 =  ( # 6(Y = ; :(+(M:CM =M 32 FVCY K CF( C=M F6 MY =6F F=
( 6CF( KK6

F( CM +

KFCK(2 M: K M
( (

M(: Y 3 . -< B     >(MK(2 = 6Y6  C6C:Y2 FV( YF FCK D=M: M6
( YV=6:
(

FV( Y  
( K 6Y( -  2 C( FV( :(+FV ) V Y F=
( Y6
CC(F CK F= ( KV (Y=M MK(
(F((M FV(Y(  #(Y ,M  YY F

MY( YTYF(Y2 FV(Y(  #(Y CVF
( = ( ( YCT =
Y( #
( YCMK( FV( # C FC=M = Y6  K( F(MYC=M CFV K=MK(MF

FC=M  T
( (F ((T  ( /V(C :(F C(: YT+F=FCK M TYCY2 K=+((MF(:
T 6T M6( CK  (Y6FY2 V Y
((M KVC(#(: CM R(:MC =# (F 2 *LL< CM T2 (Y=M MK(
(F((M CMF( M  M: K +C T

#CFT  #(Y F6 MY =6F F=
( C+=YYC
(2 CM:(+(M:(MFT = FV( C6C: :(+FV ,M:((:2 F  CM CMF= KK=6MF FV F > - -; 2 CF CY ( YT F= YV= FV F (  (

-  B : <  F MVI%  -  >(MK(2 (Y=M MK( =6: (6C ( 6CF( 6M KVC(#
( F(+(

F6 ( : =+Y CM 6Y6  C6C:Y CM ::CFC=M F= FV(  KF FV F FV( D=6YYCM(Y ++ =C FC=M 6M:( TCM =6 M TYCY (C(Y =M FV( VT+=FV(YCY -  < !0

* 3# %3 # 1%1

Z(F 6Y
C()T K=(
K F= = ( (M(

 +( Y+(KFC#( ,M FV( + (#C=6Y Y(KFC=MY2 (M(

  FV( FCK  

(= V Y
((M :(YK C
(:2 VCKV  =Y YF6:TCM CMYF
CCFC(Y CM YCM(  T( = C6C:2 (CFV( YC+T V( F(:  =
(= =
=#(2 = 6M:( =CM (# +=

FC=M F CFY 6++(  (( Y6  K( CFV FV( +=YYC
( K=+CK FC=M = V( F K=M:6KFC=M CM FV(  Y M: FV( + (Y (MK( = M CM( F K=+=M(MF @FC2 + F = YC+( (( (MK( YF F(Y  M: FV(C CM( YF
CCFT

M TY(Y  2 CF CY (M(

T C+=YYC
( F= Y=#( FV( 6 YTYF( = (6 FC=MY 52 " M:  M TFCK T2 (#(M CFV FV( YC+(YF
=6M: T K=M:CFC=MY 
36

Pierre Colinet

  3  !C (KF ! M6( CK  YC6 FC=MY = +6 (T Y6  K(F(MYC=M : C#(M K=M#(KFC=M CM Y6 ( K=MF CM( CFV CC: CMY6 FCM YC:( Y /V( :C0( (MF YC6 FC=MY (+ (Y(MF FV(  (( Y6  K( F(+(

F6 ( ;(: =
F CM(:

F( K( F CM FC(2 = # C=6Y C+=Y(: F(+(

F6 ( :C0( (MK(Y FV( K  K6 F(: H

M=MC M6
( CY CM:CK F(:

VC( FVCY K( T YV=Y FV F FV(
=#( YTYF( = (6 FC=MY M:
=6M:

T K=M:CFC=MY CY 6CF( Y6CF
( F= :(YK C
( (+( C(MFY =M  T( Y V( F(:  =
(=  M: FV( (= ( Y= CM = ( K=+( K Y(Y CMK6:CM (# +=

FC=M2 =

CMYF MK(2 Y6KV :C (KF FV ((:C(MYC=M  YC6 FC=MY ( + FCK6 T FC( K=MY6CM2 M=F =MT :6( F= FV( ;M( (YV M((:(: F= K +F6 (  :(F CY = FV( )=2
6F Y= F= FV( :C0( (MK(Y CM FC( YK (Y CMV( (MF F= FV( +VTYCK  YTYF( K=MYC:( (: ,M:((:2 M( FV( CMYF
CCFT FV (YV=: FV( (#=6FC=M CY 6CF( Y= =  ( 

M:F M6
( )6C:Y C ( YCCK=M( =CY2 CF  T
( YV=M FV F CMYF 
CCFT :(#(=+Y F( FC( = = :(   2 V( (  - 6

6  6  CY Y6+( K CFCK CFT +

(F( 2 +V(M=(M=M M ==6Y F= FV( Y=K (: K CF CK  Y=CM :=MN = (6CC
C6 +V Y( F

MYCFC=MY WM=FV( FC( YK ( CY

YY=KC F(: CFV + FF( M =  MC? FC=M =M FV(  F(

 YK ( B = FV( K=MF CM( 2 VCKV CY
=6F B) FC(Y  ( FV M FV( FC( CF F (Y = K=M#(KFC=M K(Y F= :(#(=+ 7=F( ;M T FV F C << Y= YV=Y FV F M( FV (YV=:2 FV( V( =M  + FF( M CY 6CF( (6 2 CFV ( :((KFY2 VC( FV(C M6
( CMK ( Y(Y F CMK ( YCM  ,M ::CFC=M2 FV( :CYF MK( F VCKV FV( + FF( M ((YN FV( + (Y(MK( =
=6M: C(Y Y= Y((Y F= :C#( ( M( FV( K CFCK  +=CMF2 VCKV

Interfacial Patterns and Waves in Liquid Layers and Thin Films

37

 CM YV=Y M M =T CFV FV( K=

( FC=M (MFV = +V Y( F

MYCFC=MY @6KV M =C(Y V #(
((M YF6:C(: CM :(F CY2 FV=6V

FV(  = =:(Y = DG(M : K=M#(KFC=M M: =FV( + FF( M(: YF 6KF6 (Y2 F= VCKV FV( (YF = FV(Y( (KF6 ( M=F(Y CY :(:CK F(:




%  %$   % 
% $

R FV( FV M FF(+FCM :C (KF M6( CK  YC6 FC=MY = FV(
YCK (6 FC=MY

M:
=6M: T K=M:CFC=MY ++TCM F= Y+(KC;K DG(M : Y(F6+2 F( M FC#(

++ = KV(Y (CYF2 VCKV = F= 6M:( YF M: =YF 6M: (MF  Y+(KFY = + FF( M M:  #( =  FC=M CM YTYF(Y Y6KV Y FV=Y( K=MYC:( (: CM FV(Y( (KF6 (Y 7=F( FV F =YF = FV( K=M#(KFC=M =:(Y + (Y(MF(: V( ( F( K M
( YTYF( FCK T :( C#(:  = (6 FC=MY Y6KV Y + (Y(MF(: CM 2 F ( YF CM Y=( YT+F=FCK CCF Y  Y6+( K CFCK CFT 2  (YK ( )=Y K=+ (: F= C6C: :(+FV2  >=(#( 2 FV( + (Y(MF FC=M  :( V( ( F( C ( CM

F +V(M=(M==CK  (#(2 CM = :( F= #=C: F== F(KVMCK  K K6 FC=MY2

M: F= (+V YC?( FV( =( = YT(F C(Y (M3=T(:
T FV( +VTYCK  Y(F6+ K=MYC:( (: 

.3  (.3%3 3"% % 1 , %1

4=MYC:( ; YF FV( (6 FC=M 8  -  P B <  B 9 

"*

= M = :( +

(F(  - < B =< # 5 VCKV  T = CMYF MK( (+

(Y(MF FV( F=:C(MYC=M   (( Y6  K( F(+(

F6 ( ;(: CM H

M=MC DG(M : + =
( = FV( #( FCK  C6C: #(=KCFT F C::(+FV = FV(  T(

CM R T(CVDG(M : Y(F6+ V(M 9 - L2 FVCY (6 FC=M CY FV( K((


F(: @CF>=V(M
(  @> (6 FC=M + =+=Y(:
T FV(Y( 6FV= Y @CF M: >= V(M
( 2 <552 Y +V(M=(M==CK  =:( K +F6 CM =YF = FV( (YY(MFC  ( F6 (Y = + FF( M =  FC=M CM R T(CVDG(M : K=M#(KFC=M = 9
- L2  "* CY =:C;(: @> (6 FC=M2 VCKV

FV( Y((Y F= 6 CF FC#(T ++T F= H

M=MCDG(M : K=M#(KFC=M2 Y Y((M CM FV( ==CM C YF2 ( FV F  - L CY + FCK6 YF( :T Y=6FC=M =  "*2 VCKV

KF6 T K=

(Y+=M:Y F= FV( 6MC=  YF F( = V( F K=M:6KFC=M + (# CCM F Y  P"  ,M = :( F= :(F( CM( V(FV( FVCY (( (MK( YF F( CY YF
(2 (

= Y  +( F6
FC=MY CM FV( =  = M=   C( =6 C(  =:(Y # 5 -  (+I5% (+I2  %

CFV   <

"

V( (  CY FV(  =FV

F( M:  - < B < CY F=:C(MYC=M   #(#(KF=  ,MY( FCM FVCY (+ (YYC=M CM  "* M: CM( C?CM CFV (

38

Pierre Colinet

  3  Z(F :CY+( YC=M ( FC=M = FV( @CF>=V(M
(  (6 FC=M "* RCVF :CY+( YC=M ( FC=M = +( F6
FC=MY = FV( K=M:6KFC#( YF F( CM FV( H

M=MCDG(M : + =
(2 CFV > - 32 - L M:   2 = Y(#(

 # 6(Y = 6 

Y+(KF F= FV( Y  +

(F( 2 CF CY Y((M FV F FV( (6 FC=M CY CM:((: Y FCY;(: + =#C:(:   -  <  "  VCKV CY FV( :CY+( YC=M ( FC=M = +( F6
FC=MY2 C( FV( ( FC=M
(F((M FV(C  =FV

F(  M: FV(C  #(M6
(  -  /VCY ( FC=M CY (+ ( Y(MF(: CM C <*2 M: K=+ (: CFV FV( ( KF :CY+( YC=M ( FC=M = FV( H

M=MCDG(M : + =
( :(YK C
(: CM  ,F CY Y((M  = C <* FV F
=FV :CY+( YC=M ( FC=MY YV= YCC
( V #C= M( FV( K CFCK  +=CMF  - L = 6 - 6  - 5 E 7 (T2  =6 C( =:(Y V #( M( FC#(  =FV

F(
(= FVCY K CFCK  +=CMF V(MK(2 FV( (( (MK( YF F( CY YF
(2 VC(
M: =  #(M6
( Y ( : F= +=Y CFC#(  =FV

F( Y Y==M Y CF CY (K((:(: 6MYF
( (( (MK( YF F( WF K CFCK CFT2 =MT =M(  #(M6
(  V Y ?( =  =FV

F( CF CY FV(M Y C: FV F FV( (( (MK( YF F( CY M(6F

T  =  CM T  YF
( =   L2 Y=( =6 C( =:(Y C FV( (= (
( +C;(: (+=M(MFC T2

M: FV( YY6+FC=M = Y   CM  " (#(MF6 T  CY F( Y=( FC( ,M FVCY K Y(2 M=MCM( (0(KFY F( Y  M:   YF F F= K=6+( FV( :C0( (MF =6 C( =:(Y2 CM ::CFC=M F= Y F6

FCM FV( CMCFC  (+=M(MFC   =FV = )6KF6 FC=MY @=( :C (KF M6( CK  YC6 FC=MY = FV( =:C;(: @> (6 FC=M

( YV=M CM C < WF Y  # 6(Y = 9 M: *2 FV( YTYF( YV=Y K=( CYF(MK( = V( =MY M: =Y = =M FC(2
(= ( FV( V( =M  + FF( M CM# :(Y FV( V=( YC6 FC=M := CM WF  ( # 6(Y = 92 FV( V( =MY

( YF =MT  #= (: (#(M F( 6KV YV= F( FC(2 FV=6V FV( ;M  + F F( M Y((Y F= K=MF CM = ( :((KFY  KF6 T2 Y=( +(MF =MV(+F =M + C Y K M
( Y((M2 M: FV( ( ( Y(#(

 := CMY = (6 V( =MY CFV :C0( 

Interfacial Patterns and Waves in Liquid Layers and Thin Films

39

(MF = C(MF FC=M Y(+

F(:
T 

CM
=6M: C(YN2 6CF( YCC T F= V F CY =
Y( #(: (+( C(MF T

  3 ! !C (KF M6( CK  YC6 FC=MY = FV( =:C;(: @CF>=V(M
(  (6 FC=M "* /=+ =   - L <2 9 - L *E D=FF= =   - L <2 9 - <  4=6 F(YT = H D(YF(V= M >=(#( 2 :(Y+CF( FVCY MCK( 6 CF FC#(  (((MF CFV H

M=MCDG(M : K=M#(KFC=M2 FV( ( ( Y= C+= F MF :C0( (MK(Y C YF2 CF K M
( YV=M FV F FV( :TM CKY :(YK C
(:
T  "* CY  TY (  FC=M  =M=F=MCK (#=6FC=M F= YF( :T YF F(2 CMCC?CM Y=( (M( T 6MKFC=M  >=(#( 2 Y(#(

 6FV= Y V #(  F( YV=M FV F FVCY CY M=F FV( K Y( = H

M=MC DG(M : K=M#(KFC=M2 CFV C+= F MF K=MY(6(MK(Y =M FV( :TM CKY = :((KFY Y(( ( 4=CM(F (F 2 *LL*2 M: (( (MK(Y FV( (CM @(K=M:2  "* :=(Y M=F YV= F

MYCFC=M F= Y6 ( + FF( MY F CMK ( YCM 2 (#(M FV=6V FVCY Y((Y F=
( =
6YF ( F6 ( = H

M=MCDG(M : K=M#(KFC=M K ( F (F 2 <"O K ( F M: /V(YY2 < C MFY = FV( @CF>=V(M
(  (6 FC=M V #( FV( (= (
((M K=MYF 6KF(: D(YF(V= M M:  C(: CKV2 < CM = :( F=
(FF(  FKV + FCK6 +VTYCK  Y(F6+Y WY M ( +(2 ::CM F( Y Y6KV

Y P   CM  "* CM:((: + =:6K(Y Y6 (Y + =#C:(: Y=( K=M:CFC=MY =M FV( K=(KC(MFY = FV( # C=6Y F( Y ( Y FCY;(: Y(( Y= (KF6 (Y = H D(YF(V= M 

1# 1'3 31   3% 13(
1

/V( @> (6 FC=M M: CFY # C MFY 6CF( MCK(T =:( DG(M : K=M#(KFC=M2 VCKV K M
( FF C
6F(: F= FV(  KF FV F FV( :CY+( YC=M ( FC=MY V #(

40

Pierre Colinet

YCC
(V #C= M( FV( K CFCK  +=CMF Y(( C <*2 CM ::CFC=M F= FV(  KF FV F FV(C YT(F T + =+( FC(Y ( C:(MFCK  F= FV=Y( =
(T(:
T 6 YTYF(Y = (6 FC=MY M:
=6M: T K=M:CFC=MY Y6KV Y C#(M CM  /= YC+CT FV( + (Y(MF FC=M2 ( V=(#( (YF CKF FV( ==CM K=MYC:(

FC=MY F=  "* CM FVCY Y(KFC=M QC#(M + FCK6 Y=6FC=M  # 5 = FV( =:C;(: @CF>=V(M
(  (6 FC=M2 K=MYC:( ; YF FV( F

MY F(: Y=6FC=M # 5 -    # 52 V( (  CY M
CF

T V= C?=MF  #(KF=  4( T2 FV( (6 FC=M :(YK C
CM FV( (#=6FC=M = FV( F

MY F(: Y=6FC=M  CY C:(MFCK  F= FV F :(YK C
CM FV( (#=6FC=M =   ,M:((:2 Y  CY Y=6FC=M =  "*2 ( V #( 8  # 5 -  # 5 P B <  # 5 B 9 # 5  # 5

"X

V( ( P CY FV( V= C?=MF  Z + KC M CFV (Y+(KF F= # C
(Y @6
YFCF6F CM 
T   CM  "X :C (KFT ( :Y F=  "* = 2 Y P   - P 2 CM:CK FCM FV F FV( =:C;(: @> (6 FC=M CY CM# C MF
T F

MY FC=M =

/% $$/ DG(M : Y(F6+2 CF CY CM:((: M F6

 F= (+(KF FV F MT F

MY FC=M = + FCK6 Y=6FC=M = FV( + =
( CY CFY( Y=6FC=M 4=MYC:( M= FV( =F FC=M YT(F T2
T F

MY= CM FV( + FCK6

Y=6FC=M  # 5 =MF= =F F(: Y=6FC=M # 5 -  > # 52 V( ( > CY

=F FC=M =+(

F= CFV M( - @6
YFCF6FCM 
T >  CM  "X ( :Y F= 8 # 5 - # 5 P  B < # 5 B 9# 5 # 5 7=2 6YCM FV(  F C (+ (Y(MF FC=M  K=Y > YCM -

YCM K=Y -

"E

 "5

M: :(;MCM   - > 2 ( V #(     8 8 8 8 8 8 B YCM B K=Y P
B  - K=Y B YCM 8 8= 8 8= 8 8= 8 8 - B - P

8 8= ""

M: V(MK( FV( (6 FC=M Y FCY;(:
T FV( =F F(: + FF( M  (:6K(Y F= FV( = CCM  (6 FC=M2 VCKV CY FV6Y CM# C MF
T =F FC=M  =
=FV K Y(Y = F

MY FC=M M: =F FC=M2 CF CY Y((M FV F FV( K=M:CFC=M = CM# C MK( =  "* CFV (Y+(KF F= YT(F T =+(

F= / KFCM =M 2 CY FV F FV( Z + KC M =+(

F= CY CM# C MF CFV (Y+(KF F= FVCY F

MY=  FC=M2 C( P - P 

Interfacial Patterns and Waves in Liquid Layers and Thin Films

41

,F CY YF

CVF=  : F= KV(K FV F FVCY CY #( C;(: = FV( + CFT =+(

F=  :(;M(:
T   - 2 CM  KF > -  2 M: = FV( C

= ()(KFC=M  6 -

K=Y *YCM *-

YCM * K=Y *-

 "

CFV (Y+(KF F= CM( + YYCM FV =6V FV( = CCM M: = CM M M( - CFV FV(  CY 7=F( FV F MT C

= ()(KFC=M CFV M( - K M
( =
F CM(:

Y FV( K=+=YCFC=M = C

= ()(KFC=M CFV (Y+(KF F= 2 ==(:
T

=F FC=M = M( *- C( *- CM FV( K=6MF( K=K CY( :C (KFC=M = 6F=M==6Y YTYF(Y2 FV( = CCM = FC( CY
CF

T2 Y= FV F M=FV(

CM# C MK( F=
( Y FCY;(: CY FV( FC(F

MY FC=M 5 5BP52 VCKV CY F C#C T Y FCY;(:
T  "* /= Y6 C?(2  "* V Y
((M =6M: F=
( CM# C MF F= F

MY FC=MY2

=F FC=MY CMK6:CM + CFT2 C

= ()(KFC=MY M: FC(F

MY FC=MY2 VCKV CY Y= FV( K Y( = FV( 6 YTYF( = (6 FC=MY M:
=6M: T K=M:CFC=MY + (Y(MF(: CM  /V(Y(
YCK YT(F C(Y C
( Y6KC(MF = =6 +6 +=Y(Y2

M: CF C
( YV=M CM FV( M(F Y(KFC=M FV F FV(T CM:((: YF =MT :(F(  CM( FV( M F6 ( M: YF
CCFT = M( Y=6FC=MY ++( CM
=#( CMYF
CCFT FV (YV=: ,+= F MFT2 = YTYF(Y CM 6MC=  =F FC=M =6M: #( FCK  CY2 Y=( YT(F C(Y := M=F V=: MT= ( ,M:((:2 CM FVCY K Y( K(MF C6  M: 4= C =CY = K(Y 4V M:

Y( V 2 <E< V #( F=
( CMK= +=

F(: CM FV( 7 #C(  @F= (Y (6 FC=MY  CFF(M CM FV( =F FCM 

(2 VCKV
( Y
=FV F

MY  FC=M M: ()(KFC=M YT(F C(Y  FV=6V CM# C MK(
T =F FC=M K=MFCM6(Y F= V=: /VCY C V #( C+= F MF K=MY(6(MK(Y2 Y Y((M CM FV( ==CM Y(KFC=MY 7=F( ;M T FV F =FV( = ( Y+(KC;K YT(F C(Y C
( (MK=6M F( (: CM FV( ==CM !

'3 31 % 
 3 3"%1

7( CMYF
CCFT FV (YV=:2 C( = Y6KC(MFT Y  Y6+( K CFCK CFT 2 CF CY +=YYC
( F= :(YK C
( FV( ( T M=MCM( (#=6FC=M = Y(#(

 FT+(Y = + FF( MY 6YCM (6 FC=MY :(YK C
CM FV( (#=6FC=M = +CF6:(Y = )6KF6  FC=MY =6M: FV( (( (MK( YF F( VC( FVCY K M
( :=M( C= =6YT 6YCM +( F6
FC=M (+ MYC=MY CM +=( Y = 2 (  = YTYF(Y = !'Y Y6KV Y + (Y(MF(: CM  
YCK F(KVMC6(Y ( :(YK C
(: CM 4=CM(F (F 2 *LL<2 ( C

FV( 6Y( FV( YT(F T K=MYC:(

FC=MY + (Y(MF(: CM FV(  YF Y(KFC=M F= =
F CM FV( =  = Y6KV +CF6:( (6 FC=MY CFV 6KV (YY K K6 FC=MY W CM2 ( C K=MFCM6( F= 6Y( FV( =:C;(: @> (6 FC=M Y =:(2 6  CF FC#(T (6C# (MF F= FV( = CCM  6 YTYF( = (6 FC=MY M:
=6M: T K=M:CFC=MY

42

Pierre Colinet

  #%  1 Z(F 6Y ; YF FF(+F F= Y=#(  "*
T  - A 5 (+I2  % B A 5 (+I 2  %

L

V( (  CY M
CF

T  #(#(KF= 2 M: A K=+( +CF6:( CM (M(

  M YF( CY :(M=F(Y FV( K=+( K=M36 F( = Y  A 2  "*  T
( CM( C?(: M: CY FV(M =6M: F=
( Y FCY;(: C AS - A

<

V( ( FV( :=F ( MY FV( FC( :( C# FC#(2 M:  CY C#(M
T  "  CFV  -   >( ( F( 2 ( Y((KF  - < K CFCK   #(M6
(  = YC+CKCFT2 Y6KV Y  -  4( T2  < CY Y=#(:
T M (+=M(MFC 2 Y6KV Y CM  " >(MK(2 FV( 6(YFC=M CY F= :(F( CM( V= F= K=+(F(  < Y6KV Y F= #=C: FVCY 6M+VTYCK  (+=M(MFC   =FV =   L WYY6CM = ( (M(

 (#=6FC=M (6 FC=M = FV( =  AS -  A # A 

*

V( (  CY M M TFCK  6MKFC=M = CFY 6(MFY2 ( C (+ M: FVCY

+CF6:( (6 FC=M CM / T= Y( C(Y Y AS - A B9 A B9 A  B9 A B A B A A B A A B A   V( ( FV( (+ MYC=M V Y
((M CCF(: F= K6
CK = :( F=
( 36YFC;(: +=Y F( C= C 7=2 FVCY =   T
(  ( FT YC+C;(:
T  CM 6Y( = FV( YT(F T K=MYC:(

FC=MY 4=MYC:( FV( F

MY FC=M  CM# C MK(     ,    CY Y=6FC=M2     6YF Y=
( Y=6FC=M WY Y((M  = L2 FVCY F

MY F(: Y=6FC=M  T Y=
( =
F CM(:
T FV( F

MY=  FC=M A 3 - A (+I 2  % KFCM =M FV( +CF6:( A  /V6Y2 3 YV=6: Y FCYT FV( +CF6:( (6 FC=M  C A :=(Y @6
YFCF6FCM A
T 3 CM FVCY (6 FC=M2 FV( (Y6F YV=6:
( # C: =   M: ( Y(( FV F  F( Y M=F + =+= FC=M  F= (+I 2  %2 F( Y6
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/V( =MT F( Y ( CMCM ( FV=Y( 6FC+C(:
T (+I24% 6M:( FV( F

MY =  FC=M A A (+I24% (6C# C MF F( Y 7=F( FV F FV( (6C# C MF 6MKFC=M A  A   CY FV( + =:6KF = FV( (6C# C MF 6MKFC=M A M: FV( CM# C MF 6MKFC=M  A  C( CF ( CMY CM# C MF 6M:( FV( F

MY =  FC=M

Interfacial Patterns and Waves in Liquid Layers and Thin Films

43

7=2 FV=6V F

MY FC=M  CM# C MK( V Y  ( :T  ( FT YC+C;(: FV(

+CF6:( (6 FC=M2 ( V #( F= K=MYC:( FV( (0(KF = =FV( YT(F C(Y /V(

=F FC=M  CM# C MK(    - >  C M=F
( 6Y(6 V( ( WKK= :CM F=  L2 CFY KFC=M =M FV( +CF6:( CY A A (+I2 >   %2 C( FV( +V Y( (M(

T :(+(M:Y =M 2 YCF6 FC=M (K6:(: CM FV( + (Y(MF M TYCY ,M FV( K Y( = + CFT   - >  - 2 FV( K=

(Y+=M:CM KFC=M =M FV(

+CF6:( CY A A  /V6Y2 FV( (6 FC=M   YV=6:
( Y FCY;(: 6M:(

FVCY KV M(2 VCKV CY =MT +=YYC
( C  -   M:  -    (  K=(KC(MFY CM T2 CF K M
( Y((M FV F FV( C

= ()(KFC=M YT(F T  K(Y FV( Y ( + =
(Y Y =F FC=M @F CKFT Y+( CM2 FV(  KF FV F =F FC=M M: ()(KFC=M CM# C MK(Y K MM=F
( Y FCY;(: CY K=MY(6(MK( = =6
CF

T KV=CK( =

+ FCK6 :C (KFC=M  CM FV( V= C?=MF  + M(2 VCKV K( T
( Y FV(Y( YT(F C(Y /V( +CF6:( (6 FC=M   = =Y = YF C+(Y  T
( Y=#(:
T ; YF Y(+

FCM =:66Y M: +V Y( A -  (+I24% /VCY ( :Y F= 4S - L M: F= S -    +CFKV=
C6 K FC=M2 VCKV CM:((: Y F6

F(Y FV( (+=M(MFC   =FV + =#C:(:   L Y6+( K CFCK  +CFKV=
C6 K FC=M2 Y(( C <  $FV( CY( Y6
K CFCK 
C6 K FC=M2 VCV( = :( F( Y V #( F=
( CMK6:(: CM Y  M:  

  3  @6+( K CFCK  +CFKV=
C6 K FC=M   F(+=

 (#=6FC=M = FV( +CF6:(  YF FCM  = M CMCFC  )6KF6 FC=M  - L L<2 = Y(#(

 # 6(Y = FV( CM(  =FV

F(  
 YF( :T YF F(Y M: FV(C YF
CCFT 6 CM(Y YF
( Y=6FC=MY2 : YV(: CM(Y 6MYF
( Y=6FC=MY

  #%  .3 %
3 %1 ( M= K=MYC:( FV( C+= F MF K Y( = FV( CMF(

KFC=M = FV (( =6 C( =:(Y CFV  #(#(KF= Y  2  M:  = C(MF(: F <*L + F =M FV( K CFCK  KC K( =

:C6Y  - < 7=F( FV F  B  B  - L2 C( FV(
YCK  #(#(KF= Y =   (Y=M MF F C :N2 F=

44

Pierre Colinet

(+=T FV(  M6 ( =  #(Y /V( + M=  6MKFC=M CY FV( (= (  CFF(M  - A (+I2  % B A (+I2  % B A (+I2  % B 7 7

X

M: FV( +CF6:(Y ( YY6(: F= Y FCYT (#=6FC=M (6 FC=MY = FV( =  AS -  A # A # A # A # A # A 

- <# *# 

E

VCKV YV=6:
( / T= (+ M:(: 6+ F= K6
CK = :( 2 36YF Y = = + FF( MY WKF6 T2 =YF = FV( F( Y CM FV( (Y6FCM =M (+ (YYC=MY K M ( :CT
( (CCM F(:
T K=MYC:( CM FV( F

MY FC=M CM# C MK(    2 VCKV CY (6C# (MF F= FV( KFC=M A A (+I 2  % # - <# *#  /V6Y2 CM FV( (6 FC=M = A = ( +(2 FV( CM( F(  A ( CMY Y = =Y2

M: FV( 6 :

FCK F(  A A CY (6C# C MF Y ( CF CY 6FC+C(:
T (+I2 B    % - (+I 2  % 6M:( FV( F

MY FC=M2 36YF Y AS  ,F CY ( YCT Y((M FV F FV(Y( ( FV( =MT (6C# C MF CM( M: 6 :

FCK F( Y2

M: =CM =M F= K6
CK = :( ;M T ( :Y F= 
AS - A B 9A A A A  B  B >A  B  >A  5 V( (  FV( K=(KC(MFY 2 92 2  M: > ( ( 2 :6( F= FV( + CFT CM# C

MK(   36YF Y = =Y /V( (6 FC=MY = FV( =FV( +CF6:(Y ( =
F CM(:  = KTKCK +( 6F FC=MY A A 2 A A M: A A 2 VCKV

(Y6FY  = FV( =F FC=M  CM# C MK( V( ( (YF CKF(: F= M(Y - - *< :6( F= =6 + FCK6 KV=CK( =  #(#(KF= Y VCKV
( Y FV( K=MFCM6=6Y YT(F T  =6+ = =F FC=M2 Y  Y FV( K Y( = = + FF( MY 7=F( FV F FV( K=(KC(MFY = K6
CK F( Y CM  5 V #(
((M  CFF(M CM Y6KV =  = ( Y=MY FV F C
(K=( ++ (MF V( ( F(  CM T2 CF ( CMY F= (+ (YY FV( +=YYC
( CM# C MK( F= C

= ()(KFC=MY VCKV2 Y (MFC=M(: F FV( (M: =  *2 K M
(
= (M
T 4= C=CY = K(Y CM K Y( FV( DG(M :  T( CY  :( F= =F F( 6MC= T =6M: #( FCK  CY YFC M((KFCM FV( +=YYC
(
( CM = F

MY FC=M  CM# C MK(
T K(MF C6  = K(Y2 (+(KF(: F=
( Y  4=MYC:( M=M =F FCM YTYF( = FV( =(MF  YFC2 M=F (#( T ()(K FC=M + M( CY +( CFF(:2
(K 6Y( = =6 + FCK6 Y((KFC=M =  #(#(KF= Y (  T = CMYF MK( Y((KF FV(
CYY(KF= = FV( M(
(F((M  #(#(KF= Y  M:  2 M: :(M=F( FVCY + FCK6 ()(KFC=M
T FV(  F C 6  /V(M2

Y  6 - 6   -  2 FV( ()(KFC=M =+(

FC=M  6 ++C(: F=  X CY Y((M F= K=

(Y+=M: F= FV( KFC=M A A M: A A  ,M K Y(

()(KFC=M YT(F T V=:Y2 +CF6:( (6 FC=MY YV=6:
( CM# C MF 6M:(

FVCY KFC=M2 VCKV ( :Y F= > - L CM  5

Interfacial Patterns and Waves in Liquid Layers and Thin Films

45

WY K=MK6YC=M2 > CY FV( +

(F( VCKV ( Y6 (Y FV(
( CM = ( )(KFC=M = KVC

 YT(F T
T 4= C=CY = K(Y2 M: CY FV( (= ( (+(KF(: F= CMK ( Y( CFV FV( M6 Y+((: ( M=
C()T ( CM( Y=( K=MY(6(MK(Y FVCY V Y =M FV( :TM CKY = FV( + FF( MY
=#( CMYF
CCFT FV (YV=: = > - L2 FV( 6MKFC=M  -  A  B A  B A  9 A A A B A A A    B A  B A  B A  B  A  A  B A  A  B A  A  * " CY ZT +6M=# 6MKFC=M = FV( YTYF( = +CF6:( (6 FC=MY ,M:((:2 CF CY  Y((M FV F AS - 8 8A # - <# *#  M: S - * 
8 8A  L2 FV6Y C+TCM =M=F=MCK :(K ( Y( = 6+ F= =M( = CFY =K  CMC  ,M FV( M=M =F FCM K Y( > - L2 FV(
(V #C= =  5  M: CFY KTKCK +( 6F FC=MY  CY FV6Y Y(#( (T (YF CKF(:
T FVCY + =+( FT  CM + FCK6 2 CF (K6:(Y  M=M=M=F=MCK
(V #C= Y Y6KV Y =YKC FC=MY2 = KV =FCK :T M CKY 7=F( FV F = >
- L2 CF CY M=F +=YYC
( F= ;M: 6MKFC=M Y FCYTCM FV( Y ( K=M:CFC=MY2 M: (  T FV( (= ( (+(KF = ( K=+(
(V #C= Y CM K Y( =F FC=M =KK6 Y @(+

FCM =:6C M: +V Y(Y Y A -  (+I24 % # - <# *# 2 M: Y6CM 6+ FV( C CM T + FY = +CF6:( (6 FC=MY2 ( (F M (6 FC=M = FV( Y6 = FV( +V Y(Y 4 - 4 B 4 B 4 Y         YCM 4  B B 4S - 9 

  VCKV CY KF6 T # C: = >
- L Y (2 M: YV=Y FV F FV( =MT YF( :T Y=6FC=MY = 4 ( 4 - L M: 4 - <  Y FV( 6 MFCFT
(F((M + (MFV(Y(Y CY  TY +=YCFC#(2 FV( (#=6FC=M CY =M=F=MCK F= :Y 4 - L C 9  L M: 4 - < C 9 . L /VCY +V(M=(M=M CY K (: +V Y( YTMKV =MC? FC=M2 VCKV CY + FCK6 F= FV( K Y( = (Y=M MF F C :Y2 C( CF CY :C (KFT CM (: CFV FV( 6 :

FCK F( Y = +CF6:( (6 FC=MY 5 4=(KFCM FV( (  + FY = +CF6:( (6 FC=MY2 ( (F ( FC=MY =CM F= :(F( CM( M=F =MT FV( YF( :T YF F( Y=6FC=MY2
6F Y= FV(C YF
CCFT Y(( = CMYF MK( 4=CM(F (F 2 *LL< ,M + FCK6 2 ((# MF YF( :T Y=6FC=MY

( =Y  -  - L#  -   )1 R 
M: V( =MY  -  -  -

9

 9 B  B * * B *

3 %1 >


,M ::CFC=M F= FV( (( (MK( YF F(   - L 6MYF
( =   L2 Y 6Y6 2 FV( ( ( Y= C(: Y=6FC=MY2 M=F C#(M V( ( = K=MKCY(M(YY  M:
(K 6Y(

46

Pierre Colinet

FV(T F6 M =6F F=
(  TY 6MYF
( W FT+CK 
C6 K FC=M :C 

 CY C#(M CM C <X2 = FV( M=M =F FCM K Y( > - L 7=F( FV F V( =MY CFV 4 - L 9  L ( :(M=F(:
T >B  M: K=

(Y+=M: F= V( =M  YF 6KF6 (Y CFV 6+)= F FV( K(MF( = M V( =M2 VCKV ( 6Y6 T =
Y( #(: = VCV 

M:F M6
( )6C:Y2 VC( FV=Y( CFV 4 - < 9 . L ( :(M=F(:
T >

:=M)= F FV( K(MF( 

  3 0 Z(F 
C6 K FC=M :C 

 = FV( K=+(FCFC=M
(F((M =Y R

M: V( =MY >B M: >  CM FV( M=M =F FCM K Y( > - L2 M: =     L FT+CK  K Y( CM H

M=MCDG(M : K=M#(KFC=M @F
( Y=6FC=MY (

(+ (Y(MF(: Y + CM CM(Y2 VC( 6MYF
(


MKV(Y ( : YV(: /V(


MKV(Y 
((: H K=

(Y+=M: F= M 6MYF
( C(: Y=6FC=M 4(MF(  (K=MYF 6KFC=M = FV( 6+V( =M Y=6FC=M >B  =  X RCVF (K=MYF 6KFC=M = FV( :=MV( =M Y=6FC=M >  =  X ,F CY Y((M CM C <X (F FV F FV( =MT YF
( Y=6FC=M 36YF
=#( FV( CMYF
CCFT FV (YV=:  - L CY FV( V( =M  + FF( M CFV   -   C#(M
T  
 . L2 6MFC F

MYCFC=M F= FV( (( (MK( YF F(   - L F (Y + K( >(MK(2 FV( + C T
C6 K FC=M  = FV( (( (MK( YF F( F= V( =M  + FF( MY CY

YY=KC F(: CFV M +%-//-- 
2 V=Y( C:FV :(+(M:Y =M FV( + =
( YF6:C(: CF CY Y  CM +

KFCK( = H

M=MCDG(M : K=M#(KFC=M = CMK ( YCM 2 FV( V( =M  Y=6FC=M
(K=(Y 6MYF
( V(M    9  =     L2 M: FV( =MT YF
( Y=6FC=M
=#( FVCY Y(K =M: T
C6 K FC=M F6 MY =6F F=
( FV( = + FF( M WY ( (: F FV( (M: =  <2 FVCY
(V #C= CY M=F ( CYFCK = H

M=MCDG(M : K=M#(KFC=M2 YCMK( Y6 ( + FF( M CY

FV( =
Y( #(: F CMK ( YCM Y6+( K CFCK CFT W = ( K=+(F( M TYCY 6YCM YTYF( = YC +CF6:( (6 FC=MY CY FV(M M(K(YY T2 CM = :( F= = K=
CM FC=MY = + M(  #(Y CFV = FV==M   #(#(KF= Y2 CM:((: (+ (Y(MFCM Y6 ( + FF( MY /VCY CY M=F :=M( V( ( =

K=MKCY(M(YY Y(( ( 4=CM(F (F 2 *LL< /= K=MK6:( FVCY Y(KFC=M2 =CM FV( DG(M : Y(F6+ F= =F F( =6M:

Interfacial Patterns and Waves in Liquid Layers and Thin Films

47

#( FCK  CY2 C( = >
- L2 FV( Y(K=M: T CMYF
CCFT = V( =M  + FF( MY YFC =KK6 Y F    9  =     L2
6F CY M= =6M: F=
( =YKC F= T Y(( ( KV(


C M: RC(K (2 *LLL /V( Y=K (: >=+
C6 K FC=M FV(M ( :Y F= CCF KTK(2 C(  2  M:  =YKC F( +( C=:CK T2

F( Y=( F

MYC(MF +( C=: Y(( C <E

  3  $YKC FCM V( =MY CM YTYF(Y CFV
= (M KVC

 YT(F T2 =
F CM(:
T :C (KF M6( CK  CMF(

FC=M = $!'Y 5 CFV  - L X2 9 - <2  - <2  - X2 M: > - *L /=+ (F  F

3(KF= T CM FV( +V Y( Y+ K( Y+ MM(:
T  - A # - <# *#  /=+ CVF  (#=6FC=M = +CF6:(Y :6 CM =M( =YKC FC=M +( C=: D=FF=  " YM +YV=FY = FV( (K=MYF 6KF(: + FF( M2 Y +(: :6 CM =M( =YKC FC=M +( C=:



2% (,3  3 (
 33 3"%1 
3 %1

WM=FV( K YY = =:(Y ++TCM F= # C=6Y FT+(Y = DG(M : CMYF
CCFC(Y  Y ( Y F= =FV( +VTYCK  YCF6 FC=MY K M
( :( C#(: V(M FV( + C T CMYF
CCFT V Y  ( Y+ FC  YK ( C( Y  K CFCK   #(M6
(  K= + (: F= Y=( =FV( KV

KF( CYFCK :C(MYC=M = FV( + =
( ,M FVCY K Y(2

48

Pierre Colinet

6FCYK ( YTF=FCK (+ MYC=MY K M (M(

T
( ++C(: F= FV(
YCK YTY F(Y = (6 FC=MY M:
=6M: T K=M:CFC=MY2 VCKV FT+CK T ( :Y F= =:(Y = =( :C(MYC=M CFT FV M FV( = CCM  =M( R FV( FV M F= == FVCY

=6F(2 ( C FF(+F CM FVCY Y(KFC=M F= :( C#( FV( =  = Y6KV =M #( CMYF
CCFT =:(Y  = YT(F T K=MYC:(

FC=MY2 Y  ( :T :=M( =  +CF6:( (6 FC=MY = FV( Z M: 6 FT+( K=MYC:( (: CM
 Z=M #( CMYF
CCFT =:(Y ( =F(M CM (: CFV + FCK6 K=MY( #  FC=M   = YT(F T + =+( FT = CMYF MK(2 V(M
=FV
=6M: C(Y = DG(M : Y(F6+ ( V( FCMY6 FCM2 6MC=  YVCF = FV( F(+(

F6 ( CY +=Y YC
(2 M: FV( ( CY M= = K( VCKV KFY CM (YF= CM FV( CMCFC  F(+(

F6 (   >(MK(2 FV( YTYF( CY M(6F

T YF
( F= Y6KV 6MC=  YVCF2 M: FV( CM(

YF
CCFT + =
( YV=6: +=YY(YY F ( YF =M( ?( = (C(M# 6( 7=F( FV F =

=M #( Y6  K( :(=  FC=M  =:(Y :(YK C
(: CM
* 2 FV( K=

(Y+=M:CM YT(F T CY M M CM# C MK( CFV (Y+(KF F= YVCF = FV( V(CVF = FV( CMF(  K(2 VCKV CY Y= M(6F

T YF
( Y =M Y FV( YVCF CY 6MC=  Y(( C <5


 @ (FKV C6YF

FCM V= M(6F

 YF
CCFT CFV (Y+(KF F= 6MC =  YVCF P = FV( = :( +

(F(  CY ( T
= (M
T  (YK ( M=M6MC= CFC(Y = FVCY YVCF /=+ =  Y6  K( :(=  FC=M  =:( D=F F= =  FV( =K +C T =:( 7=2 V(M FV( YVCF CY M=M6MC= 2 C( CF CY Y=T =:6 F(: =M FV( V= C?=MF  K== :CM F(Y2 Y= :TM CKY K M
( (+(KF(:2 VCKV  T K=6+( F= K=M#(KFC=M )=Y M: +=YYC
T ( : F= CMYF
CCFT = FV( V= C?=MF T 6MC=  YF F( /V6Y2 :(M=FCM FV( K=MF = +

(F(
T 2 CM ( KV K Y( FV( ( (CYFY M (C(M# 6( 
  Y6KV FV F L  - L =   ( V( ( (YF CKF

FF(MFC=M F= YTYF(Y = VCKV  CY ( 2 M: K M
( (+ M:(: Y 
  -
  
 B 





,8 ;=%  =% $  8  * %8= ;;%= ,0;#== #0
) :#= $= %8= ;= #%#, 0 ,0#%#0;   - #0 ,= * +; '- 0 '- 0 %8#; ,;= %8= * ;;%= $  #0== = #0:#0% %  ;8#*%  !   %  ==0#%#0 * %8= =;;= #0 %8= :#=2%6=; =+%#0; '5

Interfacial Patterns and Waves in Liquid Layers and Thin Films

49

V( ( +=YCFC#( K=MYF MFY CM  =MF = FV( F= F( Y V #(
((M Y(F F= 6MCFT CFV=6F =YY = (M(

CFT Y6CF
T (YK CM FC( M: + 7=2 FV( ( (CYF  MT YTYF(Y CM VCKV FV( YT(F T (Y+=MYC
( =

FV( ?( = (C(M# 6( F  - L CY =MT ++ =C F( /VCY V ++(MY = CMYF MK( V(M F=+ M:=
=FF=
=6M: C(Y V #( ;MCF(
6F Y  FV(   K=M :6KFC#CFT2 Y6KV FV F FV( V==(M(=6Y =:(  - L CFY( V Y Y= (#=6FC=M  YY6(: F=
( Y= (+=M(MFC  : +CM V( ( ,M Y6KV K Y(2  
F6 ( " 2 = FV( )6C: :(+FV @2 FV( CM( + F = FV( = :( +

(F(

(6 FC=M =   T
(  CFF(M2 K=MYCYF(MFT CFV  =V(M
(  (6 FC=M "*2 (#(M FV=6V FV( K=MF = +

(F( + V( ( ++( Y F= 0(KF FV( :C06YC=M K=(KC(MFN = FV( 6 MFCFT 2 C( FV( K=(KC(MF = FV( Z +  KC M F(  /VCY CY FV( ( Y=M VT YTYF(Y :(YK C
(:
T Y6KV CM: = CM(


(V #C= ( Y=(FC(Y K (: M( FC#( #CYK=YCFTN YTYF(Y V(M +  L W

FV( K=+(F( :CYK6YYC=M = Y6KV YTYF(Y V Y
((M (YF
CYV(: 7(+= MT YVKVT2 <X W
=#( FV( CMYF
CCFT FV (YV=:2  
FC=MY2 CM FV( Y ( Y+C CF Y CM  * M:   /V( (= (2 6YCM  CM Y "5 M: "2 MT F(  ++( CM CM FV( (6 FC=M YV=6:
( 6M 0(KF(:
T =F FC=MY  >   CMK6:CM + CFT - - <2 ()(KFC=MY  6 2 CM ::CFC=M F= F

MY FC=MY    M: FC( YVCFY 5 5 B P5  =  *2 CF CY K( FV F FV( CM(

+ F 
F= CY  CY Y= 6M 0(KF(:
T FV(Y( F

MY=  FC=MY >=(#( 2 VC( FV( =+(

F=  6M 0(KF(:
T Y+ K(  M: FC( F

MY FC=MY2 CF CY 6Y(6 = FV( ==CM  >   M:   6   6M:(

F= M=F( FV F CF F

MY= Y KK= :CM F= 

=F FC=M M: ()(KFC=M2 (Y+(KFC#(T ,M ::CFC=M2 +=YYC
( M=MCM( F( Y ( Y= (YF CKF(:
T FV( YK CMY 8  +P P 2 Y  F( Y 6YF
( = FV( Y ( = :( CM  
(F( (#=#(Y =M Y= FC( M: (MFV YK (Y 7=F( FV F M= YK CM CY C+=Y(: T(F =M FV( +CF6:( =  4=MYC:( ; YF YTYF(Y CM VCKV FV( ( CY YF CKF CM# C MK( CFV (Y+(KF & WY  FV( F( Y 6YF
( CM# C MF F= FVCY F= 6MC=  YVCFY   B  F

MY=  FC=M2 ( V #(  - L M: M=MCM( F( Y Y6KV Y  =  K MM=F

8= ;8#*% #; 0$ =0%=   #0 = % :# ,0*;#0 $#%8 %8= < ,#0  =%

50

Pierre Colinet

++( CM FV( (#=6FC=M (6 FC=M WYY6CM  F= ( CM Y 2 CF K M
(  CM:((: (Y+(KFY  YT(F C(Y2 M: FV F M= =FV( F(  KV(K (: FV F  K M
( =6M: F FVCY = :(  /VCY TC(:Y FV( K((


F(: 6

=F=@C# YVCMY T (6 FC=M  8  - +P P  B  
 :TM CKY M=F =MT CM F=2
6F Y= CM =M( :C(MYC=M Y(( C <" 4=MYC:( CM M= FV F FV( CM# C MK(   B & CY ( T
= (M
T FV( CM( F( 2 (

FV( (F FV( : +(: @ (6 FC=M  8  -  +P P  B 


VCKV V Y
((M 6Y(: Y =:( F= :(YK C
( F

MYCFC=MY  = YF( :T V(  =M  + FF( MY F= =YKC FCM M: (#(M KV =FCK YF 6KF6 (Y Y(( (  MCK=MC

M: :( 2 <5

  3 + !C (KF M6( CK  YC6 FC=MY = FV(
F( FV( (+=M(MFC   =FV = 2 WY FV( F(   FV( YK CM = FV( = :( +

(F( CY  * 2 CM = :( =  F( Y F=
( = FV( Y ( = :( CM FV( (#=6FC=M (6 FC=M 7=F( FV F M=FV( F(  = FV(

Interfacial Patterns and Waves in Liquid Layers and Thin Films

51

Y ( = :( Y FCYTCM  YT(F C(Y
6F FV( CM# C MK( F= 6MC=  YVCF2 CY FV( F(  P >=(#( 2
=FV FV(Y( 6 :

FCK F( Y := M=F ++( CM (M(

 = DG(M : + =
(Y2 6M(YY Y=( M=MD=6YYCM(Y (0(KFY = Y6  K( :(=  FC=M ( K=MYC:( (: ,M FV(  FF( K Y(2 =M( (FY FV( (6 FC=M            8  -  +P P    
2 4 YFC= M: (  %  -   B P  :( Q KC 8


(F 2 <"5 2
 7=F( FV F  I  CFVCM Y=(

M(Y = # 6(Y = FV( K=(KC(MFY  M:  2 FV( CMYF
C CFT CY Y F6

F(: M: #( T K=+( KV =FCK (C(Y =KK6 2 YCC T F= FV( 6

=F=@C# YVCMY T (6 FC=M WF FV( K=MF

T2 = M=MD=6YYCM(Y K Y(Y2
6F 6M:(= 
( CMF(  K(Y   M=
=KV2 <L2   # MCYV(Y M: =MT FV( M=MY F6

FCM F(    

( CMY F FVCY = :(  >CV( = :( M=MCM( F( Y YV=6: FV(M
( K=M YC:( (: DT YT(F T2 M(F= :( 6 :

FCK F( Y YV=6: K=MF CM F ( YF =6 

:C(MF =+(

F= Y2 Y6KV FV F FV(
 MK( CFV CM( F( Y ( :Y F=  < >(MK(2 FV( ( CY M= =M( MT ( Y=M F= K=MYC:( 6 :

FCK F( Y
(= ( K6
CK =M(Y2 M: FV( ( (  MT F( Y CM (M(

 VCKV ( K=+ FC
( CFV YT(F T (6C ((MFY = + FCK6 + =
(2 Y(#(

 = FV(  T # MCYV2 Y6KV FV F ( C M=F FF(+F V( ( F= +6 Y6( FV( :CYK6YYC=M CM (M(

 F( Y2
6F

FV( =K6Y =M Y=( + FCK6 K Y(Y = DG(M : K=M#(KFC=M
(F((M +== T K=M:6KFCM 6M:(= 
(
=6M:

C(Y2 M: CM FV( CCF =  ( 

M:F M6
( 2 CF V Y
((M YV=M M=
=KV2 <L FV F FV( (M(

 =  = FV( = :( +

(F( (6 FC=M ( :Y           P  B 8  -  +P P  B % B       
F6 (2 VC( =FV( K=(KC(MFY K M
( K K6 F(: = + FCK6

+ =
(Y 
6=T MKT: C#(M = Y6  K(F(MYC=M: C#(M K=M#(KFC=M2 =M( T(

= F= T( YTYF(2  ,M + FCK6 2 = # 6(Y = +

(F( Y K=

(Y+=M: CM F= 6Y6  H

M=MCDG(M : K=M#(KFC=M2 :C (KF M6( CK  YC6 FC=MY Y(( C < YV= #( T MCK(  (((MF CFV (K(MF (+( C(MFY @KV F?

M: 7(CF?(2 *LL< YV=CM YF =M K= Y(MCM (0(KF CMK ( Y( = FV( ( M  #((MFV = K=M#(KFC=M YF 6KF6 (Y CFV Y6+( K CFCK CFT2 CM ::CFC=M CFV FV( :(#(=+(MF = YV + 

:C(MFY F FV( K=:
=6M: C(Y = +=T=M  K=M #(KFC=M K(Y Y(( Y= /V(YY M: $ Y? 2 <XO /V(YY (F 2 <XO 4=CM(F (F 2 *LL< /VCY YV=Y FV F =M #( =:(Y Y6KV Y FV( M=
=KV (6  FC=M 
52

Pierre Colinet

F

MYCFC=MY F= VCVT Y6+( K CFCK  H

M=MC K=M#(KFC=M2 (#(M FV=6V FV(T CVF M=F
(
( F= :(YK C
( F

MYCFC=MY F= K=+( FC(:(+(M:(MF (C(Y KV

KF( CYFCK = CMF(  KC  F6
6(MK( /VCY ( CMY M =+(M + =
( Y((

Y= 7(+=MT YVKVT (F 2 *LL*

  3  Z(F  + FF( M K= Y(MCM F CMK ( YCM # 6(Y = FV( Y6+( K CFC K CFT CM H

M=MCDG(M : K=M#(KFC=M /=+  :C (KF M6( CK  YC6 FC=MY = FV( M=
=KV (6 FC=M 
 YT(F T CY
= (M
T FV( =F FC=M2 ( C K=MYC:( FV( =:(           P  B 8  -  +P P  B % B         <  9P  FV(  YF F(  = VCKV
(CM CM:((: FV( =(YF= :( F( 
( CM FVCY  6   CM# C MK( + =+( FT CFY YCM CY KV M(:
T FV( F

MY=  FC=M  7=F( FV F FV( Y ( F(  V Y Y=
((M 6Y(: CM 8=6M M: RC(K (2 *LL2 FV=6V

FV( CM K=
CM FC=M CFV =:C;(: @CF>=V(M
(  (6 FC=MY /V(Y( 6FV= Y V #( C:(MFC;(: (C( = Y+ FC=F(+=

 KV =Y VCKV CY CM:6K(:2 CM V( =M  YF 6KF6 (Y2
T +( YCYF(MF (KV MCY = M6K( FC=M = :CY=K FC=MY
T +(MF =MV(+F =M :((KFY 7=F( FV F FVCY (0(KF CY 6CF(

Interfacial Patterns and Waves in Liquid Layers and Thin Films

53

YCC F= FV F YV=M CM C <"2 FV=6V M= +(  M(MF KV =FCK (C(Y

( =
Y( #(: FV( (2 + =

T :6( F= FV(
Y(MK( = =F FC=M !C (KF M6( CK  YC6 FC=M = FV( =:C;(: M=
=KV (6 FC=M 
 F(KVMC6(Y
Y(: =M  YF =6 C( F

MY= Y W FT+CK  (Y6F CY + (Y(MF(: CM C *L2 VCKV YV=Y KV =FCK (C(Y (CMCYK(MF = FV( +( C=:CK YF 6KF6 (Y = C <E2 (#(M FV=6V  MT :((KFY ( + (Y(MF2 K=(CYFCM CFV CY M:Y = ( FC#(T = :( (: V( =M  YF 6KF6 (Y 7=F( ;M T FV F =FV( # C MFY = FV( M=
=KV =:( (CYF2 CMK6:CM K=6+CMY CFV ( M )= (0(KFY

F = 

M:F M6
( Y @VFC M M: @C# YVCMY T2 <<2 =  (( Y6  K( :(=  FC=M = F= T( YTYF(Y Q==#CM (F 2 <X

  3 $ @M +YV=FY = KV =FCK YF F( :(#(=+CM =M FV(
K  =6M: = M V( =M  + FF( M2 =
F CM(:
T :C (KF M6( CK  YC6 FC=MY = FV( M=
=KV (6 FC=M =:C;(:
T F( 
( CM FV( KVC

 YT(F T C CK CM =F FC=MCM:6K(: 4= C=CY = K(Y

#

" % %$

/V( 6FV= =6: C ( F= FV M  D6YV2 W >=Y=C2  K ( F2 W C( YKV(2 HQ ( :(2 4@ ,= C=2 H D(YF(V= M2 Q 7(CF?( M: H @KV F? =

=CM (+ =:6KFC=M = Y=( = FV(C (Y( KV (Y6FY = :

CMY CM FV(Y( (KF6 ( M=F(Y /V( 6FV= 

F(6T K M=(:(Y ;M MKC  Y6++= F = FV( =M:Y :(  R(KV( KV( @KC(MFC;6(  7R@2 = FV( 4=6M 6FG( 

MK CY(

54

Pierre Colinet

:( D(C6( FV =6V FV( WR4>,H!@ + =3(KF WR4 L LL"2 = FV( D(C M (:(

 @KC(MK( =CKT $K( M: = FV( 6 =+( M @+ K( W(MKT FV =6V FV( 4,HJ 4=M#(KFC=M M: ,MF(  KC  H YY KV M( (Y( KV + =



  
! > DG(M : Z(Y F=6
C=MY K(6 C (Y : MY 6M( M ++( C6C:( F

MY+= F(MF :(  KV (6 + K=M#(KFC=M (M G(C( +(  M(MF ,$$ + +%-2 *E*2 <L< H D(YF(V= M M: R  C(: CKV R=F FC=M T CM# C MF = :( +

(F(

(6 FC=MY = M F6

 + FF( MY CM M=M(6CC
C6 YTYF(Y +%- '/* 2 X*E **EX*2 < RD DC :2  @F( F2 M: 7 ZCVF==F $-
 +/$/$ C(T2 <EL H D=K  @6  K( F(MYC=M Y FV( K 6Y( = DG(M : K(Y M: Y6  K( :(=   FC=M CM C6C: ; /2 <5"EXLEX<2 <XE  D6 (
KV2 @Q D M =02 M: @> ! #CY 7=MCM( YF
CCFT = (# + =

FCMK=M:(MYCM C6C: ;Y !  /"+2 <X E  2 <"" @ 4V M:

Y( V  %%$" $ % $/" % !=#( 2 <E< 4, 4V CYF=# M: HQ ( :( !CYYC+ FC#( Y=CF=MY +%-" (2 "E*  52 <X  4=CM(F2 4 Z( =Y2 M: HQ ( :( $$/ (%$"-  "/& /$-$&(*/$ $-/- C(T 4>2 *LL<  4=CM(F2 WW 7(+=MT YVKVT2 M: 4 Z( =Y H6FC+CK FC=M = :((KFY CM V( =M  + FF( MY 
+%- ./2 X5 "L "E2 *LL* H4 4 =YY M: 4 >=V(M
(   FF( M =  FC=M =6FYC:( = (6CC
C6 '/*  +%-2 EX"X<<<<*2 < D KV(


C M: > RC(K ( ,MYF
CCFC(Y = V( =M  + FF( MY CFV
= (M KVC

 YT(F T +%-" (2 <5
VM 6(Y M: ;M( (C(Y : C#(M
T M (=FV( CK CMF(  KC  ( KFC=M +%- '/* ./2 "* E2 <  K ( F M: W /V(YY 7=M
=6M: :CY=K FC=MY CM V( =M  + FF( MY +(MF =M CM(Y CM Y6  K(F(MYC=M: C#(M DG(M : K=M#(KFC=M +%- '/* 2 EL <<5 <* 2 <  K ( F2 H D(YF(V= M2 M: W /V(YY @6 ( K(Y CM Y6  K(F(MYC=M : C#(M DG(M : K=M#(KFC=M (+( C(MF M: FV(= T !  /"+2 XE <XX<52 <"  K ( F2 H WK ( 2 M: 8 @VC 4V(CK  + FF( M =  FC=M : C#(M
T

M(6F

C? FC=M ( KFC=M  F ,  H(KV MCY M:
YCK ( F6 (Y +%- -2 <E"X2 *LL 

Interfacial Patterns and Waves in Liquid Layers and Thin Films

55

Z Q KC 8


2 Z 4 YFC=2 M: HQ ( :( W M=MCM( (#=6FC=M (6 FC=M = DG(M :H

M=MC K=M#(KFC=M CFV :(= 
(
=6M: T +%- ./ ,2 <**
2 Z 4 YFC=2 M: HQ ( :( DG(M :H

M=MC K=M#(KFC=M CFV :(= 
( CMF(  K( M: +== T K=M:6KFCM
=6M:

C(Y +%- -2 L*EXX*EE<2 <"5
 WW Q==#CM2 WW 7(+=MT YVKVT2 M: ZH CY(M  FF( M =  FC=M CM  (YK ( H

M=MC K=M#(KFC=M CFV :(= 
( CMF(  K( +%-" (2 "<<<5< 52 <X  Q6T=M2  >6CM2 M: Z (FCF %%$ / +%- / @ #=C Y WKF6(Y2 :CFC=MY :6 47R@2 << D > 6F M:  4=CM(F @6  K(F(MYC=M: C#(M CMYF
CCFC(Y = +6 ( C6C:  T( (# +=

FCM CMF= M CM( F  Y !  $/"/ "2 *"X*E LX2 *LLX  ,Y

( KV#CC $//" $ -"/ "/- WK :(CK  (YY2 <*  M=
=KV  FF( M Y((KFC=M CM =M #((MFV K=M#(KFC=M +%-" (2 < XL 52 <L H H C :2 Z H=FF(2 W/ 7=2 M: H C(MC RCMY M: V( =MY  :( = M M=K TYF Y  H

M=MC (0(KF ! +%- +/ 2
(F( (6 FC=MY = =M #((MFV CM YF
CCFC(Y +%-" (2 "ELX2 <X WW 7(+=MT YVKVT2 HQ ( :(2 M:  4=CM(F $/" +/$/$ $ $*/"$ 4V + M 9 > 4R42 *LL* W $ =M2 @> ! #CY2 M: @Q D M V=0 Z=MYK ( (#=6FC=M = FVCM C6C: ;Y '/*  +%-2 E<"L2 <5 H  MCK=MC M: R :(  @F FC=M T2 :TM CK  M: KV =FCK YF F(Y = FV( F=:C(MYC=M  : +(: 6

=F=@C# YVCMY T (6 FC=M +%- '/* 2 XE*5<*5*<2 <5 RW ( Y=M $M K=M#(KFC=M K(Y CM:6K(:
T Y6  K( F(MYC=M !  /"+2  "2 <X"  =MF(Y2 4, 4V CYF=#2 M: HQ ( :( 76( CK  YF6:T = + FF( MY

M: FV(C (#=6FC=M CM ;MCF( (=(F C(Y $ !  +-2 E<""<"L2 <E  =MF(Y2 4, 4V CYF=#2 M: HQ ( :( 76( CK  ++ = KV F= + FF( M Y((KFC=M CM =:( + =
( = DG(M : K=M#(KFC=M CM ;MCF( )6C:  T(  ,$$ #$* 2 <X5<5X2 < W8( R(:MC =# M: HQ ( :( 7=MCM(  #(Y M: :CYYC+ FC#( Y=C F=MY CM FVCM C6C:  T( Y ,M >4 4V M2 (:CF= 2 $$/ */- $ &+-/  6( 2 *LLL

56

Pierre Colinet

W8( R(:MC =#2  4=CM(F2 HQ ( :(2 M: 4 Z( =Y R T(CV H

M=MC =YKC F= T CMYF
CCFT CM V= C?=MF  C6C:  T( V( F(:  =


=#( K=6+CM M: =:(CCM = CMF( M  M: Y6  K( :C FC=M   #(Y !  /"+2 LXX5552 *LLL W8( R(:MC =#2  4=CM(F2 HQ ( :(2 M: 4 Z( =Y $YKC F= T FV( =K +C T CMYF
CCFT CM C6C:  T( CFV :(= 
( =+(M Y6  K( K +C T

#CFT  #(Y2 =MCF6:CM   #(Y M: =:(CCM ! $&   +/%$2 *X"< LX2 *LL< H  @KV F? M: Q 7(CF?( +( C(MFY =M FV( =K +C T CMYF
CCFC(Y ,$$ '/*  /"+2 <*52 *LL< Z @VFC M M: Q @C# YVCMY T >( =M  YF 6KF6 ( =  (YK ( H

M=MC K=M#(KFC=M +%-" (2 X* 55 ""2 << Q, @C# YVCMY T 7=MCM( M TYCY = VT: =:TM CK CMYF
CCFT =  CM

) (Y + F < :( C# FC=M =
YCK (6 FC=MY ," ,-$"2 <<55 <*LE2 <55 Q, @C# YVCMY T $M K(6 CMYF
CCFT CM FV( Y=C:C;K FC=M = :C6F(
CM T

=T +%-" (2 "* * "2 <"  @F(+V M M: 4W D6YY( WM TYCY = FV( V( F F

MY( K=(KC(MF =  ==#(: V( F +C+( (# +=

F=  Y $ ! / -- $-/2 X "<2 <* D @CF M: 4 >=V(M
(  >T: =:TM CK )6KF6 FC=MY F FV( K=M#(K FC#( CMYF
CCFT +%- '/* ,2 <X<2 <55 W /V(YY M: @W $ Y?  @6  K(F(MYC=M: C#(M DG(M : K=M#(KFC=M F CM;MCF( 

M:F M6
(  !  /"+2 *"*L<*L2 <X W /V(YY2 ! @+C M2 M: D A 6FFM(  CYK=6Y )= F CM;MCF( H

M=MC M6
(  +%- '/* ./2 5X E<  E<52 <X  /V=Y=M $M K( F CM K6 C=6Y =FC=MY =
Y( #
( =M FV( Y6  K(Y = CM(

M: =FV( K=V=CK C6=6 Y +-
+"  $/2 == 2 H @KV F?2 D @CF2 ! HK4= CK 2 M: >Z @CM M(T Z=M #((MFV Y6  K(F(MYC=M: C#(M DG(M : K=M#(KFC=M (+(  C(MF M: FV(= T !  /"+2  X X2 <5 Z (V M: > ZCM:( H

M=MCYF (YY: C#(M 'Y=CF=MCK' +( C=:CK  #( F

CMY =F FCM CM M MM6 K=MF CM( :6 CM V( F F

MY(  !  $/"/ "2 <"5<X<EX2 <5 W C( YKV(2 > ZCM:(2 M: HQ ( :( ,MF( M   #(Y (KCF(:
T FV( H

M=MC (0(KF +%- '/* 2 E*EX**EXL2 *LLL 87 8=6M M: > RC(K ( (MF V(+F :((KF KV =Y CM =:( = =F FCM V( =M  K=M#(KFC=M +%- '/* ./2 L< XL*2 *LL

% $   $ U(L:( U 6(+=LRWTJTR ,5!5(! # "!05!&3 5&0(( * 35 (3!!!5 # 5&0() 35 ')  05 35&! # !05 535(! &0!5 3 !05 #(! * !( ( 5&!(*3( 3)3!53
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58

Alexander A. Nepomnyashchy 2$ 2 22

,)(&3 # &.5 #(!3                       $ 4(5 3!!) !05) # !05 ( #(!                ((5 5.55(! # #(! (3!!53              ''

 
&



Nonlinear Dynamics of Fronts



59

  

/TDW (JG5 ( DW 
=5G "(W1
5G 5L5W5 =L(W /T( "(W GTG + =+G( =L GT( G( W5 J(1 ((JG =L(GDJ "(W1 W=5L: "(W (GJ  ( ="( L(:
R .'2# (5GD=LW U GR+DJ (+( = W5JT L (5GD=L DW GT( JWWDJ ,2 2! %1 '  6 ' - K$  /  / $ 4  KN '$ K - ' $ ' $ K -   < ,GW (L(  W=5GD=L TW GT( =  '$ 4 - '
  64 C '  C 64 /T( +TRWDJ WRWG( ="( L(:
R < DW #%02,,2 GT( (L( R DLG(     < < ' C 6 ' + * *  DW J=LW( "(: /T( (L( R J=LW( "GD=L DW GT( +TRWDJ (W=L = GT( "( + =+GD=L DGT=5G  :(JR ,L (DGR1 "(W DL G( :(JR
(J5W( = GT( "DWJ=WDGR1
5G GTG :(JR R
( GT( W= U GR+DJ (+( =  # (5GD=L DW GT( :D05WD=L (5GD=L1 ' - ' $  /  / $ 4  KN '$ K - ' N   K /T( W=5GD=L = GTDW (5GD=L DW J=LWG 5JG(: DGT GT( T(+ = GT( P ((L 5LJGD=L 
<     :   $ 4  4  -   (+  4  4  ;4  4  /TDW (5GD=L :(WJ D
(W  :(JR = GT( DLDGD :DWG5
LJ( GT( GTL "(W /T( (L( R   < ' +  * DW L=G J=LW( "(:
5G :(J (W(W + -  +4







' + / K$

TDJT DW JT JG( DWGDJ =  L=LJ=LW( "GD"(1 00,2 WRWG( ,L = :( G= (G  +( WDWG(LG "( DL  :DWWD+GD"( WRWG(1 ( T"( G= DLG =:5J( L 2%2 ' 0 #2 GTG D J=+(LWG( GT( (L( R :DWWD+GD=L /TDW W=5 J( JL
( (G(: G= L DLG DLWDJ %0' = GT( WRWG(

60

Alexander A. Nepomnyashchy

D WG1 (G 5W :DWJ5WW WD+( (+(W ="( L(:
R = :DL R :D0( (LGD (5GD=LW UL (+( =  J=LW( "GD"( WRWG( DW GT( T =LDJ =WJDG=  +  C )  - K +4 /T( (L( R -

< *

+ +4



< C )  *

DW J=LW( "(:1  GT( W=5GD=LW 4 - A J=W)4 C  :(WJ D
( =WJDGD=LW DGT  J=LWGLG +DG5:( A , GT( :DWWD+GD=L DW G(L DLG= JJ=5LG1 GT( =WJDG= DW :(WJ D
(:
R GT( (5GD=L +  + C )  - K C * +4 +4 /T( (L( R :(J (W(W

 + + - *  K +4 +4 /T( W=5GD=L

 4 - A J=W )   4 C 

:(WJ D
(W :(JR = =WJDGD=LW  / ) DW WW5(: ,L = :( G= (G WGGD=L R =WJDGD=LW DL GT( + (W(LJ( = :DWWD+GD=L1 =L( L((:W L (L( R W=5 J( UL (+( DW + ="D:(:
R GT( %%%2 0# + +  C )  - K C *< C   +4 +4 /T( W=5GD=L  - K DW %2' %02 ,L:((:1 W=5GD=LW = GT( DL( D?(: (5GD=L1 +  + C )  - K$  * +4 +4  = DGT GD(  4 - A J=W )   4 C  /T( ("=5GD=L = (L( R DW ="( L(:
R GT( (5GD=L

 + + - <     +4 +4

Nonlinear Dynamics of Fronts

61

/T5W1 GT( (L( R  =W T(L  / < L: :(J (W(W T(L   < #L( JL (+(JG GTG GT( ( DW  J=W(: =
DG =L GT( +TW( +L( $ ++4 TDJT DW WDG5G(: DL
=GT (D=LW  / < L:   < DL W5JT  R GTG GT( DL L: GT( =WW = (L( R  ( 5R J=+(LWG(: @5JT L =
DG  #'#2 D :(WJ D
( W5WGDL(: +( D=:DJ =WJDGD=LW DL GT( L=LJ=LW( "GD"( WRWG( 6= ( D ::  :D05WD=L G= GT( :RLDJW



% $ $$%$

X(G 5W J=LWD:(  WD+( :RLDJ WRWG( TDJT :(WJ D
(W GT( 5GD+DJGD=L = LDW +% -
%  +%$ +4 T( ( % DW GT( L5
( = LDW +( 5LDG  (1
DW GT( 5GD+DJGD=L G( L: + DW GT( :(GT G( ,
 +1 GT( LDW 5GD+R DL L (+=L(LGD R %4 - %K  $ T( (  -
 + DW GT( J=LWGLG  =GT G( /T( =:( :(WJ D
(: 
="( DW L=G (DWGDJ DG DW LG5  G= WW5( GTG GT(  =GT G( JG5R :(J (W(W DGT GT(  =GT = GT( L5
( = LDW1 :5( G= J = (W=5 J(W1 L: GT( 5GD+DJGD=L DW J=+(G(R WG=++(: T(L GT( :(LWDGR = LDW % (JT(W  J( GDL "5( %  UWW5( GTG GT( :(+(L:(LJ( = GT(  =GT G( % =L % DW DL(  +% - %%$ * +4 T( (

 % % -  <   % 2. 0 2 N ( T5WG 
%  < C    %  % %

5GD=L *1  TW G= (5DD
D5 +=DLGW1 5LWG
( +=DLG % - K L: WG
( +=DLG % - %  = LR DLDGD J=L:DGD=L W5JT GTG % - K1 % - % 1 W=5GD=L %4 G(L:W G= % W 4  UL =
"D=5W (WJDL = " D
(W % L: 4 =W G= ( DG( GT( WRWG( *1  DL GT( =  +' - '<  '$   +4

62

Alexander A. Nepomnyashchy

DGT (5DD
D5 +=DLGW ' - K 5LWG
( L: ' - < WG
( UWW5( L= GTG GT( :(LWDGR = GT( +=+5GD=L ' DW L=G 5LD=  L: :(+(L:W W= =L GT( J== :DLG(  UWW5( W= GTG GT( LDW +( =  

L:=  =L GT( J== :DLG( 1 TDJT JL
( :(WJ D
(:
R GT( ::D GD=L :D05WD=L G(  DL GT( DTGTL: WD:( = (5GD=L   ' - ' C '<  '

V

5GD=L V DW J(: 0.2( , (5GD=L DWT( <41 = == =0 ( <4 /TDW (5GD=L :(WJ D
(W
=GT 5GD+DJGD=L = LDW  (JGD=LM L: GT(D D GD=L :D05WD=LM1 L: T(LJ( DG DW L (+( =  2#%( 0% 0'02 U (L(  (JGD=L:D05WD=L WRWG( JL
( J=LWG 5JG(: DL GT(  (=  = JT(DJ DL(GDJW UWW5( GTG ( T"(  5GDJ=+=L(LG DG5 ( DGT GT( J=LJ(LG GD=LW = J=+=L(LGW % 1 2 - <$    $ 9 , GT( J=+=L(LGW  ( :DWG D
5G(: 5LD= R DL W+J(1 GT( (JGD=LW
(G((L GT(  ( :(WJ D
(:
R W=( DL(GDJ (5GD=LW +% -  % $    $ % $ 2 - <$    $ 9 +4 , GT( :DWG D
5GD=LW = GT( J=+=L(LGW  ( L=L5LD= 1 =L( TW G= G( DLG= JJ=5LG L=L?( = )5(W  $ 4 = J=+=L(LGW 7% -   74



C  % $    $ % $ 2 - <$    $ 9

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-

 

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T(LJ( 

 7%    %  C  % $    $ % $ 2 - <$    $ 9 74  , =L( JL :DW ( : GT( :(+(L:(LJ( =  =L % $    $ % 1 GT(L 

 7%  O% C  $ 2 - <$    $ 9 74  /T( L=L:D=L (((LGW = GT( :D05WD=L G D  ( 5W5R W( GTL GT( :D=L =L(W , ( WW5( GTG  -  8 1 ( (G 7% -  O% C  % $    $ % $ 2 - <$    $ 9 74

Nonlinear Dynamics of Fronts

63

W5R1 GT( G(W = (JGD=LW WG =LR :(+(L: =L GT( G(+( G5 (1 GT( ( = ( =L( TW G= G( DLG= JJ=5LG GT( T(G + =:5JGD=L J5W(:
R GT( (JGD=L 7% -  O% C  % $    $ % $ # $ 2 - <$    $ 9N 74

F

7# - O# C % $    $ % $ #  74

4

GT( :(+(L:(LJ( = GT( T(G :D05WD=L J=(JD(LG  =L GT( " D
(W % $    $ % 1 # DW DL= (: /T( (JGD=L:D05WD=L (5GD=LW R T"( W=5GD=LW DL GT( =  = + =+ GDL  =LGW UW L (+(1 (G 5W J=LWD:( GT( WRWG( F1 4 DGT 9 - *1 T( ( % DW GT( J=LJ(LG GD=L = GT( 5( L: % DW GT( J=LJ(LG  GD=L = GT( J=
5WGD=L + =:5JG ( WW5( GTG GT( =R(L DW 
5L:LG L: T(LJ( DGW J=LW5+GD=L DW D

(("LG /T( WRWG( = (5GD=LW JL
(  DGG(L W ==W 7% -  O%   # % $ ! 74 7% -  O% C  # % $ 74



7# - O# C  # %  74


( JL WW5( GTG  +5 ( 5( DW =JG(: G   L:  +5 ( + =:5JG DW =JG(: G   
(TDL: GT( J=
5WGD=L  =LGN GTG (:W G= GT(
=5L: R J=L:DGD=LW    % <$ % K$ # # N    % K$ % <$ # # C  6=G( GTG ( T"( G= +=WG5G( GTG  #  - K XG( ( D WG5:R GT( J=
5WGD=L  =LG W=5GD=LW DL = ( :(GD #L( = ( (+( =  (JGD=L:D05WD=L WRWG( +TW(;(: :(WJ D+ GD=L =  ; WG= :( +TW( G LWDGD=L1 D(1 W=D:D;JGD=L X(G 5W JT JG( D?( GT( D5D:W=D: WRWG(
R G= " D
(W1 G(+( G5 ( # L: +TW(= :(

+ (G(  /T( +TW( G LWDGD=L DW :(G( DL(:
R GT( GT( =:RLDJ +=G(LGD   $ #  TDJT TW G= DLD W  5LJGD=L = =  ;(: "5( = # 1 G GT( +=DLG -  D5D: L: W=D: ,L GT( +TW( G LWDGD=L +=DLG1 # - # 1
=GT DLD T"( GT( W( :(+GT    $ #  -   $ #  = #  # 1    $ #  /   $ # N = # / # 1   $ #  /    $ #  # J=5 W(1 GT( ( WT=5:
(  D5 =  DL  J( GDL +=DLG 
(G((L G=

64

Alexander A. Nepomnyashchy

DLD1  /  / 1 TDJT J=

(W+=L:W G= L 5LWG
( DLG( (:DG( WGG( /T( DL(GDJ (5GD=L GTG :(WJ D
(W GT( +TW( G LWDGD=L1 ==W W ==W 7 7 - O  &  $ # $ 74 7

<<

T( (  L: &  ( +=WDGD"( J=LWGLGW /T( T(G G LW( DW =:((:
R GT( (5GD=L 7 7# - O# C  <* 74 74 GT( WG G(  DL GT( DTGTL: WD:( DW J5W(:
R GT( G(LG T(G = GT( +TW( G LWDGD=L /T( +TW( G LWDGD=L  =LG J=

(W+=L:W G= GT( ==DL
=5L: R J=L:DGD=LW = GT( = :( + (G( 1   



N





$

L: ++ =+ DG(
=5L: R J=L:DGD=LW = G(+( G5 ( #  ED==DJ WRWG(W + ="D:( LR (+(W = (JGD=L:D05WD=L =:(W1
(WD:(W GT( === ="DWT( (5GD=L (LGD=L(: 
="( X(G 5W (LGD=L  ( = GT( = :(GDW1 W(( H5

R *KK* !  (0- " 0 -  -* " 
*0 / (GD( L: W+D  "(W = 3

D=LW  ( =
W( "(: U WD+( =:( TW
((L W5(WG(: ' - O' C A'  ' '  ''  ' 

<

*0- ULDW  ( GG JG(:
R W=( W5
WGLJ(W GTG GT(R JL W(J (G( GT(W("(W ,L GT( 
W(LJ( = G( 1 W=( =(
( =  WD(W GTG + =+G( GT =5T "R =GD=LW G (GW L: W+D W +"-. -0-0 /T( +=+5GD=L = =(W JL
( :(WJ D
(:
R G= " D 
(W1 0$ 4 GT( :(LWDGR = T(GTR LDW L: !$ 4 GT( :(LWDGR = D LDW /T( ="( LDL (5GD=LW  ( 7! - !0  ! C 
O! 74 GT( G( W DL GT( DTGTL: WD:( J=

(W+=L: G= GT( DL(JGD=L1 :(GT L:

L:= =GD=L = GT( D LDW1 70 - !0 C  O0 74 DL(JGD=L /T( =:( DW GT( WDD G= J=
5WGD=L

Nonlinear Dynamics of Fronts

65

6= ( D WG G DL"(WGDGDL GT(
WDJ (G5 (W = GT( (JGD=L:D05WD=L (5GD=L1 DGT L (+TWDW =L GT(  =LG W=5GD=LW



"% $"%  ! % 

( WG G DGT GT( WD+(WG + =
(1 ' - ' C '<  '$  /  / $ 4  KN

< 

'K$  - '   K$ ' /  W  


6=G( GTG L(GD"( "5(W = '  ( (LDL(WW 
(J5W( ' DW GT( (WJ(: :(LWDGR = GT( +=+5GD=L ,G JL
( WT=L GTG D '   K1 '$ 4  K = LR 4  K 

"# - "-

D WG1 (G 5W :DWJ5WW GT( WGGD=L R W=5GD=LW = (5GD=L <  GTG WGDWR GT( ==DL = :DL R :D0( (LGD (5GD=L ' C '<  ' - K


= R1 (5GD=L 
< <  '  ' * 

<4

" D
(  W( "(W W GT( GD( " D
( = GTG ;JGDGD=5W + GDJ( W= GTG GT( (L( RM <  - ' C   * DW J=LW( "(: #
"D=5WR1  GT( W=5GD=LW GTG  ( ;LDG( G   T"( G( LGDL WDLW L: T(LJ( +TRWDJR (LDL(WW1 (J(+G G= J=LWGLG W=5GD=LW1 ' - K DGT  - K L: ' - < DGT  - <FN W(( D* XDL( D?DL (5GD=L <   =5L:  J=LWGLG W=5GD=L ' - ' ' DW (DGT(

K = <1 ( =
GDL '% - ' % C   ' % '$  /  / N % ' / $   >( (  '  - ' <  ' 1 T(LJ(   '  - <  *'  WDL GT( =5 D( G LW = GD=L1 ( ;L:   + ' %$ 4 ' 7$ 4 $ *; 

66

Alexander A. Nepomnyashchy U

U

s

s

u2

u

0

u

1

0

u1

s

u u3

u

(a)

(b)

 0  /T( WT+(W = GT( +=G(LGD  =  WRWG( ="( L(:
R GT( DWT( === =" (5GD=LN 
 =  5GDWG
( WRWG(

u

u

(a)

ux

ux

ux

(b)

u

(c)

 0  #
DGW DL GT( +TW( +L( = G "(DL "( W=5GD=LW ="( L(:
R GT( DWT( === =" (5GD=LN  6 - KN 
 K / 6 / *N J 6  *

+7 '$ 4 - 7 '$ 4$ +4 T( ( GT(  =GT G( = GT( J=+=L(LG DGT GT( "(L5
( 1  -   '     /T5W1 = GT( W=5GD=L ' - K1  - <   N GT( (= (1 GTG W=5GD=L DW 5LWG
( DGT (W+(JG G= :DWG5
LJ(W DGT  / <1 TD( = GT( W=5GD=L ' - <1  - <   / K =  1 T(LJ( GTG W=5GD=L DW WG
(

Nonlinear Dynamics of Fronts 

67

" - "-

X(G 5W J=LWD:( L= G "(DL "(  =LG W=5GD=LW = (5GD=L <  DL GT( =  '$ 4 - '1$ 1 -   64 6 DW GT(  =LG "(=JDGR UWW5( 6  K W=5GD=LW DGT 6 / K JL
( =5L:
R (LW = GT( ()(JGD=L G LW= GD=L    = GT=W( DGT 6  K ( =
GDL (5GD=L ' C 6' C '<  ' - K


GTG :(WJ D
(W GT( =GD=L =  ;JGDGD=5W + GDJ( DL GT( W( +=G(LGD  ' :(;L(:
R <4
5G GTDW GD( DGT GT(  DJGD=L 6 WDL GT( (JTLDJ L=R1 ( J=( G= GT( J=LJ5WD=L GTG  GT( W=5GD=LW TDJT  ( ;LDG( W 1   ( GT( ==DL =L(W D 5LD=  5LWG
( W=5GD=L ' - K GT( + GDJ( DW DL GT( WGGD=L R +=DLG J=

(W+=L:DL G= GT( DLD5 = GT( +=G(LGD  1 DD 5LD=  WG
( W=5GD=L ' - < GT( + GDJ( DW DL GT( WGGD=L R +=DLG J=

(W+=L:DL G= GT( D5 = GT( +=G(LGD  1N DDD  =LG W=5GD=L '1 DGT ' - < L: ' - K GT( + GDJ( WG GW ="DL DL GT( +=DLG ' - < G 1 -  G= GT( (G L: WG=+W DL GT( +=DLG ' - K G 1 -  E(J5W( =LR W=5GD=LW DGT '  K =  1  ( +( DGG(:1 5LJGD=L '1 WT=5:
( =L=G=LDJR :(J (WDL GT( GTL =WJDGDL XDL( D?DL (5GD=L 


6  <

*K

/T( (= (1 D 6 / *1 GT( GD = GT(  =LG W=5GD=L G  ( 1 =WJDG(W JTLDL DGW WDL1 L: W5JT W=5GD=LW WT=5:
( :DWJ :(: W(( D*
 = 6  *1 GT( WR+G=GDJW = GT( W=5GD=L G  ( 1 DW '1  A (+ 1 C 3 (+
1 ,L GT( JW( 6 - *1

*<

 6 1 '1  A C 31 (+  *

=  5DGGD"( DL"(WGDGD=L = W=5GD=LW1 DG DW J=L"(LD(LG G= :(;L( 1 - ' 1 L: ( DG( (5GD=L 
68

Alexander A. Nepomnyashchy

= W  ; WG= :( = :DL R :D0( (LGD (5GD=L1 '<  ' + - 6   +'  /T( LRWDW WT=W GTG  GT(  =LG W=5GD=LW DGT 6  *  ( L=LL(GD"( L: =L=G=LDJR :(J (WDL W(( D*JN A DL *< L("( "LDWT(W1 T(LJ( GT( W=5GD=L G(L:W G= ?( = =L GT( 2%2#1 (GD"(R W= :D (JGD=L '  (+ 1

**

= 6  *1 / < /T( W=5GD=L DGT 6 - * DW W= L=LL(GD"( L: =L=G=LDJR :(J (WDL /T5W1 ( T"( =
GDL(:  DR =  =LG W=5GD=LW ="DL DGT :D0( (LG "(=JDGD(W 6  * L: T"DL :D0( (LG GD WR+G=GDJW ,G DW DLG( (WGDL G= L=G( GTG GT(  =LG "(=JDGR 6 DW :(G( DL(: W=(R
R GT( WT+( = DGW (:( :(WJ D
(:
R GT( + (G( 1 W D DG W +5(:
R GT( (:(N W5JT   =LG DW J(: 2 % UJJ= :DL G= <1 6- C

< 

E(J5W( DL GT( (D=L = GT( (:( '  <1 GT( + =
( DW JG5R =" ( L(:
R  DL( D?(: (5GD=L /T5W1 DL GT( JW( =  +5(:  =LG ( JL (+(JG (GGDL D+= GLG DL= GD=L TD( J=LWD:( DL  DL( D?(: (5 GD=L E(= ( D :DWJ5WW GT( W((JGD=L = GT(  =LG W=5GD=L TDJT W( "(W DL:((: W GT(  (GD( WR+G=GDJ = GT( W=5GD=L =  35JTR + =
( < 1 
"  *0 " 0 0

UWW5( GTG '  DW =5 D( G LW= 
(      + $ ' 7  '7  '  + '  *;   @+(JD;JR1 GTG (LW GTG '  TW  ;LDG( W5++= G = G(L:W G= ?( = W5JD(LGR WG W 1  UWW5( GTG '  DW "( R W W= GTG ( JL 5W( GT( DL( D?(: + =
(1 ' - ' C '$  /  / $ 4  KN '$ K - '  WDL GT( =5 D( G LW= GD=L1 ( ;L: GTG   + ' 7 
   '$ 4 *; 

*

Nonlinear Dynamics of Fronts

69

+ (WWD=L * :(G( DL(W GT( + =+GD=L = GT(  =LG =L GT( DL(

WG( = DGW :("(=+(LG X(G 5W :(;L( GT(  =LG "(=JDGR DL GT( ==DL R 3=LWD:( GT( ("=5GD=L = GT( ;(: '$ 4 DL  (( (LJ(  ( ="DL DGT GT( "(=JDGR 1 D( DL"(WGDG( GT( W=5GD=L'W
(T"D= =L GT( RW  - 5 C 41 5 -J=LWG ( JL (+(JG GTG D ( G(  W5JD(LGR  (1 ( D =5GWG D+ GT(  =LG1 GT5W ' D :(JR , ( G(  W5JD(LGR W1 ( D 
(TDL: GT(  =LG1 W= GTG ' D  = UWW5( GTG GT( ( (DWGW W5JT  "5( 6 GTG = GT( (+ (WWD=L * D '5 C 4$ 4 - K$   6




L: D '5 C 4$ 4 $  / 6




/T(L ( D J 6 GT( %2 % ,2#' ,L = :( G= (WGDG( GT(  ( GD( WR+G=GDJW = GT( (+ (WWD=L   + '5 C 4$ 4 '7   $ *;  .$ 4 - 2 C <   4$ ( JL 5W( GT( (GT=: = GT( WGGD=L R +TW( /T( W::( +=DLG J=

( W+=L:DL G= GT( ?( = = GT( 5LJGD=L +. DL GT( J=+( +L( = 1 DW =JG(: G  - 2* X(G 5W WTDG GT( DLG( GD=L +GT DL W5JT  R GTG DG D +WW GT =5T GT( W::( +=DLG  - C 2* /T(L ( (G

 
  

5  + 2 '5C4$ 4 - (+  ' 7 C     (+ <  4 * *; *   #
"D=5WR1 =  ( 4 =LR GT( DLG( " = DGT    < 4 (WW(LGDR J=LG D
5G(W G= GT( DLG(  >(LJ( GT( DLG(  DW (5 ++ =DG(R G=

  % +    2 '7    $ * *; 4  T( ( % DW  J=LWGLG /T5W1 ( ;L: GTG 


 5  < '5 C 4$ 4  (+  (+ <  4  L 4  * * WDL GT( :(;LDGD=L = GT(  =LG "(=JDGR D"(L 
="(1 ( ;L: GTG GT( "(=JDGR = GT(  =LG GTG DW J (G(: DL GT( LG5  R  =  (L( DJ DLDGD J=L:DGD=L DW 6 - *

70

Alexander A. Nepomnyashchy

6=G( GTG GTDW DW GT( =(WG "5( = GT(  =LG "(=JDGR TDJT DW =(:
R GT( J=L:DGD=L = GT( L=L=WJDG= R (:(
(T"D=  UG GT( W( GD(1 GT( W((JG(:  =LG DW GT( WG((+(WG =L( =L  GT(  =LGW =L GT( 2%2#
LJT - 6 X(G 5W (LGD=L W= GT( D= =5W (W5G = === ="1 (G ="WR L: DWJ=5L=" =L GT( W((JGD=L = GT(  =LG DGT 6 - * DL GT( JW( = GT( DLDGD J=L:DGD=L DGT  ;LDG( W5++= G === =0 ( <4 U WDD LRWDW JL
( :=L( =  (L(  (JGD=L:D05WD=L (5GD=L ' - ' C  '  /T( DL( J DG( D=L W((JGW GT(  =LG DGT GT( "(=JDGR 6 - *   K 

"$ 0"0! "-

=  ' :D0( (LG  = '<  '1 DG R T++(L GTG GT( J=(JD(LG A DL *< "LDWT(W =  J( GDL "5( = 6 ,L GTDW JW( GT( W(+ G D =L GT( +TW( +L( '$  TDJT ("(W GT( W::( +=DLG <$ K1 G(L:W G= GT( WG
( L=:( K$ K =L GT( %%( 2%2# WGM :D (JGD=L1 '  (+
1 , W5JT  L=L(L( DJ  =LG (DWGW1 DG DW WG((+( GTL GT(  =LG W((JG(:
R GT( DL( J DG( D=L :(WJ D
(: DL GT( + ("D=5W W5
W(JGD=L ,G G5 LW =5G GTG GT( L=L(L( DJ  =LG DLW GT( J=+(GDGD=L DGT GT( DR = (L( DJ  =LGW UW L (+( :(=LWG GDL GT( +=WWD
DDGR =  L=L(L( DJ  =LG1 (G 5W J=LWD:( GT( =:D;JGD=L = GT( DWT( === =" (5GD=L DGT  ' '<  '< C '
1 K /
/ < (;L(  - +'+1 L: ( DG( GT( + =
( ' C 6' C '<  '< C '
$ ' - <$ ' - K

* 

DL GT( =  '<  '< C '
 + - 6  $ K - < - K +' 

*V

X(G 5W ;L: L (JG W=5GD=L ' DL GT( =  =  +=RL=D /DL DLG= JJ=5LG GT(
=5L: R J=L:DGD=LW = ' L: GT( WW5+GD=L GTG '1 DW  =L=G=LDJR :(J (WDL 5LJGD=L1 ( J=( G= GT( J=LJ5WD=L GTG GT( WD+(WG G D 5LJGD=L DW  -  '<  '$  K

*F

@5
WGDG5GDL *F DLG= *V1 ( ;L: GTG GT( GG( (5GD=L DW WGDW;(: D  < < -  $ 6 - *
C   *
*


Nonlinear Dynamics of Fronts

71

/T( W=5GD=L = GT( = DDL + =
( *  DL GTDW JW( DW '1 -

 < <  GLT 1  1  $ 1 - J=LWG * *

*4

,
 <*1 GT(L / <1 GT5W GT( =
GDL(: (JG W=5GD=L
(=LW G= GT( DR = (L( DJ  =LGW1 L: GT( J=

(W+=L:DL  =LG DW (WW WG((+ GTL GT(  =LG DGT - < W((JG(:
R GT( DL( J DG( D=L >=("( 1 D
/ <*1 GT(L  <1 T(LJ( GT( W=5GD=L *4 DW  L=L(L( DJ  =LG TDJT DW WG((+( GTL  GT( (L( DJ  =LGW /T( WG((+(WG  =LG D DL GT( J=+(GDGD=L /T( ( (DWGW  GT(= ( U =LW=L L: (DL
( ( <4! GTG (L(  D?(W GT( (W5GW = === ="1 (G ="WR L: DWJ=5L=" G= GT( JW( =  5LJGD=L  ' TDJT DW L=G J=L"( ,  K -  < - K1 GT( 5LJGD=L '  TW  ;LDG( W5++= G1 L: K  '   <1 GT( W=5GD=L '$ 4 G(L:W W 4  G=  G "(DL "( DGT GT( +TW( "(=JDGR 6 W5JT GTG 6  6  6
1    6 - * + 1 T( (  '    '

+ -   K$ +
- W5+





 $ "- 000" !# -0
*-0-

,L GT( + ("D=5W W(JGD=L ( J=LWD:( (:  =LGW ="( L(:
R GT( (JGD=L :D05WD=L (5GD=L *! ' - ' C  ' DL GT( JW( T( ( GT( 5LJGD=L  ' - '<  ' T: G= ?( =W1 '  - K L: ' - <1 W5JT GTG   '   K T(LJ( GT( W=5GD=L ' - ' W 5LWG
( L:   '  / K T(LJ( GT( W=5GD=L ' - ' W WG
( /T(  =LGW W(+ G(: G= :=DLW1 =L( DGT '  =5L: ' L: L=GT( =L( DGT '  =5L: '  T(L GT(  =LG + =+G(:1 GT( 5LWG
( WGG( W =5WG(:
R GT( WG
( =L( UW ( T"( W((L1 GT( ( (DWG(:  DR =  =LG W=5GD=LW ' - '11 1 -   64 DGT :D0( (LG "5(W = 6 /TDW (G5 ( = WG
(5LWG
( +5(:M  =LGW DW J(  = GT( ==DL (JTLDJ DLG( + (GGD=L = GT( + =
( ' C 6' C  ' - K$ ' - ' $ ' - '  U + GDJ( WG GW DGW =GD=L DL GT( +=DLG ' - ' = GT( D5 = GT( += G(LGD  '  ' -   ' L: ="(W G= GT( +=DLG ' - ' = GT( DLD5 = GT( +=G(LGD  ' DL GT( + (W(LJ( = GT(  DJGD=L = J( + =+= GD=L G= 6  K E(J5W( GT( (L( R = GT( + GDJ(  - ' * C  ' :(J (W(W :5(

72

Alexander A. Nepomnyashchy

G= GT(  DJGD=L1 DG DW =
"D=5W GTG GT( + GDJ( ="DL G= GT( (G  = GT( +=DLG ' - ' - < ;LDWT(W DGW R G GT(
=GG= = GT( (1 DL GT( +=DLG ' - ' - K1 = LR 6  K , K / 6 / *1 GT( +=DLG ' - ' $  - ' - K DW L GG JGDL =J5W1 GT5W GT( + GDJ( +( = W =WJDGD=LW = 6  *1 GTG +=DLG DW L GG JGDL L=:(1 L: ' :(J (W(W =L=G=LDJR X(G 5W J=LWD:( L= GT( JW( T( ( GT( +=G(LGD  ' TW W("(   G( LGDL D L: DLD DL GT( +=DLGW ' - ' W(( D<
 ,L GT( +=DLGW = D1   '  -  '  - K L:   '  -   '  / K1 TD( DL GT( +=DLGW = DLD   '  -  '  - K L:   '  -   '   K X(G 5W DL( D?( (5GD=L *!  =5L: GT( (5DD
D5 +=DLG ' - '  ' - ' C '%1 % '  < ( =
GDL ' % - '% C   ' % '$  /  / N % ' / $   ( ;L: GTG = GT( L=  :DWG5
LJ( ' %  (+I2 C 4$1    '     /T( (= (1 GT( D = GT( +=G(LGD  ' J=

(W+=L: G= WG
( (5DD
D5 WGG(W  / K =  1 TD( GT( DLD J=

(W+=L: G= 5LWG
( (5DD
D5 WGG(W   K = W=(  /T5W1 GT( (DWG(LJ( = W("(  D = GT( +=G(LGD  ' (:W G=  5GDWG
DDGR = GT( +TRWDJ WRWG( 3=LWD:( GT( WDG5GD=L T( ( GT( +=G(LGD TW G= D ' - '  L: ' - ' L:  DLD5 ' - '
(G((L GT(1 ' / ' / '  5GD=L *! JL
( J=LWD:( (:  WD+(WG =:( = L DW=G( DJ +TW( G LWDGD=L
(G((L G= =JR WG
( +TW(W1 ' - ' = ' - '  ,L:((:1
=GT %  W=5GD=LW ' - ' L: ' - '  ( WG
( DGT (W+(JG G= W :DWG5
LJ(W # J=5 W(1 GT( ( (DWG DD(W = WG
(5LWG
(  =LGW J=

(W+=L:DL G= + GDJ( G 2(JG= D(W WG GDL DL +=DLGW ' - ' = ' - ' L: ;LDWTDL DL GT( +=DLG ' - ' 1 TDJT :(WJ D
( =5WGDL GT( 5LWG
( WGG(
R  WG
( =L( X(G 5W :DWJ5WW L=  =LGW
(G((L WG
( +TW(W UWW5( GTG ( T"( L DLDGD :DWG D
5GD=L = '$ K - '  W5JT GTG D - ' $ D - ' 







*

TDJT +TW( D DL T(L GT(R  ( DL J=LGJG D '$ 4 G(L: G= '  =

' W 4  UWW5( GTG  '    ' 1 L: (G 5W :DWJ5WW GT( (DWG(LJ( =  G " (DL "( W=5GD=L1 '$ 4 - '11 1 -   641 TDJT WGDW;(W GT( + =
( ' C 6' C   ' - K$ ' - ' $ ' - '  /TDW + =
( :(WJ D
(W GT( =GD=L = GT( + GDJ( TDJT WG GW =L GT( G=+ = GT( TD G ' - ' L: ;LDWT(W (JGR =L GT( G=+ = L=GT( TD G ' - '  ,W W5JT  =GD=L +=WWD
(

Nonlinear Dynamics of Fronts

73

UG 1 1 GT( (L( R = GT( + GDJ(  -  '  5 DL GT( =GD=L1 DG :(J (W(W1 L: GTDW :(J (W( :(+(L:W =L GT(  DJGD=L LDG5:( 6 D , GT(  DJGD=L DW W5JD(LGR W1 GT( + GDJ( D 

D"( DL GT( +=DLG ' - '  DGT GT( (L( R  ( GTL  ' 1 L: DG D +WW GT =5T GT( +=DLG ' - '  , GT(  DJGD=L DW W5JD(LGR  (1 GT( (L( R = + GDJ( D
(J=( (WW GTL  '  DL GT( J=5 W( = =GD=L1 L: GT( + GDJ( D L("( (JT GT( TD G=+ G ' - '
5G D ("(LG5R G(L: G= GT( "(R
=GG= '  >=("( 1 =L( JL (+(JG1 GTG :5( G= GT( J=LGDL5=5W :(+(L:(LJ( = GT( W=5GD=L =L GT( + (G( 61 GT( ( D
( W5JT  "5( = 6 GTG = GTDW "5(   '   W ' ' 1 W= GTG GT( + GDJ( D ="(  = =L( TD G=+ G= L=GT( =L( @5JT L (J(+GD=L G 2(JG= R1 TDJT (:W  = =L( W::( +=DLG '  $ K G= L=GT( W::( +=DLG ' $ K =L GT( +TW( +L( '$  - +'+11 D (DWG =

L (J(+GD=L "5( = GT(  =LG "(=JDGR 6 /T5W1 DL GT( J=LG :DWGDLJGD=L G= WG
(5LWG
(  =LGW1 GT( WG
(WG
(  =LG D
( JT JG( D?(:
R  :(;LDG( "5( = "(=JDGR 6 E(J5W( 6  K1 GT( +TW( ' - '  DGT   (

"5( = GT( +=G(LGD  ' D =5WG GT( +TW( ' - ' DGT  W( "5( = GT( +=G(LGD ux

u1

ux

ux

u2

u1

u3 u

(a)

u1

u3 u2

u

(b)

u3 u2

u

(c)

 0   =LG "(=JDGR W((JGD=L =  5GDWG
( WRWG( #
DGW DL GT( +TW( +L(  6 / 6 N 
 6 - 6 N J 6  6  ,  '  -  ' 1 GT( + GDJ( D ="(  = ' - ' G= ' - ' D GT( (L( R DW J=LW( "(:1 D( DL GT( JW( 6 - K /T5W1 GT(  =LG
(G((L G= +TW(W DGT (5 "5(W = GT( +=G(LGD D
( =GD=L(WW #
"D=5WR1 D  '  /  ' 1 GT(L 6 / K ( JL 25WG +( =  GT( ()(JGD=L = GT( J== :DLG(  L: (+(G GT(  5(LGW + (W(LG(: 
="(



/#
"(  "!"

/T( 5( GTG :(G( DL(W TDJT +TW( DW G 5R WG
( L: TDJT =L( DW =LR (GWG
( D( DG DW WG
( DGT (W+(JG G= W :DWG5
LJ(W
5G DG DW =5WG(:
R GT( G 5R WG
( +TW( JL
( = 5G(: DL  J( R 5WDL

74

Alexander A. Nepomnyashchy

GT( L=GD=L = GT( 1' %, %#%  ,L = :( G= "=D: GT( :D"( (LJ(W1 (G 5W J=LWD:( GT( + =
( DL   (
5G ;LDG( (D=L ' - ' C  '$ C /  / C$ 4  KN '$ K - ' $ C /  / CN K ' C$ 4 - ' C$ 4 - K$ 4  K

<

X(G 5W :(;L( GT( XR+5L=" 5LJGD=L 

!




 < '   ' +$  ' -   ' * 

 I'$  !

UJG5R1 GTG DW GT( JGD=L =   DJGD=L(WW + GDJ( ="DL JJ= :DL G= GT(  ' C   ' - K /T( " DGD=L = GT( XR+5L=" 5LJGD=L =L GT( JWW = 5LJGD=LW WGDWRDL GT(
=5L: R J=L:DGD=LW < JL
( G LW= (: G= 

!

8 - 

+I' C   '$8'

!


R DLG( GD=L
R + GW1 T(LJ( K JL
(  DGG(L W ' - 

8  8'

*

= LR W=5GD=L = *1 <1  -  I'$ 4$ DW  5LJGD=L = 4 X(G 5W JJ5G( GT( :( D"GD"( = GTDW 5LJGD=L  !  !  ! + +I' '   '' $ -  ' I' C  '$+ -  ' +  K +4 ! ! ! , 5LJGD=L  ' DW
=5L:(:  = 
="( DL GT( (D=L = DGW :(;LDGD=L1 GT(L GT( XR+5L=" 5LJGD=L  4 DW
=5L:(:  =
(= /TG (LW GTG  4 DW (DGT( J=LWGLG G GT( (5DD
D5 WGG(W =  :(J (WDL 5LJGD=L G(L:DL G=  J=LWGLG W 4  ,L +TRWDJ G( W1 GT( WRWG( G(L:W G= L (5DD
D5 WGG( W 4  /T( XR+5L=" 5LJGD=L J=

(W+=L:DL G=  5LD=  (5DD
D5 WGG( ' - ' -J=LWG DW (5 G=  I' $ - *C '  E(J5W( GT( XR+5L=" 5LJGD=L :(J (W(W DGT GD(1 GT( 5LD=  WGG( DGT TDT( "5( =  =5WGW GTG DGT =( "5( =  

Nonlinear Dynamics of Fronts



75

+0"$*" 0 "

U
WDJ =:( =  5GDWG
( WRWG(1 W5(WG(: DL GT( J=LG(G = GT( +TW( G LWDGD=L1 DW GT( /2%( .% 2! %1 ' - ' C '  '



TDJT DW JG5R (5GD=L *  DGT
- < /T( +=G(LGD  ' JL
( :(;L(: W <  ' -  <  '  W(( D  L: GT( XR+5L=" 5LJGD=L DL GT( DL;LDG( (D=L  /  /  DW 
  < <  I'$ + ' C <  '   *  X(G 5W :DWJ5WW WGGD=L R W=5GD=LW = (5GD=L  TDJT WGDWR GT( (5 GD=L ' C '  ' - KN   GT( ; WG DLG( GD=L D"(W ' C  ' -  - J=LWG * /T( +=G(LGD TW G= D5 +=DLGW1 ' - <1 DGT  < -  < - K1 L: =L( DLD5 +=DLG1 ' - K1 DGT  K - < N GT(W( +=DLGW J=

(W+=L: G= 5LD=  WGGD=L R W=5GD=LW = (5GD=L  #
"D=5WR1 W=5GD=LW ' <  ( WG
( L: W=5GD=L ' - K DW 5LWG
( U

U

U s

−1

1 s

s u

(a)

−1/4

u

s

u

u

u0 −

−u

u

*

u+

u

(b)

s

u+

(c)

 0  /T( WT+(W = GT( +=G(LGD = GT( U(L3TL (5GD=LN  , - KN 
 K / , / , N J , - ,  /T( +TRWDJ DLG( + (GGD=L = GT( =:( :(WJ D
(: 
="( JL
( W = =W ' DW GT( W+=LGL(=5W L(GD?GD=L :D (JG(: =L GT( :(;LDG( DW

76

Alexander A. Nepomnyashchy

:5( G= GT( J RWG LDW=G =+R @=5GD=LW ' - < L: ' - < J=

(W+=L: G= GT( =++=WDG( = D(LGGD=LW = GT( L(GD?GD=L DL  (

=L(G1 TD( W=5GD=L ' - K J=

(W+=L:W G= GT( 5LWG
( + L(GDJ +TW( TDJT TW L= W+=LGL(=5W L(GD?GD=L DL GT( 
W(LJ( = GT( (G( L L(GDJ ;(: /= = ( DD(W = W=5GD=LW1 TDJT J=

(W+=L: G= GT( (L( R  ' * C  ' - K1  (       '  - GLT  L: '  - GLT  $  - J=LWG * * /T(W( W=5GD=LW1 TDJT  ( J(: +%+ L: %+%+ W=5GD=LW1 (W+(JGD"(R1 :(WJ D
( GT( =GD=L(WW  =LG :=DL 
(G((L G= WG
( WGG(W ' < /T( XR+5L=" 5LJGD=L = 5LD=  WG
( W=5GD=LW ' - < DW (5 G= K1 TD( GT( XR+5L=" 5LJGD=L = DL L: LGDDL W=5GD=LW DW      < * * <        K V + J=WT +'<  '  *  * *    GT( " D
( JTL( ' - GLT    * DW 5W(: /T=5T GT( DL W=5GD=L TW   ( "5( = GT( XR+5L=" 5LJGD=L GTL GT( 5LD=  WGG(1 DG DW WG
( DGT (W+(JG G= W :DWG5
LJ(W /= WT= GTG1 (G 5W DL( D?( (5GD=L   =5L: GT( DL W=5GD=L L: =  5G( GT( (D(L"5( + =
( = GT( :DWG5
LJ(  =GT G( 

   '% C <  GLT  ' % - % ' F * /T( J=LGDL5=5W W+(JG 5 = GT( @JT A =:DL( (5GD=L F1 TDJT J=

( W+=L:W G= GT( (D(L5LJGD=LW GTG := L=G :(JR W  1 DW  *   / K ,L = :(

G= ;L: GT( :DWJ (G( W+(JG 5 = F1 (G 5W :(;L(  1 - GLT    * L: ( DG( F W GT( X((L: ( (5GD=L
 > +% ' % + ' C BB C <  ' % - K$ <  1   *1 +1 +1 <  1 T( ( B - *1 > - C * UJJ= :DL G= (L(  GT(= R = GT( X((L: ( (5GD=L1 GT( :DWJ (G( W+(JG 5 DLJ5:(W B (D(L"5(W1 J=

(W+=L:DL G= > - <$    $ B ,L =5 JW(1 ( (G G= (D(L"5(W    > - *$  - K$ '% - <  1 - J=WT  * GTDW L(5G  :DWG5
LJ( J=

(W+=L:W G= L DL;LDG(WD W+GD G LWGD=L = GT( DL L:   GLTI    *$    > - <$  -  $ ' % - 1<  1  * J=WTI    *$

Nonlinear Dynamics of Fronts

77

/T5W1 GT( (  ( L= +=WDGD"( (D(L"5(W =  U =GT( WGGD=L R W=5GD=LW DGT < /  / K1 TDJT J=

(W+=L: G= =WJDGD=LW = GT( + GDJ(1 JL
( =5L: LRGDJR
R (LW = (D+GDJ J=
D 5LJGD=LWN T=("( 1  = GT(  ( 5LWG
(



"0!" 000" %"%-

, GT( WRWG( ="( L(:
R  ("="(W WG GDL DGT  D"(L '$ K ' 1 W5JT GTG '  :=(W L=G T"( GT( W( WDL ("( RT( (1 GT( DLDGD ("=5GD=L (:W G= GT( J (GD=L = :=DLW DGT '  < W(+ G(:
R DL D( :=DL W E(J5W( (JT :=DL  D"(W  +=WDGD"( J=LG D
5GD=L V G= GT( WRWG('W XR+5L=" 5LJGD=L1 GT( WRWG( G(L:W G= :DDLDWT DGW L5
(  ,L:((:1 :2J(LG DL L: LGDDL GG JG1 ++ =JT (JT =GT(

L: ("(LG5R LLDTDG(1 TDJT (:W G= GT( :(J (W( = GT( XR+5L=" 5LJGD=L
R *  /T( :(J (W( = GT( L5
( = :=DL W (:W G= GT(  =GT = GT( :=DLW' WD?(W1 TDJT DW J(: #02%% 

v x

x

1

x

2

v (x,t)

v (x,t)

2

1

−2  0 , ,LG( JGD=L = :DWGLG DLW

,L = :( G= (WGDG( GT( G( = J= W(LDL1 (G 5W JJ5G( GT( DLG( JGD=L
(G((L  DL DGT GT( J(LG( =JG(: G  -  1 L: L LGDDL DGT GT( J(LG( =JG(: G  -  N WW5(  /  1     < ,G DW J=L"(LD(LG G= :(;L( L( " D
(  W5JT GTG ' - < C  5GD=L  DW ( DGG(L W  -   *      

4

78

Alexander A. Nepomnyashchy

/T( DL W=5GD=L DW     <$  $ 4 - GLT  *

!

GT( LGDDL W=5GD=L DW     <  $ 4 - GLT  * X(G 5W J=LWG 5JG W=5GD=L G= 4 TDJT J=LWDWGW =
=GT DLW W(( DV E(J5W( (5GD=L 4 DW L=LDL( 1 ( JLL=G 25WG :: DL L: LGDDL W=5GD=LW >=("( 1 L( GT( J(LG( = GT( DL    1 GT( LGDDL W=5GD=L DW (+=L(LGDR W1   $ 4  * (+I    *$$  L: DG JL
( J=LWD:( (: W  ( :DWG5
LJ( JT JG( D?(:
R GT( W + (G(

 8 - (+I    *$ #L( JL (+(JG GTG GT( DLG( JGD=L = GT( DLW JL (: G= GT(D W= =GD=L1 DGT GT( "(=JDGR = = :( 81 L: G= W=( :DWG= GD=LW = GT( W( = :(  @=1 ( J=LWG 5JG GT( W=5GD=L DL GT( =  $ 4 -  $ 4 C  $ 4 C

 

  $ 4$

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T( (   - 8   L: ++4 - 8 @5
WGDG5GDL  K DLG= 41 G GT( (:DL = :( ( =
GDL GT( ==DL (5GD=L =    7 7  

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 <

/T( W=5GD=L   DW
=5L:(: D GT( DTGTL: WD:( = (5GD=L  < DW =  GT==L G= GT( W=5GD=L 7 7 = GT( T==(L(=5W W(:2=DLG + =
(N GT5W1 ( =
GDL 

     +  7 7 7  C    + -F + +  * +4  7 7 7   @5
WGDG5GDL (+ (WWD=LW !1  DLG=  *1 ( ;L:   * * +  -
 

Nonlinear Dynamics of Fronts

79

@DD R1 ( JL =
GDL   * * +  - 
 

/T5W1 GT( =GD=L = G= :=DL W DW (5D"(LG G= GT( =GD=L = G=  GG JGDL + GDJ(W DGT DL"( W( =
DDGD(W * * /T(D DLG( JGD=L (L( R DW       - ! * (+I *   $  V ,G JL
( WT=L GTG GTDW DW 25WG GT( XR+5L=" 5LJGD=L J=

(W+=L:DL GT( W=5GD=L  K DL5W GT( (L( R * = DW=G(: DL L: LGDDL ,L GT( (L(  JW( GT( (5GD=LW = =GD=L = GT( :=DL  J(LG( W  (  7 * * +  - $  F  +4 7 T( ( -



   $



      - ! * (+I *   $

 4

6=G( GTG -   (  1 T( ( ( DW GT( L5
( = :=DL W E(J5W( = GT( (+=L(LGD DLG( JGD=L 1 =LR GT( GG JGD=L G= GT( L( (WG L(DT
=5 DW D+= GLG = ("( R + GDJ(M X(G 5W J=LWD:( GT( DLG( JGD=L =  +D = :=DL W ="( L(:
R GT( WRWG( = (5GD=LW  1   @(GGDL  -    1 ( ;L: GTG GT( :DWGLJ(
(G((L :=DL W DW :(G( DL(:
R GT( (5GD=L   * * + - *   !  +4 , K -   <1 GT( W=5GD=L DW    <  -  C  L <  !4  * /T( :DWGLJ(
(G((L G= :=DL W
(J=(W < G 4  4 -

<     !

 

5GD=L  ! :=(W L=G :(WJ D
( GT( WG WG(W = DLG( JGD=L =L WT= G :DWGLJ(W = = :( < L: GT( LLDTDGD=L = :=DL W1
5G DG DW J(

GTG GT( :5 GD=L = GT(W( WG(W DW = <1 T(LJ( DG DW 5JT (WW GTL

80

Alexander A. Nepomnyashchy

4  5GD=L   D"(W L (WGDG( = GT( GD( L(J(WW R = LLDTDGD=L = :=DL W = DDLR W(+ G(:
R GT( :DWGLJ(   DJ( "( W1 =

LR 4 < =LR GT( :=DL W DGT GT( = DDL W(+ GD=L  (G( GTL L !4 * JL W5 "D"( ,L   (
5G ;LDG( WRWG( DGT GT( (LGT  1 = 5   D"(W GT( ("=5GD=L GD( L(J(WW R = (JTDL  ;L 5LD=  WG(:R WGG( DGT ' - < = ' - < /T5W1 G( WG(W = GT( WRWG( ("=5GD=L G= :W  WG
( WGGD=L R WGG( JL
( :(WJ D
(: W L (+=L(LGDR W= =GD=L = :=DL W L: GT(D LLDTDGD=L

,

*-0 "-" " " 00" 0

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( +TW(W  ( (L( (GDJR L=L(5D"(LG X(G 5W G(  WRWG( DGT GT( XR+5L=" 5LJ GD=L  
< <  I'$ ' C <  '   ,' + VK *  TDJT J=

(W+=L:W G= GT( :RLDJ (5GD=L ' - ' C '  ' C ,

V<

/T( WG G(  DL GT( (+ (WWD=L VK :(WJ D
(W GT( DL)5(LJ( = L (G(  L L(GDJ ;(:1 TDJT ( GT( = D(LGGD=L = GT( L(GD?GD=L DL GT( :D (JGD=L = GT( ;(: + (( 
( /T( 5LD=  WGGD=L R WGG(W WGDWR GT( (5GD=L '  ' C , - K V* /T(  5GDWG
DDGR G(W +J( DL GT( DLG( " , / , / , 1 T( ( , *  W(( D 
 L: D J ,LGTDW DLG( "1 (5GD=L VK TW GT (( W=5GD=LW WG
( W=5GD=L ' - '
 < 1 L=GT( WG
(  W=5GD=L ' - '  / < 1 L: L DLG( (:DG( 5LWG
( W=5GD=L ' 1 <  / ' / <  #
"D=5WR1 GT( +=G(LGD <  ' -  <  '  C ,' DW TDT( 1 L: GT( XR+5L=" 5LJGD=L DW =( = GT( W=5GD=L '
GTL =

GT( W=5GD=L ' 1 D ,  K /T( (= (1 ( JL (+(JG GTG GT(  =LG
(G((L WG
( WGG(W '
L: ' D ="( = ,  K DL W5JT  R GTG GT( WGG( '
D =5WG GT( WGG( '  ,L:((:1 J=LWD:( GT( G "(DL "( W=5GD=L ' - '11 1 -   641 TDJT WGDW;(W (5GD=L ' C 6' C + '+' - K V

Nonlinear Dynamics of Fronts

81

DGT GT(
=5L: R J=L:DGD=LW ' - '
$ ' - '  H5GD+RDL (5GD=L V
R ' L: DLG( GDL ="( 1  =  G= 1 ( ;L:   6 '  + C  '    '
$ 

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(WG
(  =LG DW +5WT(:
R GT( :D0( (LJ(  '
  ' 
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( =5L: LRGDJR X(G 5W J=LWG 5JG  DL = LGDDL W=5GD=L DL GT( =  ' - ' I    $ C ,' C    $ - ,4

VV

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WGDG5G( VV DLG= V< L: J=(JG G( W = GT( W( = :( DL , UG GT( (:DL = :( 1 ( ;L: '  C '  ' - K$ GT5W ' -  GLT

       *

UG GT( ; WG = :( 1 ( =
GDL GT( ==DL L=LT==(L(=5W (5GD=L '  C '  ' ' - <  ' 

+  +

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DDGR J=L:DGD=L = GT==LDGR = GT( DTGTL: WD:( G= GT( W=5 GD=L = GT( T==(L(=5W + =
(1 '   D"(W  * * + - *$  +

=

 * * + - *,  +4

GT( 5++( WDL DW =  DL L: GT( =( WDL DW = L LGDDL

82

Alexander A. Nepomnyashchy

/T5W1 = ,  K GT( +TW( DGT ' - '
 K DW + (( 
( >=("( 1 GT( +TW( DGT ' - ' / K DW (GWG
( L: D (DWG DL GT( 
W(LJ( = GT( +TW( ' - '
 H= (="( 1 =LR  W5JD(LGR  ( L5J(5W = GT( G 5R WG
( +TW( JL DLDGDG( GT( +TW( G LWDGD=L ,L:((:1 (G DW J=LWD:(

W5JT  L5J(5WN DGW
=5L: D(W  (  DL G  -  L: L LGDDL G  -   /T( :RLDJW = L5J(5W
=5L: D(W DW ="( L(:
R GT( WRWG( = (5GD=LW TDJT G( DLG= JJ=5LG
=GT +5WTDLM GT(  =LGW =5G :W
R GT( :D0( (LJ( = +=G(LGDW L: GT( GG JGD"( DLG( JGD=L
(G((L GT( DL L: GT( LGDDL   * * + - *, C
u

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(G((L G= WG
( WGG(W + =+G(W DL  (R DLT==(L(=5W (:D51 D( GT( ="( LDL (5GD=L

Nonlinear Dynamics of Fronts

83

DW ' - ' C  '$ $  '$  -  ' C > '$ $ K / >  < ,L GT( J=LG(G = GT( +TW( G LWDGD=L1 GT( DLT==(L(DGR JL
( J5W(:
R  W+GD L=L5LD= DGR = GT( G(+( G5 ( ;(: X(G 5W J=LWD:( GT( ==DL +( G5
(: U(L3TL (5GD=L ' - ' C I< C > $'  ' $ DGT GT( XR+5L=" 5LJGD=L    < C > $' ' < '  C C   - + * * ( W( JT GT( W=5GD=L DL GT( =  ' - '     C >' C    $ - >4 UG GT( ; WG = :( DL >1 '  C '  ' - K$ T(LJ( ' -  GLT

       *

UG GT( W(J=L: = :( DL > ( =
GDL '  C <  ' ' - ' 

+   '  +

/T( W="
DDGR J=L:DGD=L D"(W  * * + + -   $  +4 + T( ( GT( +=G(LGD = GT( DLG( JGD=L
(G((L GT( :=DL  L: GT( DLT==(L(DGR DW  <     -    C <' <+< VF *  @+(JD;JR1 D GT( DLT==(L(DGR TW  8D( WT+(1   - * 8   $

84

Alexander A. Nepomnyashchy

GT( DLG( JGD=L +=G(LGD DW    -  J=WT

     *

, GT( (  ( LR :=DL W L: LR DLT==(L(DGD(W1 GT( (5GD=LW = =GD=L = :=DL W  ( J  F  * * + 7 - $  +4 7       C   $ 



T( (     DW :(G( DL(:
R GT( (+ (WWD=L  41 L:    J= 

(W+=L:W G= VF UW L (+( (G 5W J=LWD:( G= :=DL W DGT J== :DLG(W  L:  TDJT  ( L( GT( :DWGLG 8WT+(: GG JGDL DLT==(L(DGD(W   - * 8     * 8   $   K$        -   <1     - <1     - < /T( (5GD=L = =GD=L = GT( (G :=DL  DW    * * +        -
DTG DLT==(L(DGR /T( ; WG  G(  DL GT( DTGTL: WD:( = (5GD=L JL
( (WGDG(: W
DTGTL: WD:( = GT( (5GD=L DW (5 G= * *   /T5W1 ( J=( G= GT( J=LJ5WD=L GTG D   * *    (LJ(1 GT( J= W(LDL D
( WG=++(: T(L GT( :DWGLJ(W
(G((L GT( L(DT
= :=DL W WGDWR GT( DL(5DGR V! @DD R1 DL GT( JW( = :=DL W +5WT(:
R GT( WR(G R = +TW(W W(( GT( + ("D=5W W5
W(JGD=L1 ( ;L: GT( J DG( D=L = +DLLDL       , *

Nonlinear Dynamics of Fronts



85

 (0 "-  *0
*-0 "-"

,L GT( + ("D=5W W(JGD=LW ( J=LWD:( (:  =LGW :(+(L:DL =L =LR =L( W+GD " D
( 6= ( D J=LWD:(  =LGW DL G= W+GD " D
(W ( WT :( =LR DGT WG
(WG
(  =LGW DL GTDW W5
W(JGD=L X(G 5W J=LWD:( GT( (JGD=L:D05WD=L (5GD=L ' - O
' C  '$ ,$ O
-

7 7 C $  '$ , - '  ' C , 7 7<

V

DL GT( DLG( " = , J=

(W+=L:DL G= GT(
DWG
DDGR X(G ' - '
, L: ' ' ,
( WG
( W=5GD=LW = GT( (5GD=L  '$ , - K # J=5 W(1 (5GD=L V TW W=5GD=LW = GT( GR+( ' - '  "$ 4 9 DW  5LDG G=:D(LWD=L "(JG= 1 TDJT J=

(W+=L: G= +L(  =LGW DGT GT( L=  "(JG= " /T(W( W=5GD=LW  ( D:(LGDJ G= GT=W( J=LWD:( (: DL GT( + ("D=5W W(JGD=L /T(R  ( =GD=L(WW W , - K L: ="( DGT GT( :(;LDG( "(=JDGR 6, W , - K

Y

−

2

1/2

z +

u=0

L>>1 X  0  35 "(:  =LG X(G 5W J=LWD:( L=  J5 "(:  =LG W(+ GDL G= (D=LW DGT '  '
L: '  ' W(( D4 ( WT WW5( GTG GT( JT JG( DWGDJ WD?( = (JT

(D=L L: GT( J5 "G5 ( :D5W = GT(  =LG  (  ( W J=+ (: G= GT( D:GT = GT(  =LG ,LG =:5J( L( W=M " D
(W 5 - >1 * - >< ,L GT(  (=  = GT(W( " D
(W1 GT(  =LG1 TDJT TW GT( GTDJL(WW < DL GT( = DDL " D
(W1 DW  GTDL WG D+( = GT( GTDJL(WW >  =5L:  J( GDL DL(

86

Alexander A. Nepomnyashchy

* - @5 UJ =WW GTDW GTDL WG D+(1 GT( 5LJGD=L ' JTL(W +D:R /T5W1 ( T"( L %2% '2 TDJT ( D :(WJ D
( 5WDL GT( WGM " D
( 1 - *  @5> = W( = WD+DJDGR1 ( D WW5( GTG GT( + (G( , DW W= > , - , > ,G DW (W=L
( G= (+(JG GTG GT( "(=JDGR = GT(  =LG D
( > DL = DDL " D
(W L: T(LJ( >  DL L( " D
(W /T( (= (1 ( DLG =:5J(  W=M GD( " D
( # - > 4 DLR1 ( W( JT W=5GD=L = V DL GT( =  ' - '1$ 5$ *$ # N 5 - >N * - > 4N 1 -

*  @5$ #   >

/T( G LW= GD=L = GT( + GD :( D"GD"(W DL V DW W ==W 7 7 7 7 7 7 7 7 7 - >@ C > N - @ C> N C>  74 71 7# 7 71 75 7< 71 7* 5GD=L V DW ( DGG(L W  

7 7 7 7 C> C> @ 'C >@ ' C > ' - @ 71 75 71 75

7 7 C> 71 7*

 ' C '  ' C , >

/T( W=5GD=L DW J=LWG 5JG(: DL GT( =  ' - ' C >' C    $ @ - @ C    UG GT( (:DL = :( 1 ( =
GDL I< C @  $'  C '  ' - K /T(  =LG W=5GD=L DW 1 ' -  GLT  *I< C @  $

FK

( WW5( GTG GT( +TW( '  '
- < C > DW =JG(: 
="( GT(  =LG1 L: GT( +TW( '  ' - < C > DW =JG(:
(= GT(  =LG UG GT( L(G = :( 1 ( ;L: I< C @  $'  C <  ' ' -

Nonlinear Dynamics of Fronts

87

@  '  C @  '  C *@  '   *@  @  '   ,  /T( J=L:DGD=L = GT( W="
DDGR =L GT( JWW =
=5L:(: 5LJGD=LW DW GT( = GT==LDGR = GT( DTGTL: WD:( G= GT( W=5GD=L = GT( T==(L(=5W (5GD=L     7 I@  C @  $ '  +1 C @  '  +1 C *, - K 75   @5
WGDG5GDL FK L: ("5GDL GT( DLG( W1 ( =
GDL GT( ==DL (5 GD=L GTG :(G( DL(W GT( =GD=L = GT( J5 "(: WG
(WG
(  =LG GT( W5
 WJ D+G K DL @ DW =DGG(:  @  * @  ,  C * < C @  < C @ /T( G(  DL GT( (GTL: WD:( DW GT( L=  "(=JDGR = GT(  =LG  1 TD( GT( ; WG G(  DL GT( DTGTL: WD:( DW GT( J5 "G5 ( & = GT(  =LG /T5W1 GT(  = GT(  =LG =GD=L JL
(  DGG(L DL GT( ==DL DL" DLG =  DL:(+(L:(LG =L GT( JT=DJ( = GT( WRWG( = J== :DLG( L: GT( WT+( = GT(  =LG   *  - & C ,  F< * @+(JD;JR1 DL GT( JW( , - K ( (G 25WG GT( (GD=L  - &1 TDJT DW JT JG( DWGDJ =  # , 2 3  /T( +TRWDJ = DDL = GT( J5 "G5 ( J=L G D
5GD=L DLG= GT( L=  "(=JDGR = GT( J5 "(: :=DL  DW GT( +=WDGD"( J=LG D
5GD=L = :=DL  DLG= GT( XR+5L=" 5LJGD=L W(( V GT( WRWG( G(L:W G= :DDLDWT GT( (LGT = GT( :=DL  DL = :( G= :DDLDWT GT( "5( = GT( XR+5L=" 5LJGD=L UW L (+(1 (G 5W J=LWD:(  =5L: : =+(G = GT( +TW( '
DL GT( DL;LDG( W( = GT( +TW( '  UJJ= :DL G= F<1 GT( ("=5GD=L = GT( : =+(G

:D5W =4 DW ="( L(:
R GT( (5GD=L  <  * += -  C $  , +4 = * DGT  J( GDL DLDGD J=L:DGD=L =K - =  ,L GT( JW(  - K 
=GT +TW(W  ( (5D"(LG1 GT( W=5GD=L DW  = - =  *4 /T( : =+(G J=+W(W :5 DL GT( ;LDG( GD( 4  = * GT( ;L WG( = GT( ("=5GD=L T(L GT( :D5W = GT( : =+(G
(J=(W = GT( = :( = GT( :=DL  D:GT DW L=G :(WJ D
(:
R GT( WR+G=GDJ GT(= R :(WJ D
(: 
="(

88

Alexander A. Nepomnyashchy

,L GT( JW(  / K GT( =5G( +TW( DW G 5R WG
(1 GT( W=5GD=L JL
( + (W(LG(: DL L D+DJDG =  W 
= - 
=  C 41 T( ( 
= - = I" C L< C "$$ " - == $ = - < /T( 5LJGD=L 
= DW L(GD"( L: =L=G=LDJR :(J (WDL =  = >(LJ(1 DGT GT(  =GT = 4 GT( :D5W =4 :(J (W(W =L=G=LDJR L:

(JT(W ?( = G 4 - 4 - =  ,L GT( JW(   K GT( DLL( +TW( DW G 5R WG
(1 GT( W=5GD=L JL
( + (W(LG(: DL L D+DJDG =  W  = -  =  C 41 T( (  = - = I" C L <  "$$ " - == $ = - < /T( 5LJGD=L  = DW :(J (WDL = " / < L: DLJ (WDL = "  < >(LJ(1 D GT( DLDGD :D5W = GT( : =+(G = / = 1 =4 :(J (W(W L: (JT(W ?( = G 4 - 4 - =  E5G D GT( DLDGD :D5W = GT( : =+(G =  = 1 =4 DLJ (W(W L: G(L:W G=  W 4  /T( ( DW W=  W+(JD WGGD=L R W=5GD=L =4 - = TDJT :(WJ D
(W GT( J DGDJ L5J(5W = GT( G 5R WG
( +TW( DL GT( (GWG
( W(

# ,

 $  $  "  *0
0

,L GT( + ("D=5W W(JGD=L ( J=LWD:( (:  =LGW :(WJ D
(:
R =L( " D
( 6= ( D J=LWD:( L (+( T(L GT( + =
( DW :(WJ D
(:
R G= " D
(W1 G(+( G5 ( # L: 5( J=LJ(LG GD=L % 7% - O%   # %$ 74

F*

7# - O# C  # % F 74 J (5GD=L ! L: 
( W+=L:DL (+ (WWD=L D
( D"(L
(= (5GD=L FV /T( (JGD=L G( DW WW5(: G=
(  #  - A (+=#  F 

Nonlinear Dynamics of Fronts

89

/.2% 0  1 T( ( = DW GT( 5LD"( W J=LWGLG1 L:  DW GT( JGD"GD=L (L( R = R1 GT(  F  DW DLJ=+GD
( DGT GT(
=5L: R J=L:DGD=L #   - # 1
(J5W(  #  - K >=("( 1 D =#  1 GT(L  #  DW (+=L(LGDR W L: DGW L=L?( = "5( JL
( :DW ( :(: ,L GT( LRWDW + (W(LG(:
(= ( == GT( = W
R @D"WTDLWR1 HG=WR L: H =DW W((1 (1 @D"WTDLWR <441 HG=WR L: @D"WTDLWR <41 H =DW L: HG=WR <!V UW  ; WG WG(+1 ( WT J=LWD:( GT( =L(:D(LWD=L +L(  =LG W=5GD=L # - # 1$ % - %1$ 1 -  C 4 ="DL G= GT( (G   K ( ;L: % - %   # %N # - # C  # % ,LG( GDL
=GT (5GD=LW  = 1 -  G= 1 -  L: 5WDL GT(
=5L: R J=L:DGD=LW1 ( ;L:     % -   # %+$ #
 #  -   # %+ 



/T5W1 #
- # C % 

FV

X(G 5W G LW=  GT( =L(:D(LWD=L "( WD=L = (5GD=LW F*1 F G=  L=L:D(LWD=L =  # - # C B#
<  N  - # #
/
7 7  C <  B

FF


7B 7B 7B B  < C  C 8* (+ 7

7 7  C <  B

F4

>( ( C-

<     X(DW L5
( $   &(:="DJT L5
( $  =#


   8 - A (+   =#


90

Alexander A. Nepomnyashchy

/T(
=5L: R J=L:DGD=LW *  - <$ B - KN *  - K$ B - <

F!

XG( =L1 ( WT WW5( GTG   <1 TDJT DW DL:((:  GR+DJ WDG5GD=L UW ( (LGD=L(: 
="(1 GT(
=5L: R J=L:DGD=L B - K JLL=G
( JG5R WGDW;(:1
(J5W( JJ= :DL G= GT( (5GD=L

  7B   (+  - K 7

 >=("( 1 GT( "D=GD=L = GTDW
=5L: R J=L:DGD=L DW (+=L(LGDR W =

  <1 L: ( WT :DW ( : DG ,

"0 -"# "

/T( +L( WGGD=L R  =LG DW :(WJ D
(:
R GT( + =
( FF  F! DGT GT( GD( :( D"GD"( 7* 7 L: 7B7 =DGG(:

T

C

x reaction zone  0 ' /T( WG 5JG5 ( = GT( J=
5WGD=L  =LG =  ( 1 GT( (JGD=L G(  =W "( R WG DGT GT(  =GT = G( +( G5 ( E(J5W( = GTG1 GT( (JGD=L G(W +J( DL  GTDL R( J(: # 0% % T( ( GT( G(+( G5 ( DW  (:R J=W( G= GT( D5 =L(1 B  <1 TD( GT( J=LJ(LG GD=L = GT( 5( % DW WGD :D0( (LG  = ?( = W(( D! /T( GTDJL(WW = GT( J=
5WGD=L  =LG DW >1 T( ( >     E(= ( D :("(=+ GT( WR+G=GDJ GT(= R = GT( J=
5WGD=L  =LG + =+ GD=L ( D ;L: W=5GD=LW =5GWD:( GT(  =LG  2 0 %01 T( (

Nonlinear Dynamics of Fronts

91

GT( ( DW L= (JGD=L1 L: GT( W=5GD=L DLWD:( GT(  =LG %%2 0 %01 T( ( GT( (JGD=L G(W +J( HGJTDL = GT(W( W=5GD=LW D D"( 5W GT( "5( = 8 TDJT JL
( J(: (D(L"5(M = GT( L=LDL( + =
(1 D( GT(  =LG "(=JDGR ,L GT( =5G( (D=LW GT( (JGD=L G(  JL
( L((JG(: DL GT( (D=L  / K1
(J5W( GT( G(+( G5 ( DW G== =1 L: DL GT( (D=L   K1
(J5W( GT( 5( J=LJ(LG GD=L DW G== = /T( W=5GD=LW JL
( + (W(LG(: DL GT( =  =  W( D(W DL >1 * - * C >* C    $ B - B C >B C    UJG5R1 (  ( DLG( (WG(: =LR DL GT( (:DL= :( G( W ="( L(:
R GT( (5GD=LW + B + * +B +* - C $  F + + + + ,L GT( (D=L  / K1 GT( W=5GD=L TW G= WGDWR GT(
=5L: R J=L:DGD=LW *  - <$ B  - K /T( W=5GD=L DW *  - < C A ! $ B  - 3  $

4K

T( ( A L: 3  ( J=LWGLGW ,L GT( (D=L   K1 (5GD=LW F  ( W="(: DGT GT(
=5L: R J=L:DGD=LW *  - K$ B  - < /T( W=5GD=L DW 25WG *  - K$ B  - <

4<

/T( DLG(LWD"( 5( J=LW5+GD=L L: T(G ((W( (: G= 25+W = J=LJ(LG  GD=L L: T(G 3 20 J =WW GT(  =LG1 D( :DWJ=LGDL5DGR = GT( W+GD :( D" GD"(W = GT( J=LJ(LG GD=L L: G(+( G5 ( ;(:W >=("( 1 GT( :D05WD=L L: T(G :D05WD=L (5GD=LW := L=G = 25+W = GT( J=LJ(LG GD=L L: G(+( G5 ( GT(W("(W >(LJ(1 W=5GD=LW 4K L: 4< WT=5:
( GJT(: DL GT( +=DLG  - K * K  - * K
 - K$ B K  - B K
 - <

4*

/TG D"(W GT( ==DL (+ (WWD=LW = GT( =5G( W=5GD=L DL GT( (D=L  / K *  - <  ! $ B  -   4

92

Alexander A. Nepomnyashchy

6=G( GTG  = GT( +=DLG = "D( = GT( =5G( (+LWD=L1 GT( (JGD=L G(W +J( DL GT( DL;LDG(R GTDL R( 1 D( GT( =5G( W=5GD=L JL
( J=LWD: ( (: W GT( W=5GD=L = GT( (5GD=LW WRWG( + B +* +B + * - C  - 8$ C - 8$ + + + +

4 

T( ( - DW GT( DLG(  = GT( (JGD=L G(  J =WW GT(  =LG TDJT :(G(  DL(W GT( 25+W = GT( )5(W 

+B  +* - - C - $ + + T( (

I $   K
   K 

4V

@=5GD=L 4 DW =
GDL(: DGT - - < X(G 5W J=LWD:( L= GT( ;L( WG 5JG5 ( = GT( J=
5WGD=L  =LG DL GT(

(D=L  - > = GTDW =1 (G 5W DLG =:5J( GT( WGM " D
( 5 - > 1 L: WW5( GTG GT( W=5GD=L DW GT( 5LJGD=L = GT( " D
( 5 * - < 5 C >< 5 C > < 5 C    $ B - . 5 C >. 5 C > . 5 C    E(J5W( = GT( (GD=LW 4*1 < 5 - K1 . 5 - < /T( (+=L(LGD (+ (WWD=L DL GT( (JGD=L G( DW 25WG (+.  G GT( (:DL = :(  /T(
LJ(
(G((L GT( (JGD=L G(  L: GT( :D05WD=L G(  DW (JT(: D 8 > N GT( (= (1 ( D 5W( GT( (+LWD=L 8 - > 8 C > 8 C    DLR1 ( (G GT( ==DL WRWG( = (5GD=LW = < 51 . 5 C

+ <  8 < (+.  - K$ +5

4F

+ . C 8 < (+.  - K 44 +5 /T(
=5L: R J=L:DGD=LW = 5LJGD=LW < 51 . 5  ( =
GDL  = GT( GJTDL DGT GT( W=5GD=L 41 TDJT JL
(  DGG(L = W  - >5 W * - >C5 C    $ B - < C >5 C    $ L: DGT GT( W=5GD=L 4< ( ;L: 5   < 5  C5$ . 5  5N 5   < 5 K$ . 5 K

Nonlinear Dynamics of Fronts

93

U::DL (5GD=LW 4F L: 441 ( ;L: + . C C <  - K +5 /DL DLG= JJ=5LG GT(
=5L: R J=L:DGD=LW ( ;L: < - C. 

4!

@5
WGDG5GDL 4! DLG= 441 ( =
GDL GT( ==DL L=LDL( (D(L"5( + =
( + .  C8 . (+.  - K$ 4 +5 5   . 5  5N 5   . 5 K /T( ; WG DLG(  = GT( (5GD=L 4 DW <  .   8 C.  < (+.  - %$ * 

!K

T( ( % DW  J=LWGLG TDJT DW =
GDL(:  = GT(
=5L: R J=L:DGD=L G 5  % - 8 C !< "5GDL
=GT WD:(W = !K G 5 1 ( ;L: GTG 8 -

<  *C

Q(G5 LDL G= GT( = DDL " D
(W1 ( ;L: GT( ==DL = 5 = GT( J=
5WGD=L  =LG "(=JDGR 

 =#
 *A (+    #
 #  *=#
,

&#"!-  ! (0 "-

X(G 5W (L( D?( L= GT( GT(= R :(WJ D
(: DL GT( + ("D=5W W5
W(JGD=LW X(G 5( J=LJ(LG GD=L L: G(+( G5 (
( 5LJGD=LW = GT (( W+GD " D 
(W1 % $  $  $ 4 L: #  $  $  $ 4 UWW5( GTG GT( J=
5WGD=L  =LG DW  G=:D(LWD=L W5 J(1  - . $  $ 4 (;L( GT( L( WRWG( = " D
(W1  -   . $  $ 4$ < -  $ 1 -  $ - 4 WDL GT( W( G LW= GD=L G= L=L:D(LWD=L " D
(W1 W DL GT( + ("D=5W W5
W(JGD=LW D(1 5WDL GT( "(=JDGR  = GT( +L(  =LG = L=L:D(LWD=LD?GD=L1 ( =
GDL GT( ==DL WRWG( = (5GD=LW 7* 7B 7* 7B  . - C O*  $  . - OB C $ 7

7 7

7

94

Alexander A. Nepomnyashchy
 - 8* (+ >



 B<   C <  B

/T( (+ (WWD=L = GT( X+JDL DL GT( J== :DLG(W $ <$ 1 DW

 7  7  7  7 7 7 C .  O -  1 T(LJ(1 C - <  > C    ( J=LWG 5JG GT( =5G( W=5GD=L DL GT( =  * - * $ <$ 1$ C>* $ <$ 1$ C   N B - B $ <$ 1$ C>B $ <$ 1$ C    6=G( GTG JJ= :DL G= =5 WW5+GD=L = GT( 5 J=LW5+GD=L = GT( 5( G GT( (JGD=L  =LG1 GT( =5G( W=5GD=L = * DW ?( = DL  GT( = :( DL GT(

(D=L   K /T( DLL( W=5GD=L DW + (W(LG(: DL GT( =  * - < 5$ <$ 1$ C>< 5$ <$ 1$ C   N B - . 5$ <$ 1$ C>. 5$ <$ 1$ C   $ T( ( 5 - >  UW=1 ( 5W( GT( (+LWD=LW . - . C >. C    $ - - - C >- C     = * L: B 1 ( =
GDL 7* 7*  .  - O*  - 8$ 7

7 7B 7B  .  - OB C - 8N 7

7  -   * - <$ B - KN  -   * - K$ B - <

!* !

X(G 5W :: (5GD=LW !* L: ! ( ;L: GTG GT( 5LGDGR 0 - * C B WGDWR GT( (5GD=L 70 70  .  - O0 7

7 DGT
=5L: R J=L:DGD=LW  -   0 -
Nonlinear Dynamics of Fronts

95

X(G 5W J=LWD:( L= GT( DLL( W=5GD=L E(J5W( = GT( J=LGDL5DGR = GT( ;(:W = * 1 B 1 * - K L: B - < W  - K /T( (= (1 GT( DLL( (+LWD=L JL
(  DGG(L W * - >< 5$ <$ 1$  C    N B - < C >. $ <$ 1$  C     E(J5W( ( 5W(: GT( "(=JDGR = GT( +L(  =LG = L=L:D(LWD=LD?GD=L1 GT( (+LWD=L 8 - 8 > C    J=DLJD:(W DGT GTG 5W(: DL GT( + ("D=5W W(JGD=L /T5W1 DL =5 JW( 8 - <* ( =
GDL GT( ==DL WRWG( = (5GD=LW DL GT( (:DL = :(  I
7 < < 7 . <  < (+.  - K$ I
U::DL
=GT (5GD=LW1 ( ;L: GTG = GT( 5LGDGR B - < C . 1 7 B - K 75  = GT( GJTDL DGT GT( =5G( W=5GD=L =   K ( ;L: GTG B . 1 < - .   . 1 T(LJ( < I< C .  C .  $.  C I.   . $ (+.  - K *

! 

T( ( L:
(=    (LW   $ <$ 1$  H5GD+RDL
=GT WD:(W = (5GD=L ! 
R .  L: DLG( GDL  =  G= 1 ( ;L:  (+  .  .   -   < C .  C .  E(J5W( .   - K1 ( ;L: GTG GT( JTL( = GT( :( D"GD"( .  J =WW GT( (JGD=L ?=L(1  (+  .   !V .    .   -   < C .  C .  /T( GJTDL J=L:DGD=LW .  - B K
$

7B
7. 7B 7.  K $  K  75 7 75 7

= 5W G= (+ (W(LG GT( (GD=L !V W GT( 25+ = GT( :( D"GD"( = GT( =5G( W=5GD=L (+B K
* IB  $ -   < C .  C . 

96

Alexander A. Nepomnyashchy

L=GGD=L 4V DW 5W(: 6= ( JL :(G( DL( GT( 5LJGD=L - DL !*1 ! X(G 5W DLG( G( ! ="(   =  - 8 G=  - 8 L: G( GT( DDG 8 K ( ;L: < C .  C .  IB
 $ C - - K /T5W1 - -

 < C .  C .  (+B K
*$

W= GTG (5GD=L ! JL
( ( DGG(L W  7B 7B  .  - OB C < C .  C .  (+B K
*8 !F 7

7 ( J=( G= GT( J=LJ5WD=L GTG GT( (:DL = :( (5GD=L := L=G = G= (G  J=W(: = 5GD=L = GT( =5G( + =
( (5GD=L = B J=LGDLW B K
 /T5W1 ( T"( G= J=LWD:( GT( L(G = :( = GT( =5G( + =
( ,L GT( L(G = :( 1 ( =
GDL (5GD=LW 7* 7* 7*  .   .  - O*  O*   $ 7

7 7 7B 7B 7B  .   .  - OB C  $ 7

7 7 T( (  DW GT( L(G = :( (JGD=L G(  U::DL
=GT (5GD=LW L: GDL DLG= JJ=5LG GTG 0 - * C B - <1 ( ;L: GT( ==DL (5GD=L = 0  70 70  .  - O0 C OB  7

7

!4

,L GT( (D=L   K1 * - K1 T(LJ( B K
 - 0 K
 /T5W1 (5GD=L !F JL
(  DGG(L W  7B 7B  .  - OB C < C .  C .  (+0 K
*8 !! 7

7 
(J5W( 0 DW  J=LGDL5=5W 5LJGD=L1 ( JL  DG( 25WG 0 K DLWG(: = 0 K
 5GD=LW !4 L: !! DGT
=5L: R J=L:DGD=LW    B - K$ 0 - KN    B - <$ 0 - K D"(  J=W(: :(WJ D+GD=L = GT( + =
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Nonlinear Dynamics of Fronts

97

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99

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100

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Nonlinear Dynamics of Fronts

101

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Nonlinear Dynamics of Fronts

103


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186

Len M. Pismen

MA> LBFIE>LM <:L> H? : I:BK H? =KHIE>ML L>I:K:M>= :M : =BLM:G<> 9 FNK MA:G MA>BK K:=BB %  HG> <:G LHEO> MA> +:IE:<> >JN:MBHG NLBG@ MA> ;BIHE:K LRLM>F PBMA MA> ?H =KHIE>M <>GM>KL 2A> L   K>LI><MBO>ER = :K> K>E:M>= MH MA> ":KM>LB:G L = > :L =

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0

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187

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20

30

40

50

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188

Len M. Pismen

<E:LL 6   BG : @>G>K:EBS>= MA>HKR &BKHG    

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3GEBD> MA> LM:G=:K= GBG@ IKH;E>F MA> =KHIE>ML =H GHM HGER @KHP HK =><:R ;NM :ELH FHO> NG=>K MA> BGN>G<> H? MA> @K:=B>GM H? =BLCHBGBG@ IK>LLNK> BG MA> IK> IKBG =KBOBG@ ?HK<> L ?KHF MA> L>@K:E BG $J   PA>K> MA> +:IE: G>@E><M>= BG OB>P H? :MG>LL H? MA> IK> =KBOBG@ ?HK<> LM>FL ?KHF : LF:EE =BIHE> GM H?  0 <:NL>= ;R F:LL NQ>L ?KHF ?:K K>FHO>= =KHIE>ML PAB @K:=B>GM  0 2A> @K:=B >GM  0 :M MA> EH<:MBHG H? : M>LM =KHIE>M <:G ;> >QIK>LL>= BG MA> >><MBO> F>=BNF :IIKHQBF:MBHG MAKHN@A LNI>KIHLBMBHG H? MA> NQ>L ?KHF MA> K>LM H? MA> =KHIE>ML      0     0 0    9 



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Thin Film and Droplet Patterns Shaped by Surface Forces

189

'%30#  2RIB<:E MK:C><MHKB>L H? FB@K:MBG@ =KHIE>ML 1F:EE L F:KD MA> BGBMB:E IHLBMBHGL :G= IHLBMBHGL :M MA> L<>G<> >O>GML 1BS> L =N> MH F:LL >Q :K> G>@E><M>= (GBMB:EER >B@AM =KHIE>ML PBMA =BF>G LBHGE>LL K:=BB K:G=HFER =BLMKB;NM>= BG MA> BGM>KO:E   +   P>K> IE:<>= LRFF>MKB<:EER HG :  2A> =KHIE>ML IKH<>>= MH MA> KB@AM MH =BLM:GM G:E L<>G<> >O>GML

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190

Len M. Pismen

'%30#  2K:C><MHKB>L H? =KHIE>ML BGM>K:<MBG@ MAKHN@A MA> IK>Q :G= LM:KMBG@ ?KHF : K:G=HF =BL MKB;NMBHG PBMA : F>:G L>I:K:MBHG 2A> K:MBH H? KBLMB< FB@K:MBHG :G= L<>G<> MBF> BL NGBMR "BK<E>L F:KD MA> BGBMB:E IHLBMBHGL :G= MA> L<>G<> >O>GML =HML F:KD >QMBG<MBHG >O>GML

L= ;R : EBG>:K DBG>MB< >JN:MBHG =-=;  3 - 3- PA>K> - BL MA> ;NED GMK:MBHG :G= 3  3 :K> DBG>MB< E:MM>K BL :LLNF>= ?HK LBFIEB MA> L:F> BG MA> =KHIE>M :G= MA>  2:DBG@ BGMH :< =BNLBHG :G= :LLNFBG@ MA> ;NED GMK:MBHG BG MA> =KHIE>M MH ;> G>@EB@B;E> P> PKBM> MA> :=LHKIMBHG=BNLBHG >JN:MBHG ?HK MA> LNK?:<M:GM K:@> HG MA> LN;LMK:M> BG MA> =BF>GLBHGE>LL ?HKF -   - -    



PA>K>   >JN:EL MH  HNMLB=> :G=  BGLB=> MA> =KHIE>M ?HHMIKBGM 2A> LNK?:<M:GM K:@> - BL L<:E>= ;R MA> K:@> BG >JNBEB;KBNF PBMA MA> LNK ?:<M:GM GMK:MBHG BG MA>  3 - 3 MBF> ;R MA> BGO>KL> =>LHKIMBHG K:M> G@MA ;R 3 PA>K>  BL MA> LNK?:<M:GM =BNLBOBMR HG MA> LN;LMK:M> M E:M>K LM:@>L H? MA> IKH<>LL PA>G LNK?:<M:GM :<L BG MA> =KHIE>M ;NED HG> <:G NL> MA> L:F> >JN:MBHG M:DBG@ BGMH

Thin Film and Droplet Patterns Shaped by Surface Forces

191

:< =B>K>G<> H? :=LHKIMBHG K:M>L HNMLB=> :G= BGLB=> MA> =KHIE>M BL BFIHKM:GM :G= K>IE: =B>K>G<> H? MA> K>LI><MBO> O:EN>L  2A> KBLMB< O>EH BL '
 3 (? $J  BL ;KHN@AM MH : =BF>GLBHGE>LL ?HKF ;R K>IE: :=O><MBO> M>KF BL K><:LM  :L ": <
0 PA>K> ":  3 BL MA> <:IBEE:KR GNF;>K ;:L>= HG MA> KBLMB< 8FB<:E O>EH K:MBH H? MA> 8FB<:E O>EH KBLMB< 8AR=KH=RG:FB< O>EHM>KFBG>L MA> BG N>G<> H? OBLLL>L HG MA> =KHIE>M LA:I> LNLM:BG>= ;R LNK?:<> M>GLBHG 2ABL BFIEB>L MA:M MA> <:IBEE:KR GNF;>K BL MA> MKN> LF:EE I:K:F>M>K H? MA> IKH;E>F =>OB:MBHGL H? MA> LAHNE= ;> H? " ": MH CNLMB?R MA> I>KMNK;:MBHG :IIKH:ER MH AHE= NG=>K K>:EBLMB< >Q I>KBF>GM:E LNK?:<> =BNLBHG BL LEHP PABL MA:M MA> LA:I> H? FB<:EER IKHI>EE>= =KHIE>ML BL HGER P>:DER =BLMHKM>= ;R OBLLL>L $JN:MBHG  MK:GL?HKF>= MH MA> ?K:F> FHOBG@ PBMA : =BF>GLBHGE>LL  O>EH= ;R 3 :EHG@ MA> = :QBL BL <-   - -     



2A> LHENMBHG H? MABL >JN:MBHG <:G ;> >QIK>LL>= PBMA MA> A>EI H? :G :IIKH IKB:M> &K>>GL ?NG<MBHG MAKHN@A :G BGM>@K:E HO>K MA> =KHIE>M ?HHMIKBGM :K>: :G= LN;L>JN>GMER MK:GL?HKF>= BGMH : @K:E PBMA MA> A>EI H? MA> &:NLL MA>HK>F /BLF>G    %HK : GLBHGE>LL K:=BNL 2AB>E> FH=NENL +  % 3 MA> GMK:MBHG HG BML ;HNG=:KR PABM>KFBG>L MA> IKHI:@:MBHG LI>>= BL = BG MABL P:R :L  

   
        + >
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- MA> IKHI:@:MBHG O>EH= ;R LHEOBG@ MA> >JN:MBHG H;M:BG>= ;R K>  -  - :G= :EE K>E>O:GM I:K:F>M>KL >Q<>IM MA> =BF>GLBHGE>LL =KHIE>M K:=BNL + BG $J   :K> ENFI>= BGMH : LBG@E> =BF>GLBHGE>LL
192

Len M. Pismen 1
M 0.2 0.15 4 0.1

2

0.05

1 1

2

3

4

5

v

'%30#  /EHML H?  + < =>G>= ;R $J  :L : ?NG<MBHG H? < MA> O:EN>L H? + :K> F:KD>= :M MA> K>LI><MBO> L

PAB <:EE>= MA>    I:K:F>M>K !

 
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2A> I:K:F>M>K ! K>M:BGL : P>:D EH@:KBMAFB< =>I>G=>G<> HG MA> =KHIE>M K:=BNL $JN:MBHG  <:G ;> K>PKBMM>G :L !    + < :G= LHEO>= :?M>K KB<:EER MA> =HN;E> BGM>@K:E      +    
   + <    <       <  < ?NG<MBHG  + < BL IEHMM>= :@:BGLM < :M L>O>K:E O:EN>L H? + BG %B@  2A> L I>:D :M MA> HK=BG:M> :QBL :M MA> ;B?NK<:MBHG O:EN> ! MH ;> = BG MA> G>QM LN;L><MBHG

Thin Film and Droplet Patterns Shaped by Surface Forces 

193

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    = +    +   + LBG  =  LBG   + =+ 

     =  =  LBG  + LBG  +     =+  LM:KMBG@ ?KHF  +   +  +  /EN@@BG@ MA> K>LNEMBG@ >QIK>LLBHGL BG $J  RB>E=L MA> HGL>M H? FHMBHG ! 

+ 7 +  +  +  + 9  



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194

Len M. Pismen 1
M0 0.25 0.2

1
M0 0.005

0.15

0.001 0.0005

0.1

0.0001 0.00005

0.05

0.00001 0.02 2

4

0.05 0.1 0.2 6

a

8

a

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MA> K:=BNL H? FH;BE> =KHIE>ML BL ;HNG=>= ;HMA ?KHF ;>EHP :G= :;HO> :L BM BG=>>= ?HEEHPL ?KHF MA> >QBLM>G<> H? : F:QBFNF BG MA> K:=B:E =>I>G=>G<> H? MA> H? MA> I:K:F>M>K   LN>G BG %B@  !>RHG= MA> O>EH H;M:BG>= NLBG@ ?NKMA>K M>KFL BG MA> >QI:GLBHG   2A> =BIHE> GM H? MA> M>KF JN:=K:MB< BG < O:GBLA>L :G= MA>K>?HK> MA>K> BL GH K 2A> =BIHE> GM H? MA> MABK=HK=>K M>KF BL  <  PA>K>  

 +   +  +  +  + +  +  +  



2ABL GM BL G>@:MBO> A>G<> MA> ;B?NK<:MBHG BL LNI>K  :M !  !  %HK LF:EE =>OB:MBHGL !  ! !   MA> O>EH
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Thin Film and Droplet Patterns Shaped by Surface Forces

195

0.022

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4

6

8

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196

Len M. Pismen Ca 0.006 0.0055 0.005 0.0045

0.01 0.02 0.03 0.04 0.05 0.06 0.07




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2ANL MA> O>EH:L>L PBMA =KHIE>M LBS> BG MA> NGL:MNK:M>= :G= => :L>L BG MA> L:MNK:M>= K>@BF> BG :@K>>F>GM PBMA >QI>KBF>GM +>>     :G= GNF>KB<:E E>      %HK BGM>KF>=B:M> O:EN>L H?   MA> =>I>G=>G<> H? O>EH= ;R LHEOBG@ $J  GNF>KB<:EER BL IEHMM>= BG %B@ 

Thin Film and Droplet Patterns Shaped by Surface Forces 

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222

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Interfacial Phenomena in Materials Science

223

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Interfacial Phenomena in Materials Science

225

polar plot of surface energy N

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226

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Interfacial Phenomena in Materials Science

227

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Alexander A. Golovin

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Interfacial Phenomena in Materials Science

229

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Interfacial Phenomena in Materials Science

231

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Interfacial Phenomena in Materials Science

233

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Interfacial Phenomena in Materials Science

heat flows away here faster freezing

heat accummulates here slower freezing

235

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236

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Interfacial Phenomena in Materials Science

237

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238

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Interfacial Phenomena in Materials Science

239

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240

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Interfacial Phenomena in Materials Science

241

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8" . J4  4 O= JW+ ==GJGBO4 Z8 M+ . J4  0 -)<3 /O JW+ +O+  MZ+ B  M8 %+= Z8 M+4 8ZGO ZGG  8+OJZ4 BO+ B
JGOZ JWJ JW+ Z8 M+ MW+GM .BJ+OJG  0 -)<

$$

242

Alexander A. Golovin

W+ +  GZ JW+ +O M8 %J8 + $$ WZ O G.B JOJ MBOZ+8+OM+ OU M8 %+= Z8 M+4 WGMW WZ  MBOZJOJ GZBJ B.GM Z8 M+ +O+ U4 G +
U Z8 M+ =G38ZGBO JB +GJW+  Z.W+ + B  .O+4 =+.+O=GO BO JW+ GOGJG MBO=GJGBOZ $ B JW+ Z8 M+ +%B8JGBO JW+O
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WGMW GZ ZGG JB JW+ =G38ZGBO +8JGBO4
8J B JW+ WGW+ Z.JG B =+  9B4 8ZGO +@4 +J 8Z MBOZG=+ JW+ BBGO GOJ+ +ZJGO . B
+ JW+ +%B8JGBO B  #&#% *1 6BOZG=+  Z8 M+ B  Z+G!GO?OGJ+ M UZJ JWJ MBOZGZJZ B JB Z+G!GO?OGJ+  GOZ W%GO JB =G3+ +OJ
8J ZU+J GM M UZJB .WGM B G+OJJGBOZ4 Z++ G @N 2W+ .O GOJ+ M+
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OM+ B Z8 M+ O= GOJ+ MG J+OZGBOZ J JW+ .BGOJ B JW+ J G.+ .WZ+ MBOJMJ . +ZM G
+Z JW+ ZB.+Z B JW+ M UZJ Z8 M+ J JWGZ .BGOJ JB
+ MBOZJOJ4  4 JW8Z B GO   GO
B8O= U  BB%+ 2W8Z4 G GOGJGU JW+ M UZJ Z8 M+ GZ .O 4 JW+ O+J GOZJOJ   GO
B8O= U  BB%+ G B  AB+%+ 4 =8+ JB Z8 M+ =G38ZGBO4 JW+ Z8 M+ M8 %J8 + ZWB8= =+M +Z+ GO JG+ ZB JWJ JW+  BB%+ ZWB8=
+MB+ =++.+  &8 GO B G
+ JB =+J+ GO+ JW+ JG+ =+.+O=+OM+ B JW+  BB%+ =+.JW O= JW+ ZW.+ B JW+ GOJ+ M+  B8O= JW+  BB%+

Interfacial Phenomena in Materials Science

243

GRAIN BOUNDARY GROOVE

GRAIN 2

GRAIN 1

GRAIN BOUNDARY


   S GO
B8O= U  BB%+

#8+ JB JW+ . B
+ ZU+J U GJW +Z.+MJ JB JW+ .O GOJ+ M+
+! J++O JW+ JB  GOZ4 + W%+ JW+ BBGO . B
+ B JW+ Z8 M+ ZW.+ ? + ? 0 #?

- NQ ? + 0 N ? N 0 N ? N + 0 ?

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+Z C+ B Z8 M+ =G38ZGBO ,8 J JW+ J G.+ MBOJMJ .BGOJ /J GZ MBO%+OG+OJ JB ZB%+ JW+ . B
+ - 8ZGO JW+ [.M+ J OZB  X..UGO JW+ [.M+ J OZB  JB - + B
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9B + BB" B  ZB8JGBO B  GO JW+ B   0 > O= ?O= JWJ JW+ ZB8JGBO
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244

Alexander A. Golovin

2W+ MBOZJOJZ  O=  + B8O=  B JW+ JB
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 0  

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W+ + >< GZ JW+ 8+ S!8OMJGBO 2W8Z4 + Z++ JWJ JW+ =+.JW B JW+  GO!
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B8O= U  BB%+ GZ 8GJ+ M8
+ ZB+ TJW+ 4 + ZW BB" B  Z+!ZGG ZB8JGBO B JW+ . B
+ - GO JW+ BBGO B  ? + 0 #+ 1

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+ B8O=  B I O= JW+ BBGO

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U +OZ B  B
+OG8Z +JWB=4 GO JW+ B 
  1 0 @NN   

Interfacial Phenomena in Materials Science

245

D8
ZJGJ8J+ @NN GO $! JB B
JGO JW+ BBGO +M8

+OJ +JGBO B

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GJZ /"#/ GJW  =+MUGO .GJ8=+ O= JW+ %++OJW  BGO Z I#+  2W+ . +Z+OM+ B Z8 M+ BZMGJGBOZ GZ JW+ MW MJ+ GZJGM +J8 + B Z8 M+!=G38ZGBO!= G%+O Z8 M+ +JGBO JWJ BBZ  B JW+ B8 JW Z.JG =+ G%JG%+ MB

+Z.BO=GO JB JW+ [.MGO B JW+ +O M8 %J8 + 2W+ =+ G%JGBO B JW+ Z8 M+!=G38ZGBO +8JGBO O= JW+ ZB8JGBO B JW+  GO!
B8O= U  BB%+ . B
+ Z ? ZJ B
JGO+= GO JW+ MZZGM ..+


U K8GOZ K8GOZ 4 @Y74 WGMW JW+ =+ G%JGBO . +Z+OJ+= GO JWGZ Z+MJGBO MBZ+U BBZ

 ( (!  ( )
 (  ()"(()!   ( /O JWGZ Z+MJGBO + ZW =GZM8ZZ O G.B JOJ GOJ+ MG GOZJ
GGJU JWJ GZ

+J+= JB JW+ ?+= B OOBJ+MWOBBU O= +Z.BOZG
+ B JW+ B JGBO B GOJ+ +ZJGO OOB!ZM+ Z8 M+ ZJ 8MJ8 +Z M+= 8OJ8 =BJZP 6BOZG=+  ? B ZBG= M UZJGO+ J+ G =+.BZGJ+= BO  Z8 M+ B  Z8
ZJ J+ =+ B OBJW+ M UZJGO+ J+ G DBG=!BO!ZBG= =+.BZGJGBO GZ M+= +.GJU4 O= W+O JW+ =+.BZGJ+= J+ G GZ =G3+ +OJ  B JWJ B JW+ Z8
ZJ J+ GJ GZ M+= W+J+ B!+.GJU 2W+ =+.BZGJGBO . BM+ZZ GZ BJ+O =BO+
U +OZ B 
+ B JBZ B B+M8+Z WGJJGO JW+ Z8
ZJ J+4 JW+ ZB!M+= B+M8


+ +.GJU KH W+O JBZ  + =+.BZGJ+= BO  Z8
ZJ J+ JW+U ZJGM" JB JW+ JBZ B JW+ Z8
ZJ J+ ZB JWJ JW+ JBZ GOZG=+ JW+  BGO +.GJG ?  + 

O+= MMB =GOU DGOM+ JW+ M UZJ JJGM+ Z.MGOZ GO JB =G3+ +OJ J+ GZ 8Z8U =G3+  B BO+ OBJW+ "1 /",4 JW+ ?  BZ GO O +ZJGMU ZJ +ZZ+= ZJJ+4 O= WZ +ZJGM +O+ U ZJB += GO GJ /4 Z  +Z8J B ZB+ ,8MJ8JGBO4  Z .+ J8
JGBO GO JW+ B  B  Z WG ..+ Z BO JW+ ? Z8 M+4 JW+ + G
+ B + BB GOZG=+ JWGZ WG B JW+ JBZ B JW+ ? JB =58ZJ JW+G GOJ+ JBGM Z.MGO JB JW+ OB  +8GG
G8 %8+4 WGMW G += JB JW+ =+M +Z+ B JW+ +ZJGM +O+ U 2WGZ

246

Alexander A. Golovin

. BM+ZZ B JW+ WG B JGBO G JW+ +B +
+ +O+ +JGMU . ++ 
+4 WGMW G +Z8J GO O GOZJ
GGJU B  .O ? JWJ G =+MB.BZ+ GOJB  ZUZJ+ B GZO=Z 2W+ J OZ.B J +MWOGZ
U WGMW JW+ ? Z8 M+ +MBOZJ 8MJZ GJZ+ GOJB  ZUZJ+ B GZO=Z GZ Z8 M+ =G38ZGBO =GZM8ZZ+= GO JW+ . +%GB8Z Z+MJGBO 2WGZ +MWOGZ B JW+ GZO= B JGBO GZ ZMW+JGMU ZWBO GO G@@ Ge FILM Ge FILM

INSTABILITY Si SUBSTRATE

Si SUBSTRATE


   /OZJ
GGJU B O +.GJGU!ZJ GO+= ZBG= ? .+Z B Z8MW GZO=Z  + ZWBO GO G@- &O+ MO Z++ JWJ JW+U MO W%+ =G3+ +OJ ZW.+Z4 M+J+= B B8O=+= X JU.GM GZO= ZGC+ B JW+ B =+ B OOB+J+ Z4 WGMW GZ B JW+ Z+ B =+ B OGJ8=+ Z =+ H BG+ %++OJW B ++MJ BOZ GO Z+GMBO=8MJB Z 6BO?O+= GJWGO Z8MW Z8 M+ ZJ 8MJ8 +Z4 ++MJ BOZ +WG
GJ GOJ+ +ZJGO 8OJ8 . B.+ JG+Z JWJ MO
+ 8Z+= B 
GMJGBO B O+ ++MJ BOGM =+%GM+Z B JWGZ +ZBO Z8MW GZO=Z  + M+= # / #Z B +%G+Z B JW+ B JGBO O= . B.+ JG+Z B #Z Z++ DWMW8"GO O= HG
+  @Q HG
+  + -NN Q DJO + -NN  /O B =+ JB =+ZM G
+ JW+ =UOGMZ B JW+ B JGBO B #Z GO +.GJG ?Z BO+ ZWB8= 8Z+ JW+ Z8 M+ =G38ZGBO +8JGBO =+ G%+= GO JW+ . +%GB8Z Z+MJGBO AB+%+ 4 BO GJW JW+ Z8 M+ +O+ U4 JW+ Z8 M+ MW+GM .B! J+OJG ZWB8= GOM8=+ ZB JW+ +ZJGM +O+ U MM88J+= GO JW+ ? 2W8Z + W%+ ;  0   F )

@N- @ F ; W+ +  GZ  MBOZJOJ . B.B JGBO JB JW+ Z8 M+ =G38ZG%GJU B JBZ4 ) GZ JW+ Z8 M+ +O+ U4  GZ JW+ +O M8 %J8 +4 O=  GZ JW+ . J B Z8 M+ MW+GM .BJ+OJG MB

+Z.BO=GO JB JW+ +ZJGM +O+ U ZJB += GO JW+ +.GJG ? /J GZ MB.8J+= Z  BB W++Z O= BWOZBO 4 -NN   0

@  , 

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W+ +  GZ JW+ ZJ GO J+OZB 4 , GZ JW+ ZJ +ZZ J+OZB [O=8 O= [GZWGJZ 4 @$@4 O= JW+ 8Z8 MBO%+OJGBO B JW+ Z8JGBO B%+ JW+ +.+J+= GO=GM+Z

Interfacial Phenomena in Materials Science

247


   .+Z B 8OJ8 =BJZ 8..+ +J S+ #Z BO DGNN@ GGZ + @$  +. GOJ+= GJW .+ GZZGBO  B GGZ + @$4 M @$
U JW+ X+ GMO 6W+GM DBMG+JUQ 8..+ GWJ 
D+ #Z BO 
2+!BO!DG@@@ XMW
G + -NN  +. GOJ+= GJW .+ GZZGBO  B M XMW
G + -NN4 -NN
U JW+ X+ GMO WUZGM DBMG+JUQ B+ +J /OXZ #Z BO SXZNN@ S O=BZ + -NN  +. GOJ+= GJW .+ GZZGBO M  B S O=BZ + -NN4 -NN
U JW+ X+ GMO /OZJGJ8J+ B WUZGMZQ B+ GWJ MB Z+OGO B DGS+ #Z BO DGNN@  ?8 +
 MB

+Z.BO=Z JB J+ JG+ JWO  B B + -NNN  +. GOJ+= GJW .+ GZZGBO  B B B M + -NNN4 -NNN
U JW+ X+ GMO WUZGM DBMG+JU T+. GOJ+= GJW .+ GZZGBO  B D. GO+ 

248

Alexander A. Golovin

GO 8Z+= /O JW+ .. BGJGBO B GO+ +ZJGMGJU JW+B U4 , O=   + MBOO+MJ+=
U JW+ BBGO +JGBO [O=8 O= [GZWGJZ 4 @$@  
 , 0 - .  F 

@N  @F W+ +  GZ JW+ ZW+ B=88Z4  GZ JW+ BGZZBO*Z JGB4 O= . GZ JW+  BO+"+

=+J 2W+ ZJ GO J+OZB GZ =+J+ GO+=
U JW+ MB.BO+OJ B JW+ =GZ.M++OJ %+MJB  JWJ MW MJ+ GC+Z Z =+B JGBOZ B  ZBG=
B=U4  0

@ /  F /   -

@NY

A+ + / =+OBJ+Z =G3+ +OJGJGBO GJW +Z.+MJ JB JW+ MBB =GOJ+  4 $ 0 @ -  2W8Z4 GO B =+ JB ?O= JW+ +ZJGM MB.BO+OJ B JW+ ? Z8 M+ MW+GM .BJ+OJG + ZWB8= ?O= JW+ =GZ.M++OJ ?+= GO JW+ ? WGMW GZ =+J+ ! GO+=
U JW+ +.GJG ZJ +ZZ M8Z+=
U JW+ JJGM+ GZJMW
+J++O JW+ ? O= JW+ Z8
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ZJ J+ 2W+ GOJ+ M+
+J++O JW+ ? O= JW+ Z8
ZJ J+ GZ .O O= JW+ !GZ GZ OB  JB GJ4 GJW   N MB

+Z.BO=GO JB JW+ Z8
ZJ J+ O= N    ; ? + MB

+Z.BO=GO JB JW+ ?Q  0 ; ? + =+ZM G
+Z JW+ ZW.+ B JW+ ?  ++ Z8 M+4 O= ?  + JW+ MBB =GOJ+Z GO JW+ .O+ . + JB JW+ Z8
ZJ J+ DGOM+ Z8 M+ =G38ZGBO . BM+ZZ+Z  + 8MW ZB+ JWO JW+ Z.++= B ZB8O= GO  M UZJGO+ J+ G JWJ =+J+ ! GO+Z JW+ MW MJ+ GZJGM JG+ B +ZJGM +JGBO4 + MO ZZ8+ JWJ J +MW B+OJ B JG+ JW+ + GZ +MWOGM +8GG
G8 GO JW+ ? O= GO JW+ Z8
ZJ J+ WGMW GZ =+ZM G
+=
U JW+ +8JGBO / , 0 N

@NI

+ ZWB8= ZB Z+J JW+ BBGO
B8O= U MBO=GJGBOZ G ZJ +ZZ! ++ ? Z8 M+4 , 7 0 N

@N7

W+ + 7  + JW+ MB.BO+OJZ B JW+ OB  JB JW+ ? Z8 M+Q GG MBOJGO8GJU B =GZ.M++OJ ?+= GO JW+ ? O= JW+ Z8
ZJ J+ J JW+ .O Z8
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@N$  0  F 9  ?  Q N

Interfacial Phenomena in Materials Science W+ + 90







249

@N



GZ JW+ JJGM+ GZJMW
+J++O JW+ Z8
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U JW+ M UZJ JJGM+ Z.MGOZ B JW+ MB

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%GB8ZU4 GO JWGZ MZ+ JW+ ZJ +ZZ O= ZJ GO GO JW+ Z8
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+ C+ B ZGOM+  JWGO ? MOOBJ =+B  O GO?OGJ+U =++. Z8
ZJ J+ 2W+ WB GCBOJ MB.BO+OJZ B JW+ ZJ GO J+OZB GO JW+ ? GO JW+
ZGM ZJJ+  + =+J+ GO+=
U JW+ JJGM+ GZJMW4     

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+ C+ B4 , 0 N 2W+O  B JW+ ZJ +ZZ! ZJ GO +JGBO @N  + ?O= JW+ %+ JGM MB.BO+OJ B JW+ ZJ GO GO JW+ ?4 -9    0  @@ @   2W+O4 GO  B @N  + ?O= JW+ WB GCBOJ MB.BO+OJZ B JW+ ZJ GO J+OZB 4

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0 , 0 -9 @   . +ZZGBOZ @@-!@@  + M+= +.GJG ZJ GO O= @@  GZ M+= +.G! JG ZJ +ZZ + MO Z++ JWJ
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250

Alexander A. Golovin

9BJ+ JWJ JWGZ +O+ U =B+Z OBJ =+.+O= BO JW+ ZGO B JW+ JJGM+ GZJMW4 G+ GJ =B+Z OBJ =+.+O= BO W+JW+ JW+ +.GJG ZJ GO GZ MB. +ZZG%+ B

J+OZG+ XZ =GZM8ZZ+= 
B%+4  .O 4 ZJ +ZZ+= +.GJG ? ZWB8=
+ 8OZJ
+ GJW +Z.+MJ JB JW+ 8O=8JGBOZ B JW+ ?  ++ Z8 M+ JWJ MO +Z8J GO JW+ B JGBO B B8O=Z O= GO JW+ =+M +Z+ B JW+ +ZJGM +O+ U ZJB += GO JW+ ? /O B =+ JB ZWB JWJ + O++= JB .+ B   GO+ ZJ
GGJU OUZGZ B JW+
ZGM ZJJ+ =+ZM G
+= 
B%+ 2WGZ OUZGZ GZ 8GJ+ M8
+ ZB+ O= JW+ =+JGZ MO
+ B8O= GO D.+OM+ + @@4 @ A+ + + ZW =+ZM G
+ JW+ GO ZJ+.Z B JWGZ OUZGZ + MBOZG=+ GO?OGJ+ZG .+ J8
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B8O= U MBO=GJGBOZQ  G
+ B
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JGO  0 5 9 +ZJGM MBOZJOJZ

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W+ + 5 GZ ZB+ M8
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JGO JW+ =GZ.+ ZGBO +JGBO  0 5  9 +ZJGM MBOZJOJZ
+J++O JW+ .+ J8
JGBO  BJW J+ O= JW+ %+O8
+  /O JW+ BO!%+ GGJ4  ;  @4 JW+ =GZ.+ ZGBO +JGBO G +=  0   

@@$ W+ +   9 - N GZ . B.B JGBO JB JW+ JBJ +ZJGM +O+ U ZJB += GO JW+ ?4 O=  ) GO+ U =+.+O=Z BO JW+ Z8 M+ +O+ U ) GO JW+ MZ+ W+O JW+ +ZJGM MBOZJOJZ B JW+ Z8
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+ GJW +Z.+MJ JB Z8 M+ 8O=8JGBOZ 2W+ =GZ.+ ZGBO +JGBO @@$ GZ ZWBO GO G@ 2W+ +ZJGM +O+ U WZ =+ZJ
GGCGO +3+MJ W+ +Z JW+ Z8 M+ +O+ U WZ ZJ
GGCGO +3+MJ #+ZJ!
GGCGO +3+MJ GZ  + JWO JW+ ZJ
GGCGO BO+ B BO!%+ .+ J8
JGBOZ O= Z+ B ZWB J!%+ .+ J8
JGBOZ 2W+ MB.+JGJGBO B JW+ JB +3+MJZ

Interfacial Phenomena in Materials Science

251

W

+=Z JB JW+ Z++MJGBO B  MW MJ+ GZJGM %++OJW B JW+  BGO ZJ 8M! J8 +4 6 0 -14 4 W+ + 4 MB

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