The Rosensweig instability in isotropic magnetic gels

The Rosensweig ins abili y
in iso opic magne ic gels
Von de Uni e si ¨
a Bay eu h
zu E langung des G ades eines
,,Dok o s de Na u wissenscha en“ (D . e . na .)
genehmig e Abhandlung
o geleg on
S e an Bohlius
gebo en in Lich en els/Baye n
1. Gu ach e : P o . D . Helmu R. B and
2. Gu ach e : P o . D . Ha ald Pleine
Tag de Ein eichung: 28. M¨
a z 2008
Tag des Kolloquiums: 14. Juli 2008
Con en s
Lis o Figu es
Zusammen assung ii
1 In oduc ion 1
1.1 Fe o luids.................................... 1
1.2 Fe ogels..................................... 2
1.3 The Rosensweig ins abili y . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Nonlinea heo e ical desc ip ions o he Rosensweig ins abili y . . . . . . 5
1.4.1 The ene gy me hod . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.2 Func ional analysis app oaches . . . . . . . . . . . . . . . . . . . . 6
1.4.3 The Swi -Hohenbe g app oach . . . . . . . . . . . . . . . . . . . . 6
1.4.4 Nume ical esul s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 The adjoin sys em and de o mable su aces . . . . . . . . . . . . . . . . . 7
1.6 The scope o his hesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Mac oscopic ma hema ical amewo k 9
2.1 The basic hyd odynamic equa ions . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Di e en classes o mac oscopic a iables . . . . . . . . . . . . . . . 10
2.1.2 Con inui y and balance equa ions . . . . . . . . . . . . . . . . . . . 10
2.1.3 The case o iso opic e ogels . . . . . . . . . . . . . . . . . . . . . 12
2.2 Assump ions o he Rosensweig ins abili y . . . . . . . . . . . . . . . . . . 16
2.2.1 The simpli ied bulk equa ions . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 The bounda y condi ions . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Recalling he linea p oblem 21
3.1 Theg ounds a e ................................ 21
3.2 Linea de ia ions om he g ound s a e . . . . . . . . . . . . . . . . . . . . 22
3.3 Su ace wa e dispe sion ela ion . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Rosensweig ins abili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 The linea eigen ec o s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 On he no mal s ess bounda y condi ion . . . . . . . . . . . . . . . . . . . 30
4 Nonlinea discussion using he ene gy me hod 31
4.1 Su ace ene gy densi y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Linea s abili y ................................. 34
4.3 S abili y o di e en Geome ies . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.1 S ipe solu ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
i
ii CONTENTS
4.3.2 Squa es ................................. 35
4.3.3 Hexagons ................................ 37
4.4 Some d awbacks o he ene gy me hod . . . . . . . . . . . . . . . . . . . . 39
5 The ampli ude equa ion 41
5.1 In oduc ion................................... 41
5.1.1 Nonlinea expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.1.2 The sol abili y condi ion o highe o de s . . . . . . . . . . . . . . 43
5.2 The adjoin sys em o he Rosensweig ins abili y . . . . . . . . . . . . . . 46
5.2.1 Dynamic su aces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 Basic equa ions and g ound s a e . . . . . . . . . . . . . . . . . . . 46
5.2.3 The linea equa ions and he adjoin sys em . . . . . . . . . . . . . 47
5.2.4 Adjoin eigen ec o s o he Rosensweig ins abili y . . . . . . . . . . 51
5.3 The second pe u ba i e o de . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.1 The sol abili y condi ion in second o de . . . . . . . . . . . . . . . 54
5.3.2 Solu ions p opo ional o he main cha ac e is ic modes . . . . . . . 56
5.3.3 Solu ions p opo ional o he highe ha monics . . . . . . . . . . . . 58
5.3.4 The no mal s ess bounda y condi ion . . . . . . . . . . . . . . . . 61
5.4 The hi d pe u ba i e o de . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.5 Ampli udeequa ion............................... 66
5.6 On he Newell-Ope a o . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.7 Discussion and compa ison . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Rosensweig ins abili y in ilms and memb anes 77
6.1 Mo i a ion.................................... 77
6.2 Film p ope ies in iscoelas ic media . . . . . . . . . . . . . . . . . . . . . 77
6.3 Magne ic su ace p ope ies . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Non-magne ic ilm modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.5 Fe ogel ilm su ace modes . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.6 Rosensweig ins abili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.6.1 S a iona y, asymme ic case wi hou su ace magne ism . . . . . . 84
6.6.2 Pe manen -magne ic, symme ic case . . . . . . . . . . . . . . . . . 86
6.6.3 Thegene alcase ............................ 86
6.6.4 Addi ional ema ks . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.7 Discussion.................................... 87
7 The adjoin sys em o he Ma angoni con ec ion 89
7.1 In oduc ion o Ma angoni con ec ion . . . . . . . . . . . . . . . . . . . . . 89
7.2 Basic equa ions and he adjoin sys em . . . . . . . . . . . . . . . . . . . . 90
7.3 The dimensionless ep esen a ion . . . . . . . . . . . . . . . . . . . . . . . 92
7.4 The dispe sion ela ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5 The adjoin dispe sion ela ion . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.6 Discussion o he dispe sion ela ion . . . . . . . . . . . . . . . . . . . . . . 96
8 Conclusions 99
A Decoupling o he dynamic sys em 103
CONTENTS iii
B Magne ic ields 105
B.1 The Hea iside-Lo en z sys em o elec omagne ic uni s . . . . . . . . . . . 105
B.2 Expansion o highe pe u ba i e o de s . . . . . . . . . . . . . . . . . . . 107
B.2.1 The Maxwell equa ions . . . . . . . . . . . . . . . . . . . . . . . . . 107
B.2.2 The bounda y condi ions . . . . . . . . . . . . . . . . . . . . . . . . 108
B.3 Solu ions in linea o de . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.4 Solu ions in highe o de s . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.5 Magne ic ields in he case o memb anes . . . . . . . . . . . . . . . . . . . 112
B.5.1 The supe pa amagne ic case . . . . . . . . . . . . . . . . . . . . . . 112
B.5.2 The pe manen -magne ic case . . . . . . . . . . . . . . . . . . . . . 114
C The hyd odynamic bounda y condi ions 117
C.1 Expansion o he bounda y condi ions . . . . . . . . . . . . . . . . . . . . . 117
C.2 The linea pe u ba i e o de . . . . . . . . . . . . . . . . . . . . . . . . . 117
C.3 The second pe u ba i e o de . . . . . . . . . . . . . . . . . . . . . . . . . 119
C.4 The hi d pe u ba i e o de . . . . . . . . . . . . . . . . . . . . . . . . . . 120
D Eigen ec o s in he second o de 123
E Usual e o luids 131
Li e a u e 133
Acknowledgmen s 143

i CONTENTS
Lis o Figu es
1.1 Ske ches o a e o luid wi h and wi hou an ex e nal magne ic ield . . . . 2
1.2 Ske ches o an iso opic and an aniso opic magne ic gel . . . . . . . . . . . 3
1.3 The Rosensweig ins abili y in e o luids . . . . . . . . . . . . . . . . . . . 4
1.4 Quali a i e geome y o he Kel in-Helmhol z ins abili y . . . . . . . . . . 8
2.1 Quali a i e geome y o he Rosensweig ins abili y . . . . . . . . . . . . . 18
3.1 Schema ic ske ch o he di e en su ace wa e egimes . . . . . . . . . . . . 24
3.2 Dispe sion ela ion in e o luids o di e en ex e nal magne ic ields . . . 26
3.3 Dispe sion ela ion in e ogels o di e en elas ic shea moduli . . . . . . 27
3.4 S ess dis ibu ion wi hin he e ogel . . . . . . . . . . . . . . . . . . . . . 29
4.1 Conside ed plan o ms o he ene gy me hod . . . . . . . . . . . . . . . . . 35
4.2 G aphs o es ima e he alidi y egime o he ene gy me hod (squa es) . . 36
4.3 G aphs o es ima e he alidi y egime o he ene gy me hod (hexagons) . 37
4.4 Bi u ca ion scena io o he Rosensweig ins abili y . . . . . . . . . . . . . . 38
5.1 The gene al bi u ca ion scena io o ampli ude equa ions . . . . . . . . . . 44
5.2 The ela i e o ien a ion o he di e en wa e ec o s unde conside a ion . 59
5.3 Quali a i e dynamical g ow h o he su ace spikes in e ogels . . . . . . . 70
5.4 Ske ch o a physical sys em desc ibed by he sine-Go don equa ion . . . . . 71
6.1 Quali a i e geome y in he case o memb anes and hin ilms . . . . . . . 78
6.2 De i a ion o he su ace s ess bounda y condi ion . . . . . . . . . . . . . 80
7.1 Quali a i e geome y o he Ma angoni ins abili y . . . . . . . . . . . . . 90
i LIST OF FIGURES
Zusammen assung
Die o liegende A bei be ass sich mi de nich linea en heo e ischen Analyse de Ro-
sensweig Ins abili ¨
a in iso open magne ischen Gelen. Die Rosensweig Ins abili ¨
a wu de
e s mals im Jah 1967 en deck und bezeichne den ¨
Ube gang eine zun¨
achs lachen
G enz l¨
ache zwischen eine magne ischen Fl¨
ussigkei und einem nich -magne ischen Me-
dium zu eine hexagonal geo dne en S achelobe l¨
ache, sobald ein senk ech zu lachen
Obe l¨
ache angeleg es homogenes Magne eld einen bes imm en k i ischen We ¨
ube -
sch ei e . Magne ische Fl¨
ussigkei en, auch Fe o luide genann , sind kolloidale Suspensio-
nen e omagne ische Nano eilchen in eine gew¨
ohnlichen, dem Anwendungszweck en -
sp echenden T ¨
age l¨
ussigkei , wie Wasse ode Benzol. Einem angeleg en Magne eld aus-
gese z , e hal en sich Fe o luide wie gew¨
ohnliche pa amagne ische S o e, jedoch is ih e
Pe meabili ¨
a bis zu eine G ¨
oßeno dnung h¨
ohe als in ¨
ublichen pa amagne ischen S o en,
weshalb man sie auch als supe pa amagne isch bezeichne .
Mi de En deckung de Rosensweig Ins abili ¨
a wu de auch eine e s e heo e ische
Besch eibung des Ph¨
anomens o ges ell . An de eien G enz l¨
ache zwischen de Fe -
o l¨
ussigkei und dem da ¨
ube liegenden Vakuum ¨
ube wiegen ¨
u nied ige Magne elde
die s abilisie enden K ¨
a e de G a i a ion und de Obe l¨
achenspannung die des abilisie-
ende K a des Magne eldes. Zwa besi z ein homogenes Magne eld keine K a wi kung
au die Obe l¨
ache, jedoch un e lieg die G enz l¨
ache den imme o handenen he mischen
Fluk ua ionen, die das Magne eld lokal s ¨
o en und so eine esul ie ende K a e zeugen.
Bei gen¨
ugend hohen Magne elds ¨
a ken ¨
ube i diese K a die G a i a ion und die
Obe l¨
achenspannung und das Rosensweigmus e bilde sich aus.
S a e man den Ve ne zungsp ozess in eine Mischung aus Polyme en, Ve ne zungs-
eagenzien und einem Fe o luid, so e h¨
al man ein iso opes Fe ogel, ein elas isches Me-
dium, welches zus¨
a zlich supe pa amagne isches Ve hal en au weis . Fe ogele bilden eine
neue Ma e ialklasse, on de man sich Anwendungen in ielen echnischen und medizini-
schen Be eichen e ho . So gel en sie zum Beispiel als iel e sp echende Kandida en zu
He s ellung k¨
uns liche Muskeln ode on außen egelba e Medien zu geziel en Wi k-
s o eise zung im K¨
o pe . Theo e isch l¨
ass sich zeigen, dass auch die Obe l¨
ache diese
Medien in einem angeleg en Magne eld ins abil wi d, wobei die ypische Wellenl¨
ange
im Ve gleich zu gew¨
ohnlichen Fe o luiden un e ¨
ande bleib , w¨
ah end die k i ische Ma-
gne elds ¨
a ke mi wachsendem elas ischen Sche modul s eig . Expe imen ell konn e dies
be ei s quali a i bes ¨
a ig we den. Alle dings is man in Expe imen en au seh schwach
e ne z e Gele angewiesen, da die k i ische Magne isie ung ande en alls g ¨
oße als die
S¨
a igungsmagne isie ung des elas ischen Mediums is .
Nach dem ein ¨
uh enden Kapi el und de Diskussion de g undlegenden hyd odynami-
schen Gleichungen zu Besch eibung iso ope magne ische Gele in Kapi el 2, we den im
d i en Kapi el die linea en Eigenscha en de Rosensweig Ins abili ¨
a in iso open Fe -
ii
4In oduc ion
Figu e 1.3: The expe imen al ealiza ion o he Rosensweig ins abili y. On he
le hand side he applied magne ic ield (applied no mal o he su ace) is below
he c i ical alue whe eas he pic u e on he igh hand side is aken beyond he
c i ical alue. The con aine s diame e is o he o de o 20 cm and he heigh as
well as he diame e o he spikes is in he o de o 1 cm.
1.3 The Rosensweig ins abili y
The Rosensweig o no mal ield ins abili y desc ibes he phenomenon o a la e o luid
su ace becoming uns able in an ex e nal magne ic ield. The expe imen al se up consis s
o a Pe i-dish illed wi h a e o luid which de elops a la su ace in ea h’s g a i a ional
ield (as gi en in he pho og aph on he le o ig. 1.3). I one applies a homogeneous
magne ic ield o ien ed pa allel o he su ace no mal, his la su ace becomes uns able
beyond a ce ain c i ical magne ic ield s eng h and a egula pa e n o su ace spikes
a ises (as seen on he igh o ig. 1.3). Expe imen ally one obse es a hexagonal a ange-
men o hese su ace spikes a he linea h eshold [19].
Wi h i s disco e y in 1967 [19] a i s heo e ical desc ip ion o he no mal ield
ins abili y was gi en which allowed he de e mina ion o he c i ical magne ic ield and
he mos uns able mode a onse in e ms o he luid p ope ies. The model conside ed
he o ce balance a he ee su ace be ween he s abilizing o ces o su ace ension and
g a i a ion and he des abilizing magne ic o ce. Al hough he homogeneous magne ic
ield does no gene a e a o ce in he i s place, he occu ence o su ace pe u ba ions
ende he local magne ic ield inhomogeneous esul ing in a local Kel in o ce ha d i es
he ins abili y.
Fo a magne ic gel a he han a e o luid, he su ace was also p edic ed o become
uns able [20]. Addi ionally, howe e , he s abilizing elas ic o ce has o be o e come by
he magne ic ield, which is why i s c i ical alue is shi ed owa ds highe ield s eng hs.
The cha ac e is ic mode a onse , howe e , emains he same compa ed o usual e o luids
and is gi en by he capilla y mode. Recen ly he h eshold shi has quali a i ely been
shown expe imen ally o a he mo e e sible magne ic gel a he Uni e si y o Bay eu h
[21]. Since, howe e , he h eshold magne ic ield inc eases wi h inc easing shea modulus,
one is es ic ed o e y weak gels, o he wise he h eshold magne iza ion is highe han
he sa u a ion magne iza ion o he medium. To main ain a ini e and cons an shea
modulus o he he mo e e sible gel and o a oid c eep low, a ime dependen magne ic
ield was applied which complica es he compa ison be ween he expe imen s and heo y.
A he momen mo e accu a e expe imen al esul s can be ob ained using in e se
e o luids [22, 23, 24]. One calls a e o luid in e se, i non-magne ic pa icles a e also
dispe sed in he e o luid. Usually he pa icles’ diame e is o he o de o mic ome e s
and hey a e ypically made o polys y ene. The onse o he Rosensweig ins abili y in
hese luids is shi ed o highe magne ical ield s eng hs, which canno be solely explained

1.4 Nonlinea heo e ical desc ip ions o he Rosensweig ins abili y 5
by dilu ion e ec s [25], bu p obably equi es he assump ion o a ini e e ec i e shea
modulus, which may be esponsible o he h eshold shi .
Fig. 1.3 shows ha a e o luid is a da kish b own, non- anspa en medium and o
his eason quan i a i e expe imen al esul s o nonlinea pa e ns a e di icul o ob ain
op ically. Fo supe c i ical magne ic ield s eng hs he p ope ies o he mos uns able
mode and i s g ow h a e ha e been discussed heo e ically as well as expe imen ally in
[26, 27]. In 1984 Bac i and Salin [28] disco e ed he hys e e ic na u e o he ansi ion
be ween he la su ace and su ace spikes. They used a e y hin con aine (≈200 µm)
whe e he a ising pa e n was quasi one dimensional. This allowed he obse a ion o he
ins abili y om he side exploi ing he op ical con as be ween he magne ic luid and
he medium abo e. Using adioscopic me hods [29], he hys e e ic egion be ween he
la su ace and hexagons was accu a ely measu ed [30]. I he magne ic ield is inc eased
u he , he hexagonal pa e n is ound o be uns able wi h espec o egula squa es.
Also his ansi ion is accompanied by a hys e e ic egion and is expe imen ally discussed
in [31, 32].
Expe imen s on he nonlinea egime o he Rosensweig ins abili y in e ogels ha e
ecen ly been s a ed, bu up o now no publica ions a e a ailable.
1.4 Nonlinea heo e ical desc ip ions o he
Rosensweig ins abili y
Since i s disco e y, he Rosensweig ins abili y has a ac ed he a en ion o expe imen-
alis s and heo e icians, alike. The wo k o he expe imen al scien is s oge he wi h an
in ui i e linea desc ip ion has been in oduced in he p e ious sec ion. A linea analysis,
howe e , gi es us no in o ma ion abou he ampli ude and he spa ial s uc u e o he
a ising pa e n no on he nonlinea dynamic beha io . This is why nonlinea discussions
o he go e ning basic equa ions a e needed which will be he aim o his hesis. In he ol-
lowing he eade will ind an in oduc ion o p e ious nonlinea discussion o he no mal
ield ins abili y in e o luids.
1.4.1 The ene gy me hod
The ene gy me hod was he i s a emp o heo e ically access he nonlinea egime o
he Rosensweig ins abili y and was published in 1977 by Gaili is [33]. Gaili is discussed
la e ally unbounded e o luid laye s o in ini e dep h. The applica ion o his me hod o
sys ems wi h a ini e dep h was la e done by F ied ichs and Engel in [34]. The gene al
idea o his ene gy-based app oach is o ind he dependence o he su ace ene gy densi y
o he luid as a unc ion o he su ace de lec ion, he applied magne ic ield, and he
ma e ial pa ame e s o he e o luid excep o he iscosi y, which has o be neglec ed
comple ely in his app oach. This ene gy densi y unc ional can hen be minimized wi h
espec o p esc ibed su ace pa e ns. The pa e ns conside ed by Gaili is we e egula
s ipes, squa es and hexagons. The ela i e s abili y o which can hen be gi en as a
unc ion o he applied magne ic ield.
Gaili is ound, ha s ipe pa e ns a e always uns able wi h espec o one o he o he
wo pa e ns. Below he linea h eshold he la su ace is always a s able con igu a ion
whe eas a he linea onse hexagons u n ou o be ene ge ically a o ed. Upon u he
6In oduc ion
inc ease o he magne ic ield, howe e , he hexagons become uns able wi h espec o
squa es and he p e ious hexagonal pa e n ans o ms in o a squa e pa e n. Bo h an-
si ions a e ound o be accompanied by hys e e ic egions. In ha espec he heo e ical
esul s quali a i ely ma ch he expe imen al indings. Fo ini e luid laye dep hs, he
c i ical alues o he magne ic ield a e shi ed o highe s eng hs and he hys e e ic
beha io o bo h ansi ions becomes mo e p onounced [34].
In chap e 4 we will apply he ene gy me hod o he Rosensweig ins abili y in iso opic
magne ic gels o ob ain a i s es ima e o he nonlinea pa e ns a ising in e ogels.
Howe e , he e a e se e e d awbacks o his me hod. The e o e one has o use mo e
undamen al me hods o discuss he nonlinea egime.
1.4.2 Func ional analysis app oaches
The i s app oaches conside ing he basic hyd odynamic equa ions ha e been discussed
by Twombly and Thomas [35, 36] and la e on by Silbe and Knobloch [37]. Bo h g oups
conside ed only s a ic hyd odynamic equa ions unde he condi ion ha he eloci y an-
ishes, educing he Na ie -S okes equa ion o he hyd os a ic p essu e con ibu ion. In
his app oxima ion he s ess ee su ace is go e ned by he no mal s ess bounda y con-
di ion whe eas he angen ial bounda y condi ions a e i ially sa is ied. Addi ionally, he
s a ic Maxwell equa ions we e conside ed wi h he co esponding bounda y condi ions.
This se o undamen al equa ions and bounda y condi ions is hen expanded in e ms
o ǫ( he no malized di e ence be ween he applied magne ic ield and he c i ical one)
ollowing he ideas o [38]. Since no ime de i a i e is in ol ed in he se o equa ions,
i u ns ou o be sel -adjoin and he sol abili y condi ions in he highe o de s o he
expansion can be ul illed. As s a ed by Silbe and Knobloch [37], no s able pa e n was
ound a he linea onse o ealis ic magne ic pe meabili ies. Ano he d awback o his
me hod is, since i es s on he s a ic assump ion, ha i canno gi e p edic ions o he
nonlinea dynamics in e ms o an ampli ude equa ion and ha i neglec s he possibili y
o oscilla o y ins abili ies.
A di e en app oach, using again he s a ic app oxima ion o he mac oscopic se o
equa ions, was p esen ed by F ied ichs and Engel in 2003 [39]. In hei discussion hey
ocused on he no mal s ess bounda y condi ion only. Inspi ed by [40], whe e i is shown
ha o a s ong enough angen ial componen o he magne ic ield wo dimensional
pa e ns can be supp essed and only s ipes a e he s able solu ion, hey ocused on he
nonlinea discussion o s ipes only.
Malik and Singh [41, 42, 43] we e he i s o discuss an ǫ−expansion o he undamen-
al hyd odynamic equa ions allowing o dynamic p ocesses. To ci cum en he gene al
sol abili y condi ion o he highe o de s o he expansion, whe e one needs o know he
adjoin sys em o equa ions, hey es ic ed hei discussion o po en ial low only. This
is an unphysical app oxima ion, as we will see la e on in ou discussion, since only he
o ici y con ibu ions o he low can gua an ee ha he angen ial bounda y condi ions
a he ee su ace a e sa is ied.
1.4.3 The Swi -Hohenbe g app oach
To compensa e o he lack o dynamic desc ip ions, Kubs up, He e o and P´e ez-Ga c´ıa
discussed he no mal ield ins abili y in e ms o a phenomenological Swi -Hohenbe g
1.5 The adjoin sys em and de o mable su aces 7
equa ion [44, 45]. In pa icula , hey discussed he dynamics o s able on s be ween
hexagons and squa es. A gene al ansa z o he su ace de lec ion, which is no so di e en
om he one used in he ene gy me hod app oach, is subs i u ed in o a gene alized Swi -
Hohenbe g equa ion and a se o ampli ude equa ions is ob ained. These ampli ude
equa ions a e hen sol ed nume ically. This app oach nicely e eals he s abili y dynamics
be ween hexagons and squa es in he nonlinea egime, bu only on a phenomenological
basis. The main disad an age o his s udy is, ha he coe icien s in he ampli ude
equa ion ha e no ela ion wha soe e o eal ma e ial p ope ies. Ne e heless, a good
quali a i e ag eemen can be ob ained i ing he phenomenological coe icien s o he
expe imen al esul s [31].
1.4.4 Nume ical esul s
The me hods discussed so a desc ibe he dynamics o he Rosensweig ins abili y in he
weakly nonlinea egime. These me hods allow one o discuss he dynamics and he a ising
pa e n close o he h eshold as long as he ampli udes s ay small. A p edic ion o he
inal shape o a single su ace spike canno be ob ained by hese me hods. Using a ini e
elemen me hod, La o a e al. de e mined he shape o one o he su ace spikes [46] by
in eg a ion o he basic hyd odynamic equa ions. In [30] he expe imen al and nume ical
esul s we e compa ed and a e y good ag eemen was obse ed.
1.5 The adjoin sys em and de o mable su aces
A a close inspec ion o sec ion 1.4.2 and he nonlinea app oaches discussed he ein, i is
clea ha he e is s ill a c ucial piece missing in he desc ip ion o he nonlinea egime o
he Rosensweig ins abili y in he spi i o [38, 47, 48], namely he knowledge o he adjoin
sys em o equa ions oge he wi h i s bounda y condi ions. A i s a emp o de i e his
was made by Lange in [49] who used a pa icula o m o a scala p oduc known om
discussions o he Ma angoni ins abili y. The a emp ailed since his scala p oduc
es s on he assump ion o an unde o mable su ace; an assump ion no app op ia e o
he case o he Rosensweig ins abili y. In he p esence o a de o mable su ace and o
a dynamic sys em, he adjoin sys em wi h i s bounda y condi ions was unknown. A
de ailed discussion and he de i a ion is gi en in his hesis in sec ion 5.2.
As indica ed al eady in he p e ious pa ag aph, he Rosensweig p oblem is no he
only ins abili y ha in ol es a de o mable su ace. Ano he e y p ominen phenomenon,
he Ma angoni con ec ion, sensi i ely depends on he de o mabili y o he su ace [50, 51].
In he Ma angoni con ec ion, small empe a u e luc ua ions a he in e ace be ween he
unde lying luid and he medium abo e induce luc ua ions o he local su ace ension
ha in u n cause he su ace o de o m. An analy ical weakly nonlinea desc ip ion o
he Ma angoni con ec ion accoun ing o he de o mabili y o he su ace is s ill missing.
Many au ho s [52, 53, 54, 55, 56] ea ed he nonlinea sys em assuming a la unde o med
bounda y be ween wo luids. The eason is mainly due o he missing solu ion o he
adjoin sys em in he p esence o a de o mable su ace. The case o he Ma angoni in-
s abili y is o pa icula in e es o us, since in con as o he Rosensweig ins abili y, he
d i ing o ce o he ins abili y ac s pu ely angen ially o he su ace. The me hod we
used o de i e he adjoin sys em o he Rosensweig ins abili y can he e o e be used o
8In oduc ion
luid 1
luid 2
~ 1
~ 2
Figu e 1.4: A quali a i e ske ch o he geome y app op ia e o he Kel in-
Helmhol z ins abili y. Two luids mo e a di e en eloci ies wi h espec o each
o he . The ini ially la in e ace becomes uns able agains de o ma ion beyond a
c i ical eloci y di e ence.
any a bi a y di ec ion o he d i ing o ce.
A u he si ua ion whe e a de o med bounda y becomes impo an o he nonlinea
egime and mo e p ecisely, whe e he ac ual posi ion o he bounda y depends on he
dynamics o he sys em as a whole, is he Fa aday ins abili y. Ce ain modes become
uns able upon pe iodic no mal ib a ions o he medium. The weakly nonlinea analysis
o his p oblem also c ucially depends on he knowledge o he adjoin sys em. We will
no deal wi h his phenomenon since a comp ehensi e nonlinea s udy o his p oblem has
been gi en ecen ly by Skeldon and Guidoboni in [57].
Two u he examples whe e he de o mabili y o he bounda y is essen ial in he
nonlinea egime a e gi en by he Rayleigh-Taylo ins abili y and he Kel in-Helmhol z
ins abili y. In he Rayleigh-Taylo p oblem he s abili y o a dense liquid on op o a
ligh e liquid, bo h subjec o a g a i a ional ield, is analyzed. In he Kel in-Helmhol z
p oblem he s abili y o he in e ace be ween wo luids ha mo e wi h di e en eloci ies
wi h espec o each o he is discussed (c . ig. 1.4). The la e p oblem is o pa icula
in e es o he c ea ion o low p essu e sys ems in he global wea he sys em as i occu s
in he a mosphe e a he bounda y be ween he ai a he cold pola caps and he wes
wind zone. The me hod we applied in inding he adjoin sys em o he Rosensweig and
he Ma angoni case can also be applied o hese sys ems.
1.6 The scope o his hesis
The nonlinea beha io o he Rosensweig ins abili y ei he in e o luids o in magne ic
gels s ill con ains many unsol ed ques ions. In his hesis I will ocus on he nonlinea
analy ic desc ip ion o hese phenomena in e ogels pu ing pa icula a en ion on he
de i a ion o he ampli ude equa ion. A c ucial s ep o his de i a ion will be he de-
e mina ion o he adjoin sys em wi h i s bounda y condi ions. To se he s age o
he special p ope ies o he Rosensweig ins abili y, we will ex ensi ely discuss he lin-
ea beha io and he possible pa e ns a ising in he nonlinea egime using he ene gy
me hod. In addi ion we will discuss he ob ained ampli ude equa ion o he special case
o e o luids, because also in his case he ampli ude equa ion de i ed om he basic hy-
d odynamic equa ions is s ill unknown. Since he bounda y condi ions play an impo an
ole in he discussion, we will also deal wi h hin ilms and memb anes and discuss hei
linea s abili y p ope ies in ex e nal magne ic ields. To conclude, ou conside a ions a e
applied o de i e he adjoin sys em o equa ions o he Ma angoni ins abili y.
Chap e 2
Mac oscopic ma hema ical
amewo k
In o de o gi e a comp ehensi e heo e ical desc ip ion o he Rosensweig ins abili y
in iso opic e ogels, we i s ha e o de ine he physical and ma hema ical amewo k.
We know, om expe imen al esul s, ha he ypical leng h scale o he ins abili y is o
he o de o cen ime e s and he ypical g ow h o he su ace spikes akes place on a
a he long ime scale, say seconds ( he g ow h can be ollowed by he naked eye). This
sugges s a iewpoin whe e we conside he medium con inuous and mac oscopic. The
mos sui able heo y we can use is hus he gene alized hyd odynamic heo y [58, 59]. In
his chap e he hyd odynamic app oach will be in oduced and we discuss he basic se o
hyd odynamic equa ions one ob ains o iso opic magne ic gels. This pa is mainly based
on, and summa izes, he esul s o he wo k o Ja ko a e al. [60] and will be he basis
o he nonlinea discussion. Addi ionally we speci y he simpli ica ions and ex ensions o
his gene al se o equa ions ha a e app op ia e o he Rosensweig ins abili y.
2.1 The basic hyd odynamic equa ions
The gene alized hyd odynamic app oach u ilizes simple symme y and he modynamic
a gumen s o de i e a gene al se o dynamic equa ions o ce ain mac oscopic a iables.
These a iables ha e o be iden i ied o he pa icula sys em unde conside a ion and
his pa icula choice d ama ically educes he numbe o deg ees o eedom one akes
in o accoun . In a mic oscopic desc ip ion, o ins ance, one migh model a oms o a
ce ain species as poin masses in e ac ing wi h each o he ia speci ied po en ials. To
conside sys ems o mac oscopic size like a glass o wa e , howe e , he numbe o poin
masses needed o model his sys em is o he o de o A ogad o’s cons an . In ac , oo
many deg ees o eedom o be handled. In a mac oscopic heo y one conside s he mass
densi y ield ins ead, which educes he numbe o ee a iables d as ically. Calcula ing
mac oscopic p ope ies o he sys em becomes easible.
One big ad an age o he hyd odynamic me hod is gi en by i s gene ali y, which
allows i s applica ion o a as numbe o sys ems as long as we a e able o ea hese
sys ems mac oscopically. Howe e , since we a e age o e e y many deg ees o eedom,
phenomenological coe icien s ha e o be in oduced ha a e speci ic o he pa icula
sys em. One o hese phenomenological coe icien s is, o example, he well known hea
9

10 Mac oscopic ma hema ical amewo k
conduc i i y. These coe icien s con ain he in o ma ion o all he mic oscopic p ocesses
aking place in he medium, and hey could heo e ically be de e mined ia G een-Kubo
ela ions i all mic oscopic p ocesses we e known, bu p ac ically his is only possible
in some limi ing cases, as, o example, small densi ies. The e o e hese coe icien s a e
usually aken om expe imen al measu emen s.
2.1.1 Di e en classes o mac oscopic a iables
One can dis inguish h ee di e en classes o mac oscopic a iables. The i s class includes
a iables associa ed wi h global conse a ion laws. Assume a sys em wi h a conse ed
quan i y, which by de ini ion canno decay o g ow locally bu which is allowed o be
anspo ed. The equilib a ion o his conse ed quan i y a di e en posi ions in space
akes he longe , he a he hese wo posi ions a e sepa a ed om one ano he . In ecip-
ocal space his s a emen can be exp essed as: he equency o a p ocess ends o anish
(ω→0) i i s wa enumbe anishes (k→0). This is exac ly he ma hema ical o mula-
ion o he hyd odynamic limi which es s on he ac , ha his o ically hyd odynamics
deal wi h conse ed quan i ies.
The second class o a iables, as a i s gene aliza ion o hyd odynamics, includes a i-
ables ha a e connec ed o spon aneously b oken con inuous symme ies. Wha we e e
o as a b oken symme y is he possibili y ha he Hamil onian o a sys em is o highe
symme y han i s eigens a es. One o he bes known examples o his phenomenon is
e omagne ism, whe e he Hamil onian is indeed in a ian unde o a ion al hough he e
is an easy axis assigned o he sys em which is e lec ed in he eigens a es. In gene al
he e is no conse ed quan i y connec ed o his kind o a iables ( e omagne ism is an
excep ional case whe e he magne iza ion is a conse ed quan i y) [59], bu hey allow
o exci a ions wi h in ini e li e ime in he long wa eleng h limi , he so called Golds one
modes [59]. The e o e i seems easonable o addi ionally include hese a iables in o a
mac oscopic desc ip ion.
The hi d and inal class accoun s o a iables connec ed o mic oscopic deg ees o
eedom whose dynamics akes place on a imescale la ge enough ha i en e s he mac o-
scopic egime. Consequen ly one can include hese speci ic mic oscopic deg ees o eedom
in o he mac oscopic desc ip ion. O he wise he alidi y o he desc ip ion would be e-
s ic ed o ime scales e en la ge o gua an ee ha his speci ic a iable has elaxed o i s
equilib ium alue. Bu i is wo h men ioning, ha hese a iables do no show he hyd o-
dynamic limi , ω6→ 0 i k→0, and long wa eleng h exci a ions wi h a ini e equency
may be e ained. While he iden i ica ion o he a iables o he i s wo classes is com-
ple ely sys ema ic, he iden i ica ion o a iables o his las class is no s aigh o wa d
and in ol es a deepe knowledge o he sys em.
2.1.2 Con inui y and balance equa ions
I he mac oscopic a iables o a pa icula sys em a e speci ied, one can u n o he
de i a ion o equa ions ha desc ibe he dynamics o hese a iables. Assume i s a
mac oscopic sys em in global he modynamic equilib ium. The s a e o he sys em is hen
comple ely gi en by he nume ical alues o he mac oscopic a iables and one can de ine
a he modynamic po en ial ha is a unc ion o hese mac oscopic a iables. Changes
o he po en ial as a unc ion o he mac oscopic a iables a e ela ed by he i s law o
2.1 The basic hyd odynamic equa ions 11
he modynamics. In he ollowing o mula ion E ep esen s he in e nal ene gy, T he
empe a u e, S he en opy, p he p essu e, V he olume, µ he chemical po en ial and
N he numbe o pa icles
dE =TdS −pdV +µdN (2.1)
The conjuga ed ields T,pand µcan be ob ained by pa ial di e en ia ion o he ene gy
densi y wi h espec o he associa ed a iable while he o he a iables a e kep cons an .
Wi h he help o Eule ’s ela ion we can ake he he modynamic limi V→ ∞ and
elimina e he olume om (2.1) which gi es he local mani es a ion o he i s law o
he modynamics [58]
dε =Tdσ +µdρ (2.2)
whe e εand σdeno e he ene gy densi y and he en opy densi y, espec i ely.
In he scope o a gene alized hyd odynamic app oach whe e we addi ionally accoun o
a iables associa ed wi h b oken con inuous symme ies and slowly elaxing a iables, he
Gibbs ela ion (2.2) also has o be gene alized. This can be done by exploi ing he ac ha
(2.2) is a o al di e en ial which allows one o in oduce conjuga ed ields associa ed o he
addi ional mac oscopic a iables. Howe e , we hen need he unc ional dependence o he
ene gy densi y on he addi ional mac oscopic a iables. In ou discussion we will always
assume, ha he sys em is close o he modynamic equilib ium. This allows us o expand
he ene gy densi y in e ms o he mac oscopic a iables and hei g adien s. In o de o
do so, we ha e o conside he cha ac e is ics o he ene gy densi y. The equilib ium s a e
is a s able s a e so ha we ha e o p o ide a con ex unc ional dependence. Fu he mo e
he ene gy densi y should be in a ian unde in e sion o space and ime, unde igid
ansla ion and igid o a ion and i should be co a ian upon Galilean ans o ma ion.
I he sys em is close o he mal equilib ium, i will y o achie e he equilib a ed
s a e by dynamical p ocesses. Fo he i s class o a iables he co esponding exp essions
can be de i ed easily exploi ing he ac , ha hey ep esen conse ed quan i ies in he
sys em. Le us assume a scala ield αwhich is he olume densi y o a conse ed quan i y
wi h i s co esponding lux densi y jα. Since he amoun o his pa icula quan i y in an
a bi a y olume Vis conse ed, empo al changes o his amoun ha e o be balanced
by a lux h ough he closed bounding su ace o ha olume. Ma hema ically speaking
one obse es
d
d Z
V
αdV =−I
∂V
jα·d (2.3)
whe e d ep esen s he su ace a ea elemen o he closed su ace1. Using Gauss’ heo em
one can ans o m he las exp ession in o i s local o m and ob ain he con inui y equa ion
o he mac oscopic a iable α
∂
∂ α+∇·jα= 0 (2.4)
1Th oughou his hesis ec o s a e displayed in bold and hei componen s using la in le e s as
indices, ∇deno es he ec o ∇= (∂x, ∂y, ∂z) and we will imply summa ion o e epea ed indices excep
o he wise s a ed. In he la e con ex δij is he K onecke symbol and ǫijk he Le i-Ce i `a enso .
12 Mac oscopic ma hema ical amewo k
Wha we a e le wi h is o ind an explici exp ession o he lux densi y jαwhich can be
cons uc ed as a powe se ies in e ms o he he modynamic o ces (usually he g adien s
o he conjuga ed ields), whe e he same symme y a gumen s ha e o be applied as in he
case o he ene gy densi y. Usually one can dis inguish wo di e en ypes o con ibu ions
o he cu en s: One con ibu ion ha accoun s o e e sible p ocesses p ese ing he
en opy densi y and one con ibu ion due o he i e e sible p ocesses ha lead o an
inc ease o en opy. Fo a se o mac oscopic a iables usually c oss-coupling con ibu ions
a e allowed, o example in a bina y luid mix u e an applied empe a u e g adien no
only causes a hea lux bu also a concen a ion lux. Onsage s a ed [61, 62, 63], ha in
hese cases he co esponding symme ic con ibu ions ha e o be p esen as well. Picking
up he las example, his co esponds o a hea lux caused by an applied concen a ion
g adien .
Fo he o he wo kind o a iables a s aigh o wa d de i a ion o he dynamic equa-
ions is no possible. Howe e one can assume a simila dynamical beha io balancing
he empo al change o he a iable wi h a so called quasi cu en
∂ β+Xβ= 0 (2.5)
The quasi cu en i sel can be cons uc ed in he same way as he cu en s o he
conse ed mac oscopic a iables.
2.1.3 The case o iso opic e ogels
We can now apply he hyd odynamic me hod discussed in he p e ious sec ion o he spe-
cial case o iso opic magne ic gels. The i s de i a ion o he gene alized hyd odynamic
equa ions was gi en by Ja ko a e al. [60]. We will ollow hei wo k and gi e hei esul s
needed o he discussion o he Rosensweig ins abili y.
The ene gy unc ional and he Gibbs ela ion
We s a wi h he iden i ica ion o he mac oscopic a iables. In he case o iso opic
magne ic gels, he i s class o a iables consis s o he mass densi y ρ, he momen um
densi y g, he ene gy densi y εand he concen a ion o he magne ic pa icles c. To
accoun o he elas ic deg ees o eedom we in oduce he elas ic s ain ield ǫij which
belongs o he second class o a iables. The s ain ield in amo phous solids is de i ed
om c ys als, whe e he long anged posi ional o de gi es ise o he displacemen ec o
ield uas a hyd odynamic symme y a iable. In ou desc ip ion we will es ic ou sel es
o linea elas ici y ǫij =1
2(∂iuj+∂jui). In usual e o luids he magne iza ion elaxes o he
equilib ium alue se by he ex e nal magne ic ield. The app op ia e elaxa ion ime is
much la ge han all he o he mic oscopic ime scales. The same is ue in magne ic gels.
The e o e he magne iza ion Mis aken as an addi ional mac oscopic a iable belonging
o he hi d class o mac oscopic a iables2. The ans o ma ion beha io unde ime ǫT
and spa ial ǫPin e sion is summa ized in able 2.1.
In he modynamic equilib ium all mac oscopic a iables a e elaxed o hei equilib-
ium alues and one inds he Gibbs ela ion ha ela es in ini esimal changes o he
2The dynamics o he Rosensweig ins abili y akes place on a ime scale la ge han he ime scale o
he dynamics o he magne iza ion. In ha special case i is su icien o exclude he magne iza ion again
om he mac oscopic dynamics as will be done in sec ion 2.2.1.
2.1 The basic hyd odynamic equa ions 13
mac oscopic a iable ime in e sion ǫTspa ial in e sion ǫP
ρ+1 +1
ε+1 +1
c+1 +1
gi−1−1
ǫij +1 +1
Mi−1 +1
Table 2.1: Table o he mac oscopic a iables impo an o he desc ip ion o
iso opic magne ic gels wi h hei ans o ma ion beha io unde ime and spa ial
in e sion
mac oscopic a iables o in ini esimal changes o he en opy densi y σ
dε =Tdσ +µdρ +µcdc + idgi+HidBi+hM
idMi+ Ψijdǫij (2.6)
The co esponding he modynamic conjuga ed ields a e he empe a u e T, he chemical
po en ial µ, he ela i e chemical po en ial µc, he eloci y , he magne ic molecula ield
hM
iand he elas ic s ess Ψij and a e de ined as pa ial de i a i es o he ene gy densi y
wi h espec o he app op ia e a iable whils he o he s a e kep cons an . The magne ic
lux densi y B oge he wi h he magne ic ield Hha e been in oduced o accoun o
he s a ic Maxwell equa ions in ou discussion.
In o de o gi e explici exp essions o he he modynamic conjuga ed a iables in-
oduced abo e and o de e mine he he modynamic o ces we ha e o gi e an explici
exp ession o he ene gy densi y. Assuming an expansion a ound he equilib ium alue
one inds
ε=ε0+1
2B2−B·M+1
2µijklǫijǫkl −1
2γijklMiMjǫkl +1
2αM2
i
+ǫii(χρδρ +χσδσ +χcδc) (2.7)
whe e ε0 ep esen s he ene gy densi y o a bina y luid mix u e. The coe icien αac-
coun s o he dependence o he induced magne iza ion on he s a e o he medium, o
example i s empe a u e. Addi ionally αis a unc ion o he applied magne ic ield mod-
eling he nonlinea magne iza ion beha io . In eq. (2.7) one can clea ly dis inguish he
con ibu ions due o he magne ic ene gy, he elas ic ene gy, he c oss coupling o he la -
e and he coupling be ween comp ession and he scala ield a iables. T unca ed a he
quad a ic o de , his expansion is only alid o small elas ic de o ma ions o he medium.
Fo la ge de o ma ions one should ex end he expansion o highe o de s o ǫij accoun ing
o nonlinea elas ic de o ma ions. The elas ici y enso µijkl and he magne os ic i e
enso γijkl ake he iso opic o m whe e we gi e, as an example, he elas ici y enso
µijkl =µ1δijδkl +µ2δikδkl +δilδjk −2
3δijδkl(2.8)
wi h i s wo in a ian s gi en by he elas ic comp essibili y µ1and he elas ic shea modulus
µ2.
20 Mac oscopic ma hema ical amewo k

Chap e 3
Recalling he linea p oblem
In his chap e we will ocus on he linea aspec s o he Rosensweig ins abili y in iso opic
magne ic gels. Pa s o his chap e can be unde s ood as a summa y o p e ious wo ks
[70, 20], howe e i will also p o ide an easy in oduc ion o he a he special na u e o
he kinema ic bounda y condi ion and he possible ma hema ical p oblems in he p esence
o dynamical de o mable su aces. This will help us in unde s anding he nonlinea egime
and especially he way we ea i ma hema ically.
3.1 The g ound s a e
The sys em o equa ions and bounda y condi ions (2.32-2.34) and (2.41-2.42) always has
he i ial g ound s a e solu ion, whe e he su ace is la (ξ(x, y, )≡0,n0=ez), low
and de o ma ions a e absen ( = 0, ǫij = 0), and he ields a e cons an (M0=M0ez
wi h M0= (1 −1/µ)B0). The con inui y equa ion (2.32) and he dynamic equa ion o
he s ain ield (2.34) a e hen sa is ied iden ically whe eas he Na ie -S okes equa ion
eads
∂jp0δij −B0iH0j+1
2B0kH0kδij=−ρGδiz (3.1)
Fo he la e al dimensions in x−and y−di ec ion hese equa ions a e easily sa is ied by
any p essu e p0=p0(z), which is ob ained by using i=zas
p0(z) = −ρGz +p0(z= 0) (3.2)
Fu he mo e we ha e o gua an ee a s ess ee su ace. The angen ial s ess bounda y
condi ions a e iden ically sa is ied o his g ound s a e solu ion, whe eas he no mal
bounda y condi ion eads
p0(z= 0) = −1
21−1
µB2
0(3.3)
which esul s upon subs i u ing in o (3.2) he inal g ound s a e p essu e
p0(z) = −ρGz −1
21−1
µB2
0(3.4)
21
22 Recalling he linea p oblem
The sys em o equa ions he e o e equi es a non-ze o, cons an s ess con ibu ion due
o he magne ic ield, −(1/2)(1 −1/µ)B2
0, o he hyd os a ic p essu e, which is o mino
ele ance, since in an incomp essible1sys em he p essu e has no physical meaning any-
mo e and me ely se es as an auxilia y quan i y ha gua an ees ∇· = 0 o all imes, i
low is p esen .
I is wo h men ioning he e, ha he g ound s a e is he only s a e whe e he g a i a-
ional o ce con ibu es ia he bulk equa ions. Fo he pe u bed s a es, he g a i a ion
only en e s he analysis ia he bounda y condi ions.
3.2 Linea de ia ions om he g ound s a e
Fo ini e empe a u es he sys em will be subjec o he mal luc ua ions which cause
he su ace o undula e andomly. These luc ua ions can be iewed as a spec um o
p opaga ing and damped su ace wa es (c . ig. 2.1, p. 18) wi h a wa e ec o k=
(kx, ky,0) and wi h he equency consis ing o a eal ωand an imagina y pa −σ
ξ(x, y, ) = ˆ
ξe−ikxx−ikyy+iω +σ (3.5)
and whe e ˆ
ξdeno es he ampli ude which is unde e mined in he linea heo y. In case
o ω= 0, a s a iona y spa ially pe iodic pa e n is ob ained. Gene ally ωis a complex
unc ion o k. Fou ie modes o he ype (3.5) can be supe imposed as app op ia e,
and de ia ions om he g ound s a e o all he o he a iables ha e o be p opo ional o
ξ(x, y, ). Linea de ia ions o he su ace no mal om he g ound s a e due o undula ions
a e gi en by2n(1) ≡n−n0= (−∂xξ, −∂yξ, 0).
The ac ha he sys ems o hyd odynamic bulk equa ions decouples om he mag-
ne ic bulk equa ions enables us o sol e he wo bulk sys ems sepa a ely. A de ailed
de i a ion o he magne ic ields can be ound in appendix B whe eas he e we jus e-
pea he inal esul s. The linea de ia ions o he magne ic ield and induc ion om he
g ound s a e alue, b(1) ≡B(1) −B0and h(1) =H(1) −H0, bo h o he e ogel and he
acuum, s ill obey he linea elec os a ic equa ions, b(1) =µh(1), di b(1) = 0 = cu lh(1).
This allows o he in oduc ion o a magne ic scala po en ial [69] h(1) =−∇Φ(1) ha is
de e mined by he Laplace equa ion wi h he app op ia e solu ions
Φ(1) =−M0
1 + µξ(x, y, )ekz (3.6)
Φ(1) ac =µM0
1 + µξ(x, y, )e−kz .(3.7)
o he lowe ( e ogel) and uppe ( acuum) hal plane, espec i ely and k2=k2
x+k2
y.
1In sec ion 2.2.1 we s ill allowed o a comp essible supe pa amagne ic medium. This assump ion is
necessa y o de i e he se o adjoin linea equa ions wi h i s co esponding bounda y condi ions as we
will see in sec ion 5.2. Fo he discussion o he Rosensweig ins abili y, howe e , we can sa ely assume an
incomp essible medium.
2The supe sc ip (1) is jus added o a consis en no a ion wi h chap e 5 and desc ibes he de ia ions
om he g ound s a e in linea o de .
3.3 Su ace wa e dispe sion ela ion 23
The sys em o hyd odynamic bulk equa ions eads in linea ized o m
ρ∂ (1)
i+∂ip(1) −ν2∂j(∂i (1)
j+∂j (1)
i)−2µ2∂jǫ(1)
ij = 0 (3.8)
∂ ǫ(1)
ij −1
2(∂i (1)
j+∂j (1)
i) = 0 (3.9)
∂i (1)
i= 0 (3.10)
The usual way o sol e his sys em is o dis inguish he i o a ional low con ibu ions
om he o a ional ones, (1) = (1)po + (1) o , whe e bo h pa s can be deduced om a
scala po en ial ϕ(1) and a ec o po en ial Ψ(1), espec i ely
(1)po =∇ϕ(1) and (1) o =∇×Ψ(1).(3.11)
The incomp essibili y o he medium equi es ∆ϕ(1) = 0 and leads o he ansa z
ϕ(1) = ˆϕ(1) ξ(x, y, )ekz (3.12)
o he scala eloci y po en ial. The ec o eloci y po en ial can be w i en as
Ψ(1) =ˆ
Ψ(1) ξ(x, y, )eqz (3.13)
whe e he ampli udes ˆϕ(1) and ˆ
Ψ(1) and he decay leng h q−1a e s ill unde e mined. Since
only wo o he h ee ampli udes ˆ
Ψ(1) can be independen , we se ˆ
Ψ(1)
z= 0 wi hou loss
o gene ali y esul ing in (1) o = (−qΨ(1)
y, qΨ(1)
x,−ikxΨ(1)
y+ikyΨ(1)
x).
The s ain ǫ(1)
ij can be exp essed by he eloci y ia eq. (3.9) and he linea p essu e
de ia ion, p(1) ≡p−p0is de e mined by eq. (3.8). Wi h he help o eq. (3.9), iω∂jǫ(1)
ij =
(1/2)∆ (1)
i, eq. (3.8) akes he linea o m
iωρ (1)
i+∂ip(1) −ν2+µ2
iω∆ (1)
i= 0 (3.14)
Taking di and cu l o eq. (3.14) we ge [71]
p(1) =−iωρϕ(1) + cons .(3.15)
and q2=k2−ρω2
µ2+iων2
(3.16)
espec i ely, whe e he unimpo an cons an in he p essu e can be igno ed.
3.3 Su ace wa e dispe sion ela ion
We a e le wi h h ee ampli udes, ˆϕ(1),ˆ
Ψ(1)
x,ˆ
Ψ(1)
y, ha ha e o be ela ed o he undula ion
ampli ude, ˆ
ξ, by he s ess bounda y condi ions (2.41) and (2.42) and he kinema ic
bounda y condi ion (2.44). Fo he linea analysis we could, wi hou loss o gene ali y,
choose he in-plane wa e ec o k o be pa allel o he x−axis, as done in [20]. Wi h
he nonlinea analysis in mind, i is wo h conside ing an a bi a y di ec ion o he wa e
ec o . Fo linea de ia ions om he g ound s a e and wi h he solu ions ob ained o
24 Recalling he linea p oblem
ω∝ik2
ω∝k
ω∝k3/2
ln k
kc
ω∝k1/2
ln µ2
ω∝k
ln k
ω∝k1/2
uns able
ln M
Figu e 3.1: Schema ic plo s o he di e en su ace wa e egimes desc ibed by
eq. (3.20) as in [71]. One encoun e s g a i a ional wa es wi h ω∝k1/2, Rayleigh
elas ic wa es wi h ω∝kand capilla y wa es wi h ω∝k3/2as depic ed on he
le . On he igh an uns able egion de elops o s ong magne ic ields.
he magne ic ields, he s ess bounda y condi ions can be w i en in e ms o he low
po en ials as (c . appendix C)
˜µ2(∂2
z−∂2
y)Ψ(1)
x+ ˜µ2(∂y∂x)Ψ(1)
y+ 2˜µ2∂y∂zϕ(1) = 0 (3.17)
˜µ2(∂x∂y)Ψ(1)
x+ ˜µ2(∂2
z−∂2
x)Ψ(1)
y−2˜µ2∂x∂zϕ(1) = 0 (3.18)
−(2˜µ2∂z∂y+Gρ∂y+σTk2∂y−µ
1 + µM2
0∂z∂y)Ψ(1)
x
+(2˜µ2∂z∂x+Gρ∂x+σTk2∂x−µ
1 + µM2
0∂z∂x)Ψ(1)
y
+(2˜µ2∂2
z+Gρ∂z+σTk2∂z−ρω2−µ
1 + µM2
0∂2
z)ϕ(1) = 0 (3.19)
all aken a z= 0 and wi h he equency dependen ˜µ2(ω)≡µ2+iων2desc ibing
(kinema ic) elas ici y and iscosi y.
To ha e a non i ial solu ion o equa ions (3.17-3.19) he de e minan o coe icien s
mus anish. This leads o he dispe sion ela ion o su ace wa es o e ogels
ρω22˜µ2(ω)k2−ρω2+ρω2σTk3+ρGk + 2˜µ2(ω)k2−µ
1 + µM2
0k2
−4˜µ2
2(ω)k4"1−1−ρω2
˜µ2(ω)k21/2#= 0 (3.20)
In he absence o an ex e nal magne ic ield (M0= 0) eq. (3.20) educes o he dispe sion
ela ion o non-magne ic gels [71]. I also con ains, as a special case, he su ace wa e
dispe sion ela ion o e o luids (in an ex e nal ield) by choosing ˜µ2=iων2. I can
be gene alized o iscoelas ic e o luids, whose elas ici y elaxes on a ime scale τ−1, by
eplacing µ2wi h iωτµ2/(1 + iωτ) [71].
The dispe sion ela ion (3.20) is e y complica ed and i is impossible o sol e i
analy ically o ω(k). Fo non-magne ic gels i is known ha he e a e basically h ee
wa e egimes (neglec ing dissipa ion o damping) (c . ig. 3.1): ρω2=σTk3(capilla y
wa es), ρω2= ˜αµ2k2(Rayleigh elas ic wa es), and ω2=Gk (g a i y wa e wa es) o
3.4 Rosensweig ins abili y 25
small wa eleng hs (k≫µ2/σT,pρG/σT), in e media e ones (ρG/µ2≪k≪µ2/σT),
and la ge ones (k≪ρG/µ2,pρG/σT), espec i ely, whe e ˜αis a numbe o o de uni y.
Fo ypical ma e ial alues (µ2≈1 kPa, σT≈0.02 kg/sec2) wa es a wa eleng hs o
10−4m and below (wi h equencies o 50 kHz and abo e) a e o pu ely capilla y ype,
while o wa eleng hs abo e 1 m (and equencies below 10 Hz) he g a i y cha ac e
domina es; his egime is, hus, i ele an o usual e ogel samples. In be ween, o
ypical wa eleng hs o 10−2m and equencies o 100 - 1000 Hz he elas ic na u e o he
wa e is p e ailing. This scena io also applies o iso opic e ogels in he absence o a
ield. The e ec o a no mal ex e nal magne ic ield on he su ace is a des abilizing one
[3]. F om eq. (3.20) i is e iden ha an ex e nal ield leads o an e ec i e educ ion o
he su ace s i ness (p o ided by su ace ension, g a i y o elas ici y) and dec eases he
equency (squa ed) o he p opaga ing wa es in all egimes by ∼M2
0k2. I he ield is
la ge enough, his educ ion is he domina ing e ec and can lead o ω= 0 and hus, o
he b eakdown o p opaga ing wa es. In he nex sec ion i is shown ha his is indeed
ela ed o he Rosensweig ins abili y.
3.4 Rosensweig ins abili y
As men ioned al eady, eq. (3.20) is a complica ed ela ion be ween he equency o a
su ace wa e implici ly gi en as a unc ion o i s wa e ec o . Fo a be e unde s anding
o he Rosensweig ins abili y i is wo h conside ing he simpli ica ion o eq. (3.20) o he
case o an in iscid (ν2= 0) magne ic luid
ρω2=ρGk −µ
1 + µM2
0k2+σTk3(3.21)
as has been done in [3, 19]. This is no a physical assump ions which we will ha e o
co ec la e on, bu i al eady e eals he s a ic na u e o he Rosensweig ins abili y
and he esul ing dispe sion ela ion (3.21) can be ea ed analy ically. Upon minimizing
(3.21) wi h espec o he equency ωand he wa e ec o kone s aigh o wa dly ob ains
he linea h eshold o a s a ic ins abili y (ω≡0)
M2
c= 21 + µ
µpρGσT(3.22)
beyond which he la su ace is uns able wi h espec o pe iodic pa e ns wi h a cha -
ac e is ic wa e ec o (c . ig. 3.1)
kc= ρG
σT
(3.23)
The ac ha he ins abili y is s a ic is, howe e , a he singula . This can be obse ed
by plo ing (3.21) as is shown in ig. 3.2 (wi h he ma e ial p ope ies aken om he
comme cial e o luid EMG 901 as gi en o example in [72]). Wi hou a magne ic ield
one ob ains he beha io o an ideal luid whe e he g a i a ional wa e egime can be dis-
inguished e y clea ly as well as he ansi ion o he capilla y egime. Upon inc easing
he magne ic ield, he con ibu ion due o he magne iza ion becomes mo e and mo e
dominan and a a magne iza ion o abou 6600A/m a b anch o anomalous dispe sion
a ises. Mo e in e es ing o he Rosensweig ins abili y is he necessa y minimum ha

26 Recalling he linea p oblem
M= 0 A/m
M= 6100 A/m
M= 6600 A/m
M= 6800 A/m
M= 6931 A/m
k[m−1]
ω2[Hz2]
10008006004002000
8000
7000
6000
5000
4000
3000
2000
1000
0
Figu e 3.2: The dispe sion ela ion o su ace wa es in a usual magne ic luid o
di e en alues o he ex e nally applied magne ic ield. The nume ical pa ame e
alues a e aken om he e o luid EMG 901 (ρ= 1.53·103kg
m3,σT= 29.5·10−3N
m
and µ= 28.0·10−7T
m).
comes along wi h i . This ela i e minimum is shi ed o lowe ω alues as he ex e nal
magne ic ield is inc eased and e en ually i becomes he absolu e one ouching he ab-
scissa a he cha ac e is ic wa e ec o o he c i ical magne ic ield. In physical e ms we
can in e p e his in a sligh ly di e en way. Below he c i ical magne ic ield all modes
show a ini e oscilla ion in ime. As soon as we app oach he c i ical magne ic ield he
oscilla ion o he cha ac e is ic mode dies ou esul ing in he g ow h o a s a ic pa e n.
The in e p e a ion o he Rosensweig ins abili y as he limi ing case o su ace wa es wi h
a anishing equency will be o impo ance o he nonlinea egime as we will see la e .
Wi h he same p ocedu e we can discuss he dispe sion ela ion o magne ic gels
(3.20). The cha ac e is ic mode u ns ou o be he same as o he case o e o luids,
gi en by he capilla y mode (3.23). This can be unde s ood ecalling he ac ha he
elas ic con ibu ions en e he dispe sion ela ion wi h he same k−o de as he magne ic
con ibu ions. The c i ical magne ic ield is ins ead shi ed owa ds highe magne ic ields
[20] acco ding o
M2
c= 21 + µ
µpρGσT+µ2(3.24)
which is less su p ising, since elas ici y inc eases he su ace s i ness. In he special cases
o ealis ic e o luids wi h a ini e iscosi y (µ2= 0 and ν26= 0) and o e o ubbe s
(ν2= 0 and µ26= 0) i can be shown analy ically ha only a s a ic ins abili y ω= 0 is
possible a he linea onse [70, 20]. Fo he gene al case (3.20) o ealis ic magne ic gels
wi h a ini e iscosi y and a ini e shea modulus nume ical calcula ions show he absence
o an oscilla o y ins abili y a onse .
Also in he gene al case o e ogels, he s a ic cha ac e o he ins abili y is he limi ing
case o dynamical p ocesses a he su ace. To illus a e his, we can plo he dispe sion
3.5 The linea eigen ec o s 27
µ2= 2 Pa
µ2= 1 Pa
µ2= 0 Pa
k[m−1]
ω2[Hz2]
10008006004002000
8000
7000
6000
5000
4000
3000
2000
1000
0
Figu e 3.3: The dispe sion ela ion o su ace wa es on an iso opic magne ic
gel a a ixed ex e nal magne ic ield o 0.98Mc i .and o di e en alues o he
elas ic shea modulus. The nume ical pa ame e s o he o he ma e ial p ope ies
a e aken om he e o luid EMG 901 (ρ= 1.53 ·103kg
m3,σT= 29.5·10−3N
m,
µ= 28.0·10−7T
mand ν2= 6.5·10−6m2
s).
ela ion in he gene al iscoelas ic case ( ig. 3.3). The di e en g aphs ha e been ob ained
as nume ical solu ions o (3.20) whe e he ma e ial pa ame e s a e again aken om he
e o luid EMG 901 wi h a a ying alue o he elas ic shea modulus µ2and whe e he
magne ic ield is kep cons an a abou 98% o he c i ical magne ic ield. Due o he
nume ical esolu ion we only had access o elas ic moduli up o 2 Pa. Howe e , i emains
illus a i e ha he elas ic con ibu ions ac opposi e o he magne ic ield and ha he
limi ing case o a s a ic ins abili y is app oached ei he o inc easing magne ic ields o
o dec easing shea moduli. An elas ic shea modulus o 2 Pa is ex emely weak, bu
ig. 3.3 addi ionally illus a es ha al eady a low shea modulus in luences he dynamic
beha io o he su ace d as ically close o he linea h eshold.
3.5 The linea eigen ec o s
The condi ion o ind a solu ion o he linea ized se o hyd odynamic equa ions has
been discussed in he p e ious wo sec ions. A solu ion can be ound by i s conside ing
he wo angen ial s ess bounda y condi ions (3.17,3.18) which allow one o exp ess he
ampli udes o he ec o po en ial Ψ(1) in e ms o he ampli ude o he scala po en ial
ˆ
Ψ(1)
x=−2k(−iky)
q2+k2ˆϕ(1) and ˆ
Ψ(1)
y= +2k(−ikx)
q2+k2ˆϕ(1) (3.25)
The kinema ic bounda y condi ion can hen be used o ind he explici exp ession o
he ampli ude o he scala po en ial
ˆϕ(1) =iω q2+k2
k(q2−k2).(3.26)
28 Recalling he linea p oblem
In he limi ing case o ω→0, which co esponds o he app oach o he onse o he
Rosensweig ins abili y i k=kc, he ampli udes o he po en ials di e ge wi h 1/ω. Since,
howe e , he po en ials a e a ma hema ical ool o sol e he sys em o equa ions, we need
no wo y abou ha , ye .
Subs i u ing he exp ession o he po en ials in o eq. (3.11), we e en ually ob ain he
h ee componen s o he eloci y ield
(1)
x=X
i
(−ikix)ekiz−2qiki
q2
i+k2
i
eqiziω(q2
i+k2
i)
ki(q2
i−k2
i)ξi(3.27)
(1)
y=X
i
(−ikiy)ekiz−2qiki
q2
i+k2
i
eqiziω(q2
i+k2
i)
ki(q2
i−k2
i)ξi(3.28)
(1)
z=X
i
kiekiz−2k2
i
q2
i+k2
i
eqiziω(q2
i+k2
i)
ki(q2
i−k2
i)ξi,(3.29)
whe e he index iaccoun s o he ac ha in a linea analysis o he ins abili y he
di ec ion o he uns able mode emains degene a e and a whole se o cha ac e is ic modes
wi h di e en di ec ions may g ow.
We can exploi eq. (3.9) and e en ually ind he exp essions o he componen s o
he s ain enso , whe e again he index iaccoun s o he di e en possible modes o he
same modulus
ǫ(1)
zz =X
i
(q2
i+k2
i)ekiz−2qikieqiz
q2
i−k2
i
kiξi(3.30)
ǫ(1)
xz =X
i
(−ikix)ekiz−eqizq2
i+k2
i
q2
i−k2
i
ξi(3.31)
ǫ(1)
yz =X
i
(−ikiy)ekiz−eqizq2
i+k2
i
q2
i−k2
i
ξi(3.32)
ǫ(1)
xy =−X
i
kixkiy
q2
i+k2
i
ki(q2
i−k2
i)ekiz−2qiki
q2
i+k2
i
eqizξi(3.33)
ǫ(1)
xx =−X
i
k2
ix
q2
i+k2
i
ki(q2
i−k2
i)ekiz−2qiki
q2
i+k2
i
eqizξi(3.34)
ǫ(1)
yy =−X
i
k2
iy
q2
i+k2
i
ki(q2
i−k2
i)ekiz−2qiki
q2
i+k2
i
eqizξi.(3.35)
We ealized p e iously, ha he in oduced po en ials di e ge when app oaching he linea
onse . The eloci y and s ain ield, as he obse ables o he sys em, should ins ead
acqui e a physical solu ion in he limi o a s a iona y ins abili y. Taking he limi ω→0,
3.5 The linea eigen ec o s 29
k= 2
k= 1
s ain ǫzz
dep h z
0.80.70.60.50.40.30.20.10
0
-2
-4
-6
-8
-10
Figu e 3.4: The absolu e alue o he zz−componen o he s ain ield o wo
pa icula modes, k= 1 and k= 2 espec i ely, acco ding o (3.37) as a unc ion
o dep h and o an a bi a y in ini esimal de lec ion o he su ace a a gi en
poin (x, y). The su ace modes a e measu ed in uni s o he cha ac e is ic wa e
ec o kc, leng hs in uni s o i s in e se, k−1
c.
we ob ain he ollowing eigen ec o s a he linea onse
(1)
i= 0 (3.36)
ǫ(1)
zz =−X
i
k2
izekizξi(ω= 0) (3.37)
ǫ(1)
xz =−X
i
(−ikix)kizekizξi(ω= 0) (3.38)
ǫ(1)
yz =−X
i
(−ikiy)kizekizξi(ω= 0) (3.39)
ǫ(1)
xy =X
i
kixkiyzekizξi(ω= 0) (3.40)
ǫ(1)
xx =X
i
k2
ixzekizξi(ω= 0) (3.41)
ǫ(1)
yy =X
i
k2
iyzekizξi(ω= 0) (3.42)
The eloci y ield anishes iden ically whe eas he s ain ield acqui es a ini e s a iona y
alue. Fig. 3.4 gi es he s ain (and co espondingly he s ess) dis ibu ion in he medium
o wo di e en su ace modes as a unc ion o he dep h s a ing om he su ace. The
wa e ec o s a e measu ed in uni s o he cha ac e is ic wa e ec o kc(3.23) and he
dep h is measu ed in uni s o he in e se cha ac e is ic wa e ec o k−1
c.
A his poin , he c ucial di e ence be ween he Rosensweig ins abili y and o he
commonly discussed ins abili ies becomes ob ious. Media ed by he kinema ic bounda y
condi ion, he eloci y ield anishes. Wha one obse es is no a s a iona y ini e low ield
36 Nonlinea discussion using he ene gy me hod
µ2= 0.100
µ2= 0.010
µ2= 0.001
µ2= 0.000
magne ic suscep ibili y χ
con ol pa ame e ˜ǫ
32.521.510.50
1
0.1
0.01
0.001
Figu e 4.2: G aphs sepa a ing egions in he χ−˜ǫ−plane in which he ampli udes
o squa es a e smalle o highe han 0.25 o di e en alues o he elas ic shea
modulus µ2. The c i ical magne ic suscep ibili y is enhanced wi h highe shea
modulus.
whe e we ha e again neglec ed ˜ǫin he denomina o o he ou h o de e ms. Minimizing
eq. (4.22) leads o he ampli ude AS=AS(˜ǫ, µ2, η)
AS=s4(˜ǫ−µ2)(1 + 4µ2)3−2√2(1 −µ2)
NS
(4.23)
wi h he denomina o NSgi en by
NS= (1 + 4µ2)74√2−103 + (26√2−64)µ2
+16η212 −11√2 + (48 −46√2)µ2+ 16(2√2−3)µ2
2(4.24)
Ob iously ˜ǫ−µ2>0 is a necessa y condi ion o he s abili y o squa e solu ions.
In ig. 4.2 he alue o he con ol pa ame e ˜ǫis plo ed as a unc ion o he magne ic
suscep ibili y χ o AS= 0.25 and di e en alues o he shea modulus µ2. The g aphs
sepa a e con igu a ions in he pa ame e space wi h ampli udes smalle (below he cu e)
and highe (abo e he cu e) han AS= 0.25 and he di e gence o ini e magne ic
suscep ibili ies indica es ha o a in ini esimal small con ol pa ame e ˜ǫ he ampli udes
a e al eady in ini ely la ge. In [34], o µ2= 0, he plo has been used o es ima e
he maximum magne ic suscep ibili y a which he me hod di e ges. As can be seen in
ig. 4.2, o ini e shea modulus he alidi y ange o he me hod is inc eased o la ge
magne ic suscep ibili ies. Fo hexagons (c . sec. 4.3.3) a simila esul is ob ained. The
plo addi ionally illus a es he ac al eady s a ed by Gaili is himsel ha his ene gy
me hod is igo ously alid only in he limi o a anishing magne ic suscep ibili y.
Following he me hod o Gaili is [33] i.e. s a ing wi h a mo e gene al ansa z o he
wa e ec o s ha include s ipes as a special case, we ind ha e en o µ2>0 s ipes
a e always uns able wi h espec o squa es.

4.3 S abili y o di e en Geome ies 37
µ2= 0.100
µ2= 0.010
µ2= 0.001
µ2= 0.000
magne ic suscep ibili y χ
con ol pa ame e ˜ǫ
21.510.50
1
0.1
0.01
0.001
Figu e 4.3: G aphs sepa a ing egions in he χ−˜ǫ−plane in which he ampli udes
o hexagons a e smalle o highe han 0.25 o di e en alues o he elas ic
shea modulus µ2. The c i ical magne ic suscep ibili y is enhanced wi h highe
shea modulus.
4.3.3 Hexagons
We now discuss a egula hexagonal pa e n gene a ed by he h ee main wa e ec o s
wi h angles o 2π/3 (c . ig. 4.1). The di e ence in su ace ene gy densi y wi h espec o
he la su ace, eq. (4.14), becomes
U=−1
2(˜ǫ−µ2)(A2
1+A2
2+A2
3)−3
4ηA1A2A3+1
45
16 −η2
1 + 4µ2(A4
1+A4
2+A4
3)
+"−15
32 +3
8√3 + 11
8−7√3
8!η2−3
4−√32η2
1−√32+ 2√3µ2#(A2
1A2
2+A2
2A2
3+A2
1A2
3)
(4.25)
In he ollowing we will e e o he exp essions w i en in he i s and second squa e
b acke as β(0, µ2) and β(2π/3, µ2), espec i ely, since in he limi o anishing elas ici y
hey a e iden ical o he ones gi en by Gaili is [33]. Fo he same eason we will also e e
o 3/4ηas γ.
Fo he egula hexagonal pa e n we ob ain om (4.25) by minimiza ion
AH=A1=A2=A3=γ±pγ2+ 4(˜ǫ−µ2)[β(0, µ2) + 4β(2π/3, µ2)]
2[β(0, µ2) + 4β(2π/3, µ2)] (4.26)
The hexagonal solu ions exis only i he squa e oo in eq. (4.26) is eal, and hey a e
s able, i he second de i a i e o eq. (4.25) wi h espec o he ampli udes is nega i e,
leading o he condi ions
−1
4[β(0, µ2) + 4β(2π/3, µ2)] <˜ǫ−µ2
γ2<2β(2π/3, µ2) + β(0, µ2)
[2β(2π/3, µ2)−β(0, µ2)]2(4.27)
38 Nonlinea discussion using he ene gy me hod
˜ǫ−µ2
AH, AS
AS
AS
AH
AH
Figu e 4.4: Quali a i e ske ch (no o scale) o he e olu ion o he ampli udes
o squa es (AS) and hexagons (AH). The dashed lines co espond o he case o
a e o luid while he solid lines quali a i ely desc ibe he beha io o e ogels
wi h ini e shea modulus µ2. The do ed lines ep esen he ene ge ically uns able
b anches.
Since he β- alues a e posi i e (a leas o χ≤1), hexagons can exis al eady below he
linea h eshold ˜ǫ=µ2. This exis ence ange sh inks, howe e , o e ogels compa ed o
e o luids, since β(θij, µ2)> β(θij,0) o bo h θij = 2π/3 and 0. Fo he same eason
he ampli ude o he hexagonal pa e n (4.26) dec eases wi h inc easing elas ic modulus.
Clea ly, elas ici y s abilizes a sys em agains he Rosensweig ins abili y, which is mani es
no only in an inc ease o he (linea ) h eshold, bu also in a dec ease o he spike heigh .
One can discuss he magni ude o he ampli udes as a unc ion o he pa ame e s ˜ǫand
χ o he hexagonal solu ion in he same way as done o he squa e solu ion in sec ion
4.3.2. Fig. 4.3 p esen s he co esponding plo o hexagons wi h he same quali a i e
esul , ha wi h inc easing shea modulus he alidi y ange o his me hod is ex ended
owa ds highe magne ic suscep ibili ies.
Following he me hod o Gaili is supe posing hexagons and squa es, we in es iga e he
ela i e s abili y o hexagons and squa es. We ind ha squa es a e uns able wi h espec
o hexagons unde he condi ion
˜ǫ−µ2
γ2<β(0, µ2) + 2β(π/2, µ2)
[2β(2π/3, µ2) + 2β(π/6, µ2)−2β(π/2, µ2)−β(0, µ2)]2,(4.28)
which is jus Gaili is’ exp ession, bu wi h µ2dependen β-abb e ia ions
β(π/2, µ2) = −9
16 +1
√2+15 −13√2−4(3 −2√2)µ2
6−4√2 + 4√2µ2
η2(4.29)
β(π/6, µ2) = 3
32(4√6−7) + 116 −41√6−(64 −28√6)µ2
16(2 −√6−2µ2)η2(4.30)
Figu e 4.4 shows he magni ude o he ampli udes as a unc ion o he con ol pa-
ame e o a ini e and a anishing shea modulus, espec i ely. We no e a dec ease in
size o he hys e e ic egion ( o nega i e ˜ǫ−µ2) wi h inc easing shea modulus. While
in he case o no elas ici y he lowe bounda y is a −0.25, i is shi ed o −0.24 o a
4.4 Some d awbacks o he ene gy me hod 39
shea modulus o µ2= 0.1 (bo h alues aken o a magne ic suscep ibili y o χ= 0.1)2.
The second hys e e ic egion o he ansi ion be ween squa es and hexagons also sh inks
wi h inc easing µ2. Fo ins ance, he lowe bounda y o he hys e esis loop a 5.7 ( o he
igh hand side o eq. (4.28)) o µ2= 0 inc eases o 6.9 o µ2= 0.1, while he uppe
bounda y o he hys e esis loop a 540 o µ2= 0 is educed o 480 o µ2= 0.1 (χ= 0.1).
This esul should be expe imen ally de ec able, a leas quali a i ely.
4.4 Some d awbacks o he ene gy me hod
One o he majo d awbacks o his me hod is, ha i is only alid in he asymp o ic
limi o a anishing magne ic suscep ibili y χ. This can be easily ealized by inspec ion o
ig. 4.2. Fo high enough magne ic suscep ibili ies he ampli ude di e ges o a bi a ily
small con ol pa ame e s ˜ǫ. This is why we had o scale he s abili y bounda ies gi en in
eqs. (4.27,4.28) wi h γin o de o compensa e he di e gence. Fo a nonlinea heo y o
magne ic luids his alidi y limi is no sa is ying.
The me hod elies on he s a ic ene ge ic compa ison o di e en su ace de o ma ions
wi h espec o each o he . I allows he de e mina ion o possible su ace pa e ns bu
i does no ell us which pa e n will be inally achie ed. The selec ion p ocess o a
eal pa e n is addi ionally go e ned by dissipa i e p ocesses. The Rosensweig ins abili y,
howe e , di e s om o he ins abili ies in ha i s inal s a e is s a ic and no dissipa ion
occu s. Bu du ing he g ow h o he su ace spikes ene gy is dissipa ed and his may
play a ole in he selec ion p ocess. Addi ionally we ealized in he discussions so a , ha
he occu ence o s a ic su ace spikes is he limi o a p e iously dynamic su ace mode
ha eezes in i s dynamics. Fu he mo e, as soon as he con ol pa ame e is beyond i s
c i ical alue and as long as he inal s a e is no achie ed, he sys em is ou o equilib ium
and he de ini ion o he po en ial (as is he ene gy) is no s aigh o wa d.
As men ioned al eady, he ene gy me hod gi es us i s hin s o become amilia wi h
he Rosensweig phenomenon. Wha we would like o ha e ins ead is a ull weakly nonlinea
analysis o he basic hyd odynamic equa ions ha also cap u es he dynamical p ocesses
du ing g ow h and ha addi ionally conside s dissipa ion. The de i a ion o a dynamic
ampli ude equa ion ha may sol e he add essed p oblems will be he subjec o he
ollowing chap e 5.
2Recall ha we in oduced dimensionless uni s on page 32.
40 Nonlinea discussion using he ene gy me hod
Chap e 5
The ampli ude equa ion
The physical, chemical, and biological sys ems [. . . ] a e o en qui e
complica ed and he equa ions and bounda y condi ions desc ibing
hem a e no always known p ecisely. E en when hey a e known,
as is he case o many hyd odynamic ins abili ies, a linea anal-
ysis al eady equi es nume ical e alua ion and a di ec analy ical
app oach is impossible beyond h eshold. The pe u ba ion me h-
ods desc ibed below a e a pa ial esponse o his si ua ion, hough
calcula ion o he app op ia e coe icien s can be di icul e en i
he s a ing equa ions a e known p ecisely.
M. C. C oss and P. C. Hohenbe g [48]
The discussion om he p e ious chap e s p o ides us wi h a i s unde s anding o he
Rosensweig ins abili y. Chap e 3 augh us, ha we should in e p e he s a iona i y
o he no mal ield ins abili y a he as a limi ing p ocess whe e he equency o he
cha ac e is ic mode anishes. In chap e 4 we discussed he ene ge ically a o ed su ace
pa e ns, bu we ealized some p oblems wi h he ene gy me hod. In his chap e we will
discuss he nonlinea egime s a ing om he undamen al hyd odynamic equa ions and
use an ǫ−expansion o access he weakly nonlinea egime1. In his con ex ǫdeno es he
no malized di e ence be ween he ac ually applied magne ic ield and i s c i ical alue.
The in o ma ion we ob ained so a will be o g ea impo ance on how we inally access
his egime.
5.1 In oduc ion
We will pe o m a weakly nonlinea analysis o he basic se o hyd odynamic equa-
ions as pionee ed by Schl¨
u e , Lo z and Busse [38] and Newell and Whi ehead [47]
o he Rayleigh-B´ena d sys em o ob ain a se o ampli ude equa ions in he case o he
Rosensweig ins abili y. The weakly nonlinea analysis is a pe u ba i e app oach and es s
on he assump ion ha close o he linea h eshold he nonlinea s a e can be exp essed
by a small pe u ba ion om he g ound s a e which is expanded in e ms o he con ol
pa ame e ǫ. The lowes o de in ǫde e mines he linea s abili y as al eady discussed
in chap e 3. The ampli udes o hese dis u bances, which a e s ill unde e mined in he
1Pa s o his chap e ha e been published in [74], o he s a e p epa ed o publica ion [75].
41

42 The ampli ude equa ion
linea pe u ba i e o de (c . chap e 3), ha e o ul ill ce ain equa ions in he highe
pe u ba i e o de s o gua an ee he sol abili y o he nonlinea hyd odynamic equa ions.
These ampli ude equa ions in u n a e nonlinea dynamic di e en ial equa ions desc ibing
he coope a i e dynamics o a se o c i ical modes subjec o nonlinea in e ac ions. In
his in oduc o y sec ion we will in oduce his me hod and will ocus on key p oblems
one encoun e s when applying his me hod o he Rosensweig ins abili y. Fo a comp e-
hensi e in oduc ion o he weakly nonlinea analysis and on ampli ude equa ions, he
eade is e e ed o [48, 76, 77].
5.1.1 Nonlinea expansion
Pe o ming a weakly nonlinea analysis o he s a iona y s a e e ol ing sligh ly beyond
he linea h eshold Mc, we ha e o expand he mac oscopic a iables in e ms o ǫ, he
no malized di e ence be ween he ac ual applied magne ic ield and he c i ical one
{p, B,H,M}={p0,Bc,Hc,Mc}+ǫ{p(1),B(1),H(1),M(1)}+... (5.1)
{ , ǫij, ξ}= 0 + ǫ{ (1), ǫ(1)
ij , ξ(1)}+... (5.2)
The magne ic ield, howe e , is an ex e nally gi en pa ame e ac ing as he con ol pa-
ame e . The se ies expansion o H(5.1) can he e o e be ein e p e ed as he de ini ion
o ǫ. No e, ha his de ini ion o ǫis no he same de ini ion as used in chap e 4, which
is supposed o be he app op ia e de ini ion in he case o he Rosensweig ins abili y. We
will see la e ha he de ini ion o ǫused he e leads consis en ly o he con ol pa ame e
as used in he p e ious sec ions.
In ou linea discussion (chap e 3), we modeled he su ace de lec ion ξ(x, y, ) us-
ing plane wa es ξ(x, y, ) = ˆ
ξeiω −ik· , whe e he ampli ude ˆ
ξis a common ac o in all
con ibu ions o he linea ized hyd odynamic equa ions and he e o e emains unde e -
mined. In a nonlinea discussion, we ha e o ex end his ansa z o add ess he possibili y
o nonlinea in e ac ions be ween a se o c i ical modes. The mos gene al ansa z as
a s a ing poin o a nonlinea discussion is o assume No hese cha ac e is ic modes
wi h di e en o ien a ions. Each o hese modes iconsis s o a igh and a le a eling
con ibu ion deno ed by he subsc ip s Rand L, espec i ely. Since he su ace de lec ion
as an obse able has o be eal, we ha e o add he co esponding complex conjuga e
which is deno ed by an as e isk
ξ(1) =
N
X
i
ξiR +ξiL +ξ∗
iR +ξ∗
iL
=
N
X
i
ˆ
ξiReiωi −iki· +ˆ
ξiLe−iωi −iki· +ˆ
ξ∗
iRe−iωi +iki· +ˆ
ξ∗
iLeiωi +iki· (5.3)
Besides he expansion o he hyd odynamic a iables, we can also conside o escale
ime and space in o de o u he sepa a e he dynamics o cap u e he long wa eleng h
and he long ime scale coope a i e dynamics o he indi idual cha ac e is ic modes. One
can in e p e e his escaling in he sense ha we pe mi he ampli ude ˆ
ξ o be slowly
dependen on ime2and space, espec i ely. In he p esen discussion we will disca d he
2The ime as used in eq. (5.3) is hen conside ed he “ as ” ime scale o he su ace wa es (o o
5.1 In oduc ion 43
possible escaling o space and ocus on su ace pa e ns ha a ise homogeneously and
ha do no show any long wa eleng h a ia ion. As men ioned al eady, he escaling o
ime can be in e p e ed in he sense ha he dynamics o he ampli udes ˆ
ξi sel akes
place on he slow imescales
(1) =ǫ and (2) =ǫ2 (5.4)
so ha ˆ
ξiR →ˆ
ξiR( (1), (2),...) and co espondingly o he le a eling con ibu ions
and hei co esponding complex conjuga es. This escaling o ime will lead o he
subs i u ion o he ime de i a i e
∂ −→ ∂(0)
+ǫ∂(1)
+ǫ2∂(2)
+... (5.5)
These a e, as we will see la e , he ime scales o he g ow h o he su ace spikes.
5.1.2 The sol abili y condi ion o highe o de s
F edholm’s heo em and he adjoin sys em
Wi h he escaling o ime and he expansion o he mac oscopic a iables in e ms o ǫ,
he whole sys em o di e en ial equa ions can be expanded in e ms o ǫ. Le L0be he
linea di e en ial ope a o and |φi=|φ(0)i+ǫ|φ(1)i+... he mac oscopic s a e ec o .
The basic hyd odynamic equa ions as gi en by eqs. (2.32-2.34) hen ead in gene al o m
L0|φ(1)i= 0 (5.6)
L0|φ(2)i=|N(φ(1), φ(1))i+|T(∂(1)
φ(1))i(5.7)
.
.
. = .
.
.
whe e e e y o de in ǫneeds o be sa is ied sepa a ely.
The i s equa ion (5.6) ep esen s he linea ized se o equa ions as used in he linea
s abili y analysis o chap e 3, whe e he explici ansla ion o equa ion (5.6) is gi en by
eqs. (3.8-3.10). Fu he mo e, eq. (5.6) de ines he ke nel o he linea ope a o L0, gi en
by he linea eigen ec o s |φ(1)i. In he second pe u ba i e o de he se o equa ions
(5.7) becomes inhomogeneous due o he nonlinea na u e o he basic se o equa ions
( ep esen ed by N(·,·)) and due o he escaling o ime ( ep esen ed by T(·)). In he
case ha hese inhomogenei ies ep oduce elemen s o he ke nel o he linea ope a o
L0, equa ion (5.7) canno be sol ed. The necessa y condi ion ha he inhomogenei ies
need o be o hogonal o he subspace spanned by he linea eigen ec o s |φip o ides
us wi h an addi ional sol abili y condi ion. This condi ion is named a e F edholm and
eads o he second o de
hφ| N(φ(1), φ(1))i+hφ| T(∂(1)
φ(1))i= 0 (5.8)
whe e h·|·i deno es a sui able scala p oduc abou which we will alk sho ly.
gene al he modynamic luc ua ions in he case o o he hyd odynamic ins abili ies) e en hough i is
al eady a mac oscopic ime scale. The dynamics o he ampli udes ˆ
ξis in u n assumed o ake place on
an e en slowe ime scale. The same a gumen s hold i we escale spa ial coo dina es.
44 The ampli ude equa ion
ǫ
AH, AS
AS
AH
ǫAǫSǫB
Figu e 5.1: The gene al bi u ca ion scheme o ampli ude equa ions o he o m
(5.9) acco ding o [78, 48]. The analy ical exp essions o he limi s o s abili y
ǫA,ǫBand ǫSa e gi en in he main ex . The solid lines ep esen he s able
b anches whe eas he do ed lines co espond o he uns able ones.
Ampli ude equa ions in gene al
The sol abili y condi ion (5.8) p o ides an addi ional equa ion and he s ill unde -de e -
mined sys em o equa ions is closed o ix he ampli ude ˆ
ξ. I his condi ion, alid in
he second o de , is combined wi h he co esponding condi ion in he hi d o de , one
ob ains he so called ampli ude equa ions which de ine possible nonlinea solu ions o
he ampli udes o he c i ical modes ha a e necessa y o sa is y he basic hyd odynamic
equa ions. In he absence o he in e sion symme y ˆ
ξi−→ −ˆ
ξi(which is he case o
he Rosensweig ins abili y), he usual s uc u e o his ampli ude equa ion is, w i en in
app op ia e uni s, gi en by [48]
∂ ˆ
ξ1=ǫˆ
ξ1−ˆγˆ
ξ∗
2ˆ
ξ∗
3−h|ˆ
ξ1|2+g1(θij)|ˆ
ξ2|2+|ˆ
ξ3|2iˆ
ξ1(5.9)
oge he wi h i s cyclic pe mu a ions 1 →2→3→1 and whe e θij deno es he angle
be ween wo di e en c i ical modes ha become uns able a he linea h eshold. The
quad a ic coe icien ˆγin eq. (5.9) is nonze o only o he hexagonal pa e n (θij = 2π/3).
In his case his con ibu ion domina es close o he h eshold and ende s he bi u ca ion
ansc i ical.
In 1990 Cilibe o e al. [78] discussed he se o equa ions gi en by (5.9) using a linea
s abili y analysis o de e mine he s abili y egimes o a hexagonal pa e n wi h espec
o a s ipe pa e n. La e , addi ionally he possibili y o a squa e pa e n was conside ed
[79, 80]. Fo a nonze o quad a ic coe icien ˆγ, he hexagon solu ion is he s able su ace
pa e n a he linea h eshold. The bi u ca ion om he la su ace s a e is ansc i ical
and hexagons emain s able e en below he linea h eshold as long as
ǫ > ǫA=−ˆγ2
4(1 + 2g1(θij = 2π/3)) (5.10)
The hexagon solu ion is s able o all con ol pa ame e s ǫ > ǫAi g1(θij = 2π/3) <1 and
1 + 2g1(θij = 2π/3) < g1(θij =π/2) + 2g1(θij =π/6). Fo g1(θij = 2π/3) >1 hey loose
s abili y wi h espec o ei he s ipes o squa es a
ǫB=ˆγ2(2 + g1(θij = 2π/3)
(1 −g1(θij =π/2))2(5.11)
5.1 In oduc ion 45
whe eas i 1 + 2g1(θij = 2π/3) > g1(θij =π/2) + 2g1(θij =π/6) his happens a
ǫB=ˆγ2(g1(θij =π/2) + 2g1(θij =π/6))
(1 + 2g1(θij = 2π/3) −g1(θij =π/2) −2g1(θij =π/6))2(5.12)
S ipes and squa es u n ou o be mu ually exclusi e pa e ns. In he case ha
g1(θij =π/2) <1, g1(θij =π/6)+g1(θij = 2π/3) <1+g1(θij =π/2) and o la ge enough
con ol pa ame e s
ǫ > ǫS=ˆγ2(1 + g1(θij =π/2))
(1 + g1(θij =π/2) −g1(θij = 2π/3) −g1(θij =π/6))2(5.13)
squa es a e he s able su ace pa e n. O he wise he s ipe pa e n u ns ou o be s able
o con ol pa ame e s la ge han
ǫS=ˆγ2
(1 −g1(θij = 2π/3)) (5.14)
A schema ic bi u ca ion diag am acco ding o hese conside a ions is d awn in ig. 5.1,
whe e AHdeno es he ampli ude o hexagons and AS he ampli ude o ei he squa es o
s ipes, depending on which o hese pa e ns is he p e e ed one.
The scala p oduc
We ha e some eedom o choose a scala p oduc ha is sui able o ou discussion o he
nonlinea egime. In chap e 2 we ound, ha he bulk magne ic equa ions comple ely
decouple om he bulk hyd odynamic equa ions, lea ing us wi h no con ol pa ame e in
he bulk o he nonlinea egime. A a i s glance, his ci cums ance impedes he de i a-
ion o an ampli ude equa ion as known o ins ance o he Rayleigh-B´ena d con ec ion,
i we used he usual scala p oduc gi en by
h·|·i = lim
L→∞
1
4L2
L
Z
−L
dx
L
Z
−L
dy
ξ
Z
−∞
dz
τ
Z0
d ¯
·· (5.15)
To ci cum en his p oblem, an ex ended scala p oduc can be in oduced ha , ad-
di ionally o he bulk equa ions, is applied o ce ain bounda y condi ions, in pa icula
o hose con aining a con ol pa ame e [54]. Wi h a scala p oduc like ha , Lange [49]
ied o de i e he adjoin sys em in he case o he Rosensweig ins abili y. He ailed in
doing so, since he could no ansla e he su ace con ibu ions a he de o mable su ace
in o a se o adjoin bounda y condi ions.
The kind o scala p oduc used by Lange was p e iously in oduced by Dauby e
al. [54] o desc ibe he nonlinea egime o pu ely su ace ension d i en con ec ion,
he Ma angoni ins abili y. The app oach was success ul, since he au ho s assumed a
la , unde o mable su ace. In he case o he Ma angoni ins abili y his assump ion is
comp ehensible, since wha one is a e is he low ield ha de elops in he bulk and
no he de o ma ion o he su ace. An assump ion ins ead, ha is no app op ia e o
he Rosensweig ins abili y since he low ield o he s a ic nonlinea egime anishes
iden ically and he only obse able is he de o med su ace.
52 The ampli ude equa ion
case o a s a iona y ins abili y
¯ x= ¯ωkx
kekz −2¯qk
¯q2+k2e¯qz¯q2+k2
¯q2−k2¯
ξ(5.48)
¯ y= ¯ωky
kekz −2¯qk
¯q2+k2e¯qz¯q2+k2
¯q2−k2¯
ξ(5.49)
¯ z=i¯ωekz −2k2
¯q2+k2e¯qz¯q2+k2
¯q2−k2¯
ξ(5.50)
Fo he adjoin s ain ield we ge
¯ǫzz = 2µ2kekz −2¯qk
¯q2+k2e¯qz¯q2+k2
¯q2−k2¯
ξ(5.51)
¯ǫxx = 2µ2
k2
x
kekz −2¯qk
¯q2+k2e¯qz¯q2+k2
¯q2−k2¯
ξ(5.52)
¯ǫyy = 2µ2
k2
y
kekz −2¯qk
¯q2+k2e¯qz¯q2+k2
¯q2−k2¯
ξ(5.53)
¯ǫxy =−4µ2
kxky
kekz −2¯qk
¯q2+k2e¯qz¯q2+k2
¯q2−k2¯
ξ(5.54)
¯ǫxz =−4iµ2kxekz −e¯qz¯q2+k2
¯q2−k2¯
ξ(5.55)
¯ǫyz =−4iµ2kyekz −e¯qz¯q2+k2
¯q2−k2¯
ξ(5.56)
Ob iously he adjoin s ain componen s ha e he same s uc u e as he componen s in
he o iginal case and hey also show a ini e s a ic limi . Howe e , hey do no ha e he
same uni s. While he s ain ield in he o iginal case is dimensionless, he adjoin s ain
ield is p opo ional o he shea modulus µ2. This is consis en wi h he scala p oduc
(5.23), whe e all con ibu ions need o ha e he same dimension. One could a oid he
dimension o he adjoin s ain ield by de ining a scala p oduc wi h a me ic con aining
uni s in he i h o he en h componen .
The easons why p e ious a emp s o sol e he adjoin p oblem ha e ailed a e, a
leas , wo old. One c ucial pa in ou discussion is o ea he medium as comp essible.
This ensu es e.g. he p esence o he con ibu ion ∼∂j∂i (1)
jin he Na ie -S okes equa ion.
Du ing he p ocess o adjoining, commu a i i y o g adien s in his e m equi es ha
he su ace e ms ∼¯ i∂i (1)
jand ∼¯ j∂i (1)
ia e equi alen , which would be iola ed i
incomp essibili y is applied be o e. The assump ion o an incomp essible luid is he e o e
oo s ong a es ic ion. An e en mo e impo an poin is o ea he sys em as a dynamic
one. The sub le eason o ha is mani es in he dynamic bounda y condi ion o he
su ace de lec ion. Assuming s a iona i y om he beginning would imply an always
unde o med su ace because he e ical eloci y a he su ace would anish in any case.
Howe e , his eloci y componen needs o be ini e o allow he su ace o de o m. The
ma ginal poin whe e he spikes a e abou o de elop (o he inal poin whe e he spikes
ha e ully de eloped) a e hen ob ained as he s a ic limi ω→0 o he ull dynamic
beha io .

5.3 The second pe u ba i e o de 53
5.3 The second pe u ba i e o de
The ac ha wi hin ou assump ions he magne ic bulk equa ions comple ely decouple
om he hyd odynamic bulk equa ions, has wo impo an consequences. On he one
hand his allows us o discuss and sol e hese wo sys ems subsequen ly, i.e. we i s
sol e he magne ic pa in a gi en pe u ba i e o de o a gi en su ace de lec ion ξ,
and eed back his solu ion in o he espec i e o de o he hyd odynamic sys em. The
de ailed discussion o he magne ic ield is gi en in appendix B. On he o he hand,
howe e , we ha e o ace he p oblem ha he con ol pa ame e ( he magne iza ion o
he magne ic ield in ou case) does no occu in he hyd odynamic bulk equa ions and
ha he bulk equa ions o he magne ic sys em a e homogeneous in all pe u ba i e
o de s, which makes i impossible o ob ain he con ol pa ame e in he nex o de by
F edholm’s heo em, only. The coupling be ween hese wo sys ems is, howe e , media ed
by he su ace, and mo e p ecisely by he no mal s ess bounda y condi ion. Sa is ying he
no mal s ess bounda y condi ion p o ides us wi h an addi ional condi ion supplemen ing
F edholm’s heo em as we will see in sec ion 5.3.4.
Acco ding o he gene al exp ession (5.7) he se o hyd odynamic bulk equa ions is
gi en in he second pe u ba i e o de by
ρ∂(0)
(2)
i+∂ip(2) −2µ2∂jǫ(2)
ij −ν2∂j∂i (2)
j+∂j∂j (2)
i
=−ρ∂(1)
(1)
i−∂jρ (1)
i (1)
j−2µ2ǫ(1)
jk ǫ(1)
ki (5.57)
∂(0)
ǫ(2)
ij −1
2∂i (2)
j+∂j (2)
i=−∂(1)
ǫ(1)
ij − (1)
k∂kǫ(1)
ij (5.58)
∂i (2)
i= 0 (5.59)
The s uc u e o hese equa ions sugges s wo kind o solu ions. One con ibu ion is
p opo ional o he main cha ac e is ic modes ξ(1) and a second one p opo ional o he
second ha monics ξ(2) gi en by
ξ(2) =kcX
i,j
(ξiξj+ξiξ∗
j+c.c.) (5.60)
The co esponding bounda y condi ions a he su ace z=ξa e expanded in he
same manne as he bulk hyd odynamic equa ions ( o a de ailed discussion c . app. C).
Addi ionally, howe e , one has o conside ha he linea eigen ec o s a e dependen on z
wi h con ibu ions ei he ∼ekczo ∼eqz (eqs. (3.27-3.35)). The bounda y condi ions ha e
o be e alua ed a z=ξand since ξi sel is expanded in e ms o ǫ, one has o expand
he exponen ial unc ions ekczand eqz i s wi h espec o z o subs i u e a e wa ds he
se ies expansion o ξ(eq. (5.2)). As a esul we ob ain e ec i e bounda y condi ions ha
ha e o be e alua ed a z= 0. Fo he angen ial con ibu ions hese e ec i e bounda y
condi ions ead
2µ2ǫ(2)
yz +ν2∂z (2)
y+∂y (2)
z= Ω(2)
yz (5.61)
2µ2ǫ(2)
xz +ν2∂z (2)
x+∂x (2)
z= Ω(2)
xz (5.62)
whe e he inhomogenei ies a e abb e ia ed by Ω(2)
ij and a e lis ed in app. C, eqs. (C.11)
and (C.10). The inhomogenei ies o he angen ial s ess bounda y condi ions a e solely
54 The ampli ude equa ion
p opo ional o he second ha monics ξ(2) which is di e en o he no mal s ess bounda y
condi ion
2µ2ǫ(2)
zz + 2ν2∂z (2)
z−p(2) +Gρξ(2) −µHc∂zΦ(2) +µ0H ac
c∂zΦ(2) ac
= Ω(2)
zz −σT∆ξ(2) +µ
1 + µM(1)Mckcξ(1) (5.63)
wi h Ω(2)
zz gi en in eq. (C.12). Finally, he kinema ic bounda y condi ion desc ibing ex-
plici ly he de o mable su ace eads in second o de
∂(0)
ξ(2) +∂(1)
ξ(1) + ( (1)
i∂i)ξ(1) = (2)
z+ξ(1)∂z (1)
z(5.64)
The las con ibu ion in eq. (5.64) is due o he ac , ha in second o de he su ace, a
which he bounda y condi ions ha e o be e alua ed, is al eady de lec ed.
5.3.1 The sol abili y condi ion in second o de
The gene al sol abili y condi ion discussed in sec ion 5.1.2 is applied o he se o second
o de equa ions (5.57-5.59) and explici ly eads
h¯ i|−∂(1)
(ρ (1)
i)−∂j(ρ (1)
i (1)
j−2µ2ǫ(1)
jk ǫ(1)
ki )i+h¯ǫij |−∂(1)
ǫ(1)
ij − (1)
k∂kǫ(1)
ij i= 0 (5.65)
A his poin one migh be emp ed o use he ac ha he Rosensweig ins abili y is
a s a ic one (in linea app oxima ion) and subs i u e ω(0) =σ(0) = 0 as well as he
s a ic limi s o he adjoin and o iginal eigen ec o s in o condi ion (5.65). The sol abili y
condi ion would hen educe o
h¯ǫij |−∂(1)
ǫ(1)
ij i= (±iω(1) +σ(1))h¯ǫij |ǫiji= 0 (5.66)
co esponding o he solu ion ω(1) = 0 = σ(1). He e, we ha e eplaced ∂(1)
by ±iω(1) +
σ(1) ( o igh - and le - a eling wa es, espec i ely) implying a no mal mode ansa z o
he ime dependence o he ampli udes. O cou se, ω(0) = 0 is he co ec solu ion in
he s a iona y limi . Howe e , in ha limi he connec ion be ween bulk equa ions and
bounda y condi ions is los (c . eqs. (2.34) and (2.44)) and an ampli ude equa ion canno
be de i ed. The e o e, one mus s ill ea he sys em as ully dynamic a leas a hose
places ela ed o he kinema ic bounda y condi ion and o he eloci y/s ain ela ion, and
sa is y F edholm’s heo em wi h he ime de i a i e ∂(0)
being ini e. One can, howe e ,
a non-c ucial ins ances simpli y he calcula ions by he ac ha ω(0) is small, bu only
a he e y end one can ake ω(0) ≡0.
The sol abili y condi ion (5.65) consis s o wo di e en pa s. One con aining spa ial
de i a i es and he o he he (scaled) ime de i a i e ∂(1)
. We i s discuss he la e pa .
The in eg a ion upon xand yis s aigh o wa dly done and only e ains con ibu ions
ha a e p opo ional o δ(ki−kj). A e in eg a ion wi h espec o zwe end up wi h
5.3 The second pe u ba i e o de 55
he ollowing exp ession,
h¯ i|∂(1)
(ρ (1)
i)i+h¯ǫij |∂(1)
ǫ(1)
ij i
=iω(1)n8µ2
kc(k2
c+q2)2
q(kc+q)3−ρ([ω(0)]2−[σ(0)]2)4k6
c+ 6k5
cq+ 6k4
cq2+ 6k3
cq3+ 2k2
cq4
qk3
c(kc+q)3o
×ˆ
ξ∗
iL ˆ
ξiRe2iω −ˆ
ξiL ˆ
ξ∗
iRe−2iω e2σ
+σ(1)n8µ2
kc(k2
c+q2)2
q(kc+q)3−ρ([ω(0)]2−[σ(0)]2)4k6
c+ 6k5
cq+ 6k4
cq2+ 6k3
cq3+ 2k2
cq4
qk3
c(kc+q)3o
×ˆ
ξ∗
iL ˆ
ξiRe2iω +ˆ
ξiL ˆ
ξ∗
iRe−2iω +ˆ
ξiR ˆ
ξ∗
iR +ˆ
ξiL ˆ
ξ∗
iLe2σ (5.67)
Fo he second o de con ibu ions we inally ge
h¯ i|∂(1)
(ρ (1)
i)i+h¯ǫij |∂(1)
ǫ(1)
ij i
=iω(1)4µ2kc(ˆ
ξ∗
iL ˆ
ξiR −ˆ
ξiL ˆ
ξ∗
iR) + σ(1)4µ2kc(ˆ
ξ∗
iL ˆ
ξiR +ˆ
ξiL ˆ
ξ∗
iR +ˆ
ξiR ˆ
ξ∗
iR +ˆ
ξiL ˆ
ξ∗
iL) (5.68)
whe e he s a ic limi has sa ely been pe o med.
Up o now i has been possible o do he calcula ions wi hou speci ying he ac ual
numbe o modes con ibu ing o he nonlinea pa e n and he esul s a e applicable o
any alue o Nand in pa icula o any angle be ween hese modes. This is changed
when he second pa o eq. (5.65), con aining he spa ial de i a i es, is conside ed. Two
o hese e ms u n ou o be i ele an o he second o de sol abili y condi ion since
hey a e a leas p opo ional o [∂(0)
]2and he e o e anish in he s a ic limi . The only
ele an e m, 2µ2h¯ i|∂(0)
j(ǫ(1)
jk ǫ(1)
ki )i, gene ally anishes, excep when h ee linea modes
o ien ed a 2π/3 ela i e o each o he a e in e ac ing. This hexagonal o de is en o ced
by he in eg a ion upon xand y. In eg a ing wi h espec o zyields in lowes o de o
ω(0) and σ(0)
2µ2h¯ i∂(0)
j(ǫ(1)
jk ǫ(1)
ki )i=−3iω(0)µ2k2
cˆ
ξ1Rˆ
ξ2Rˆ
ξ3R−ˆ
ξ1Lˆ
ξ2Lˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Rˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Lˆ
ξ3R
−ˆ
ξ1Lˆ
ξ2Rˆ
ξ3R−ˆ
ξ1Lˆ
ξ2Lˆ
ξ3R−ˆ
ξ1Lˆ
ξ2Rˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Lˆ
ξ3L−c.c.
−3σ(0)µ2k2
cˆ
ξ1Rˆ
ξ2Rˆ
ξ3R+ˆ
ξ1Lˆ
ξ2Lˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Rˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Lˆ
ξ3R
+ˆ
ξ1Lˆ
ξ2Rˆ
ξ3R+ˆ
ξ1Lˆ
ξ2Lˆ
ξ3R+ˆ
ξ1Lˆ
ξ2Rˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Lˆ
ξ3L+c.c.(5.69)
Eqs. (5.68) and (5.69) a e he wo pa s ha en e he sol abili y condi ion eq. (5.65),
which we a e now going o sol e. The imagina y pa yields he condi ion
4iω(1)(ˆ
ξ∗
iL ˆ
ξiR−ˆ
ξiL ˆ
ξ∗
iR) = −3iω(0)kc(ˆ
ξ1Rˆ
ξ2Rˆ
ξ3R−ˆ
ξ1Lˆ
ξ2Lˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Rˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Lˆ
ξ3R
−ˆ
ξ1Lˆ
ξ2Rˆ
ξ3R−ˆ
ξ1Lˆ
ξ2Lˆ
ξ3R−ˆ
ξ1Lˆ
ξ2Rˆ
ξ3L+ˆ
ξ1Rˆ
ξ2Lˆ
ξ3L−c.c.) (5.70)
This condi ion is iden ically ul illed by he ansa z
ˆ
ξiL =ˆ
ξiR =ˆ
ξiand ˆ
ξ∗
iL =ˆ
ξ∗
iR =ˆ
ξ∗
i(5.71)
which is he solu ion one expec s o he s a iona y case, since in ha limi one canno
dis inguish igh om le a eling wa es.
Using his esul o e alua ing he eal pa , we ob ain
2σ(1) X
i
ˆ
ξiˆ
ξ∗
i=−3σ(0)kc(ˆ
ξ1ˆ
ξ2ˆ
ξ3+ˆ
ξ∗
1ˆ
ξ∗
2ˆ
ξ∗
3) (5.72)
56 The ampli ude equa ion
which ob iously is sol ed by
σ(1) ˆ
ξ1=−σ(0)kcˆ
ξ∗
2ˆ
ξ∗
3and |ˆ
ξ1|2=|ˆ
ξ2|2=|ˆ
ξ3|2(5.73)
and all i s cyclic pe mu a ions 1 →2→3→1 and hei complex conjuga es. Equa ion
(5.73) ells us, ha he slow a iable σ(1) scales in he bulk wi h σ(0), indica ing ha
σ(1)/σ(0) s ays ini e in he s a ic limi . This beha io is media ed by he he kinema ic
bounda y condi ion d ξ= z(2.44). As a consequence, he eloci y ield as well as he
adjoin eloci y ield a e p opo ional o he ime de i a i e as we ealized in eqs. (3.27-
3.29) and (5.48-5.50). This is physically easonable, since in he case o he Rosensweig
ins abili y he eloci y ield anishes i he su ace pa e n has ully de eloped and he
hyd odynamic bulk equa ions a e i ially ul illed by ≡0, he same solu ion as o
he ini ially unde o med g ound s a e. This singula beha io , unique o he Rosensweig
ins abili y, is scaled ou by he choice o a dimensionless ime de i a i e ˜
∂(1)
T=σ(1)/σ(0) o
he bulk hyd odynamic equa ions. Using his ime de i a i e, eq. (5.73) can be ew i en
as
˜
∂(1)
Tˆ
ξ1=−kcˆ
ξ∗
2ˆ
ξ∗
3(5.74)
Equa ion (5.74) gi es he ela ion among he h ee ampli udes o he second o de
de lec ion, ξ(1), cha ac e is ic o hexagon pa e ns. Fo any o he egula pa e n he
igh hand side o eq. (5.69) is ze o implying, ha he e is no nonlinea in e ac ion be ween
wo di e en modes in he second o de o hose pa e ns.
Wha is missing in eq. (5.74), which in a sense can be iewed as a p imi i e o m
o an ampli ude equa ion, is a con ibu ion p opo ional o he con ol pa ame e M(1).
This is due o he ac , ha he wo bulk sys ems o magne ic and hyd odynamic equa ion
decouple comple ely. The con ol pa ame e en e s he ampli ude equa ion ia he no mal
s ess bounda y condi ion, he only way magne ic and hyd odynamic subsys ems a e
in e ac ing.
5.3.2 Solu ions p opo ional o he cha ac e is ic modes ξ(1)
Be o e we can exploi he no mal s ess bounda y condi ion in sec ion 5.3.4, we ha e
o de e mine he solu ion o he hyd odynamic con ibu ions, eqs. (5.57-5.62). F om
F edholm’s heo em we lea ned, unde wha condi ions we can ind a solu ion o he sys em
o equa ions in he second pe u ba i e o de . We dis inguish solu ions o he sys em o
equa ions ha a e ei he p opo ional o ξ(1) o p opo ional o ξ(2). In his subsec ion
we concen a e on he pa p opo ional o ξ(1). Inspi ed by he linea discussion, we
use a scala ϕ(2,1) and a ec o po en ial Ψ(2,1) o he po en ial and he o ici y low,
espec i ely. Fo he con ibu ions p opo ional o he main cha ac e is ic modes ξ(1), he
go e ning equa ions ead
∆ϕ(2,1) = 0 (5.75)
ρ∆∂(0)
ϕ(2,1) + ∆p(2,1) =−ρ∆∂(1)
ϕ(1) (5.76)
ρ(∂(0)
)3Ψ(2,1)
i−˜µ2∆∂(0)
Ψ(2,1)
i=−µ2∆∂(1)
Ψ(1)
i−ρ(∂(0)
)2∂(1)
Ψ(1)
i(5.77)
wi h he abb e ia ion ˜µ2=µ2+ν2∂(0)
. On he igh hand side o hese equa ions he i s
o de (linea ) po en ials ac as inhomogenei ies.
5.3 The second pe u ba i e o de 57
The app op ia e bounda y condi ions o he low po en ials a e de i ed in appendix
C.3 and ead o angen ial s ess
˜µ2∂2
z−∂2
yΨ(2,1)
x+ ˜µ2∂y∂xΨ(2,1)
y+ 2˜µ2∂z∂yϕ(2,1) = 0 (5.78)
˜µ2∂2
z−∂2
xΨ(2,1)
y+ ˜µ2∂x∂yΨ(2,1)
x−2˜µ2∂z∂xϕ(2,1) = 0 (5.79)
The physical bounda y condi ions ha e o be aken a z=ξ(1) in he second o de . This
leads o addi ional con ibu ions in ξ(1), which ha e al eady been aken in o accoun in
he e ec i e bounda y condi ions eqs. (5.78) and (5.79). The la e he e o e ha e o be
aken a z= 0.
The kinema ic bounda y condi ion now in ol es he slow imescale (1) and eads
(2,1)
z=∂(1)
ξ(1) (5.80)
We s a wi h he pa icula inhomogeneous solu ions o eqs. (5.76) and (5.77) o he
ec o po en ial Ψand he p essu e p, espec i ely. I can be checked ha he ollowing
ields sa is y he inhomogeneous bulk equa ions
Ψ(2,1)
i=ˆ
Ψ(2,1)inhom
iξ(1)zeqz and p(2,1)inhom =−ρ∂(1)
ϕ(1) (5.81)
wi h he ampli udes o he ec o po en ial gi en by
ˆ
Ψ(2,1)inhom
x=−µ2+ ˜µ2
˜µ2q∂(1)
∂yand ˆ
Ψ(2,1)inhom
y=µ2+ ˜µ2
˜µ2q∂(1)
∂x(5.82)
The inhomogeneous solu ions do no ye sa is y he bounda y condi ions (5.78) and (5.79).
Subs i u ing Ψinhom in o eq. (5.78) esul s in an addi ional sou ce o angen ial s ess a he
bounda y due o he inhomogeneous solu ions, which can be balanced by he homogeneous
ones
˜µ2∂2
z−∂2
yΨ(2,1)hom
x+˜µ2∂y∂xΨ(2,1)hom
y+2˜µ2∂z∂yϕ(2,1) =∂y˜µ2
µ2+˜µ2
˜µ2
∂(1)
ξ(1)(5.83)
I we use he ollowing ansa z o he homogeneous solu ions o he low po en ials Ψ(2,1)hom
and ϕ(2,1)
Ψ(2,1)hom
x=−∂yˆ
Ψ(2,1)eqzξ(1),Ψ(2,1)hom
y=∂xˆ
Ψ(2,1)eqzξ(1) and ϕ(2,1) = ˆϕ(2,1)ekczξ(1)
(5.84)
he ampli udes ˆ
Ψ(2,1) a e gi en by
ˆ
Ψ(2,1) =2kc
q2+k2
c
ˆϕ(2,1) −2µ2+ ˜µ2
˜µ2(q2+k2
c)∂(1)
(5.85)
No e ha qis he in e se decay leng h o he linea ans e se modes wi h q2=k2
c+
ρ[∂(0)
]2/(µ2+ν2∂(0)
) (chap e 3) and ∂(1)
is a sho hand no a ion o ±iω(1) +σ(1), as
be o e.
The homogeneous solu ion o he p essu e p(2,1)hom is s aigh o wa dly gi en by eq. (5.76)
p(2,1)hom =−ρ∂(0)
ϕ(2,1) (5.86)

58 The ampli ude equa ion
and i we exploi he kinema ic bounda y condi ion (5.80), he solu ion o he scala low
po en ial ϕ(2,1) can be de e mined as
ˆϕ(2,1) =q2+k2
c
kc(q2−k2
c)∂(1)
−2k2
c
µ2+ ˜µ2
˜µ2(q2+k2
c)∂(1)
(5.87)
Wi h he help o he low po en ials he eloci y ields a e de e mined
(2,1)
z=1
q2−k2
cq2−2µ2+ ˜µ2
˜µ2
k2
cekcz+ 2µ2
˜µ2
k2
ceqz −µ2+˜µ2
˜µ2
k2
c(q2−k2
c)zeqz
q∂(1)
ξ(1)
i
(5.88)
(2,1)
x=iki,x
˜µ2(q2−k2
c)L(z)∂(1)
ξ(1)
iand (2,1)
y=iki,y
˜µ2(q2−k2
c)L(z)∂(1)
ξ(1)
i(5.89)
wi h he abb e ia ion
L(z) = ˜µ2(q2−k2
c)−2µ2k2
cekcz
kc
+2µ2q2−(µ2−˜µ2)(q2−k2
c)(1 + qz)eqz
q(5.90)
om which he s ain ields ollow
ǫ(2,1)
zz =−µ2+˜µ2
˜µ2
k2
cL+(z)∂(1)
∂(0)
ξ(1)
i(5.91)
ǫ(2,1)
ab =µ2+˜µ2
˜µ2
ki,aki,bL−(z)∂(1)
∂(0)
ξ(1)
i(5.92)
ǫ(2,1)
az =iki,a
µ2+˜µ2
2˜µ22
q2−k2
c2k2
cekcz−(q2+k2
c)eqz+1+qz+k2
c
qzeqz∂(1)
∂(0)
ξ(1)
i(5.93)
o {a, b} ∈ {x, y}wi h
L±=2
q2−k2
ckcekcz−qeqz±1 + qz
qeqz (5.94)
This concludes he de i a ion o he second o de eigen unc ions ha a e p opo ional
o ξ(1). These solu ions sa is y e e y condi ion excep he no mal s ess bounda y con-
di ion. The la e will be used o de e mine he s ill unknown i s o de co ec ion o
he con ol pa ame e , M(1), which inally en e s he ampli ude equa ion as he linea
con ibu ion. We pos pone he ac ual de i a ion o hese con ibu ions o sec ion 5.3.4.
5.3.3 Solu ions p opo ional o he highe ha monics ξ(2)
We a e le wi h sol ing he sys em o hyd odynamic equa ions in he second pe u ba i e
o de , eqs. (5.57-5.59), o he highe ha monic con ibu ions p opo ional o ξ(2). The
app op ia e se o bulk equa ions eads, i we use again he ep esen a ion wi h a scala
po en ial and a ec o po en ial,
∆ρ(∂(0)
)2ϕ(2,2) +∂(0)
p(2,2)=∂i−2µ2∂j( (1)
k∂kǫ(1)
ij )−∂(0)
∂j(ρ (1)
i (1)
j−2µ2ǫ(1)
jk ǫ(1)
ki )(5.95)
ρ(∂(0)
)2−˜µ2∆∂i∂mΨ(2,2)
m−∆Ψ(2,2)
i(5.96)
=ǫijk∂j−2µ2∂m( (1)
l∂lǫ(1)
km)−∂(0)
∂l(ρ (1)
k (1)
l−2µ2ǫ(1)
lm ǫ(1)
km)
∆ϕ(2,2) = 0 (5.97)
5.3 The second pe u ba i e o de 59
k1
k2
k3
k4
k5
k6
Figu e 5.2: The ske ch shows he ela i e o ien a ion o he wa e ec o s unde
conside a ion in he ampli ude equa ions (5.150,5.152). I allows o discuss he
s abili y o hexagons and squa es and hei in e ac ion.
The i s equa ion de e mines he p essu e con ibu ion p(2,2). Since he p essu e appea s
only in he no mal s ess bounda y condi ion, his is deal wi h in sec ion 5.3.4. Nex we
cons uc a pa icula inhomogeneous solu ion o eq. (5.97) o he ec o po en ial Ψ.
The mos gene al ansa z necessa y eads
Ψ(2,2)inhom
k=−ǫzkl X
N,M X
i,j
∂lΨinhom
NMij(z)ξiN ξjM +˜
Ψinhom
NMij(z)ξ∗
iN ξjM +c.c.(5.98)
He e, summa ion o e all ele an modes i, j is implied (e.g. {i, j} ∈ {1,2,3},{i, j} ∈
{1,5}, and i=j= 1 o hexagons, squa es, and olls, espec i ely, ig. 5.2) as well as
o e igh and le a eling wa es {N, M} ∈ {R, L}, c . eq. (5.3). Subs i u ing his
ansa z in o he dynamic equa ions and ma ching he coe icien s wi h he inhomogeneous
con ibu ions o he o ici y equa ion (5.96) yields he unc ions Ψinhom
NMij(z) and ˜
Ψinhom
NMij(z).
Since hei gene al o m is ex emely bulky, in appendix D only he coe icien s Ψinhom
NMij(z)
and ˜
Ψinhom
NMij(z) o hexagonal (ij =ji = 13 = 23 = 31) and squa e pa e ns (ij =ji = 15)
as well as o s ipe solu ions (ij = 11) a e lis ed.
The gene al solu ion is he sum o he pa icula inhomogeneous and a gene al homo-
geneous solu ion, Ψ(2,2)
k= Ψ(2,2)inhom
k+ Ψ(2,2)hom
k. I has o sa is y he e ec i e angen ial
bounda y condi ions (c . appendix C.3)
˜µ2(∂2
z−∂2
y)Ψ(2,2)
x+ ˜µ2∂y∂xΨ(2,2)
y+ 2˜µ2∂z∂yϕ(2,2) =
∂yX
N,M X
i,j
(ˆ
F′
NMijξiN ξjM +˜
ˆ
F′
NMijξiN ξ∗
jm +c.c.) (5.99)
wi h sui ably abb e ia ed ampli udes ˆ
F′
NMij. The special o m o he igh hand side is
ob ained, i in eq. (C.14) he i s o de exp essions o he a iables a e explici ly pu in.
Subs i u ing he inhomogeneous solu ions Ψ(2,2)inhom
iin o eq. (5.99) a modi ied bounda y
60 The ampli ude equa ion
condi ion o he homogeneous solu ion esul s
˜µ2(∂2
z−∂2
y)Ψ(2,2)hom
x+ ˜µ2∂y∂xΨ(2,2)hom
y+ 2˜µ2∂z∂yϕ(2,2) =
∂yX
N,M X
i,j
(ˆ
FNMijξiN ξjM +˜
ˆ
FNMijξiN ξ∗
jm +c.c.) (5.100)
since he inhomogeneous solu ion does no sa is y he bounda y condi ion. In pa icula ,
on he igh hand side he inhomogeneous pa o he bounda y condi ions a z= 0 is
modi ied
ˆ
FNMij =ˆ
F′
NMij +ˆ
Finhom
NMij (5.101)
wi h ˆ
Finhom
NMij ξiN ξjM =−˜µ2(2∂2
x−∂2
z)Ψinhom
NMij(z)|z=0 ξiN ξjM (5.102)
Simila ly one ob ains he ycomponen o he angen ial bounda y condi ion s a ing om
eq. (C.15).
Now he gene al homogeneous solu ions o Ψ(2,2)hom
kand ϕ(2,2) can be ob ained by using
he ansae ze
ϕ(2,2) =X
N,M X
i,j
( ˆϕNMij ek1ij zkcξiN ξjM +˜
ˆϕNMij ek2ij zkcξ∗
iN ξjM +c.c.) (5.103)
Ψ(2,2)hom
k=−ǫzkl X
N,M X
i,j
∂l(ˆ
Ψhom
NMij eq1ij zkcξiN ξjM +˜
ˆ
Ψhom
NMij eq2ij zkcξ∗
iN ξjM +c.c.) (5.104)
whe e he cha ac e is ic wa e ec o kcis jus used o gi e he ampli udes ˆϕNMij,˜
ˆϕNMij,
ˆ
Ψhom
NMij and ˜
ˆ
Ψhom
NMij he same physical uni s as he co esponding ampli udes in he linea
discussion and whe e, again, he i s summa ion is o e igh and le a eling wa es
and he second one o e he undamen al modes in ol ed. In o de o sa is y he Laplace
equa ion (5.97), he in e se decay leng hs k1ij and k2ij o he second o de scala po en ial
a e gi en by
k1ij =kcp2 + 2 cos θij (5.105)
k2ij =kcp2−2 cos θij (5.106)
and depend on he angle be ween he i- h and he j- h mode. The in e se decay leng h
o he o a ional low con ibu ions ead co espondingly
q2
1ij =k2
1ij +ρ[D(0)
]2
µ2+ν2D(0)
(5.107)
and acco dingly q2ij by subs i u ing k2
2ij o k2
1ij in eq. (5.107). He e, D(0)
is an abb e i-
a ion o he Fou ie ans o med ime de i a i e and akes he alues iω(0) +σ(0),σ(0),
and −iω(0) +σ(0) when applied o RR, RL o LR, and LL modes, espec i ely. The bulk
equa ions and bounda y condi ions a e ul illed o he ampli udes
ˆ
Ψhom
RRij =q2
1ij
˜µ2kc(q4
1ij +q2
1ijk2
1ij)ˆ
FRRij −2˜µ2k1ijkcˆϕRRij(5.108)
5.3 The second pe u ba i e o de 61
and
ˆϕRRijξiRξjR =q2
1ij +k2
1ij
kck1ij(q2
1ij −k2
1ij)nkcD(0)
ξiRξjR −k2
1ij
ˆ
FRRij
˜µ2(q2
1ij +k2
1ij)ξiRξjR −2ξ(1)∂z (1)
z
+1
q2
1ij +k2
1ij h(k2
1ij∂2
z+q2
1ij[∂2
x+∂2
y])ˆ
Ψinhom
RRij ξiRξjRiz=0o(5.109)
Fo he las exp ession we explici ly used he kinema ic bounda y condi ion o he second
pe u ba i e o de , eq. (5.64). In appendix D hese solu ions o he low po en ials a e
speci ied o hexagons, eqs. (D.12) and (D.17), and squa es, eqs. (D.13) and (D.18). The
ampli udes wi h a ilde a e ob ained om hose wi hou one by eplacing k1ij o q1ij
by k2ij o q2ij, espec i ely. Fo ˜
ˆϕRRijξiRξjR his leads o a denomina o ∼k2ij, which
anishes o i=jacco ding o eq. (5.106). Ne e heless, all physical quan i ies de i ed
om ha po en ial, like eloci ies and s ain componen s, s ay ini e. The ampli udes in
eqs. (5.108) and (5.109) o he RL and LL (ins ead o RR) componen s a e ob ained by
choosing he app op ia e exp essions o q1ij and D(0)
, acco ding o he ules gi en abo e.
The only emaining condi ion no ye sa is ied is he no mal s ess bounda y condi ion,
which we will discuss in he nex sec ion.
5.3.4 The no mal s ess bounda y condi ion
To ind he solu ions o he hyd odynamic bulk equa ions (5.57-5.59), i was no necessa y
o use he no mal s ess bounda y condi ion. The same si ua ion appea s in he de i a ion
o he linea eigen ec o s. The e, subs i u ing he eigen ec o s in o he no mal s ess
bounda y condi ion yields he dispe sion ela ion es ic ing he linea solu ion o hose
wi h a speci ic ω(k) ela ion. The second o de no mal s ess bounda y condi ion, as will
be shown below, leads o he de e mina ion o M(1), he i s co ec ion o he con ol
pa ame e en e ing he inal ampli ude equa ion in linea o de .
The second o de no mal s ess bounda y condi ion has been de i ed in appendix C.3
and is gi en as eq. (C.12). I consis s o wo pa s, one is p opo ional o ξ(1), eq. (5.110)
and he o he o ξ(2). The la e equa ion can easily be ul illed by spli ing he p essu e
p(2,2) =p(2,2)B+p(2,2)Sin o one pa , p(2,2)B, ha is de e mined by he bulk equa ion
eq. (5.95) and he o he , p(2,2)S, by he ξ(2)-bounda y condi ion. This ansa z wo ks, i
∆p(2,2)S= 0 in he bulk. Indeed, p(2,2)S∼ξiξjek1ij zo ∼ξiξ∗
jek2ij zleads o he equi ed
esul . This addi ional p essu e con ibu ion is due o he inhomogenei ies a ising in he
no mal s ess bounda y condi ion, in pa icula he one due o su ace ension. Since he
su ace ension always ac s no mal o he su ace, his is he only poin , whe e i can en e
he nonlinea dynamics. I jus con ibu es o he Laplace p essu e, which is p opo ional
o he cu a u e o he su ace, a qui e in ui i e esul .
Howe e , his addi ional p essu e con ibu ion is o no impo ance because o wo
easons. Fi s he p essu e always en e s linea ly he hyd odynamic bulk equa ions and
he e o e i will ne e gi e ise o inhomogeneous con ibu ions, which ha e o be ac-
coun ed o by F edholm’s heo em. Second he p essu e en e s only he no mal s ess
bounda y condi ion, which is ac ually he go e ning equa ion o he app op ia e p essu e
con ibu ion in he nex o de . In addi ion, also p(2,2)Bis no needed in he ollowing and
is he e o e no shown he e.
68 The ampli ude equa ion
and he second o de equa ion (5.143) by ǫ2, we ob ain
(ǫ2˜
∂(2)
T+ǫ˜
∂(1)
T)ǫˆ
ξ1+µ2kc
2(ρG+µ2kc)ǫ2[˜
∂(1)
T]2ǫˆ
ξ1=kcµ(2ǫ2McM(2) +ǫ2[M(1)]2+2ǫMcM(1))
4(1+µ)(ρG+µ2kc)ǫˆ
ξ1
−kc
2ǫ2ˆ
ξ∗
2ˆ
ξ∗
3−A′
32µ2kc
ǫ3|ˆ
ξ1|2ˆ
ξ1
−B′(θij =2π/3)
64µ2kc
ǫ3(|ˆ
ξ2|2+|ˆ
ξ3|2)ˆ
ξ1(5.144)
By he de ini ion o ǫand he se ies expansion o he magne iza ion
M2−M2
c=Mc+ǫM(1) +ǫ2M(2) +...2−M2
c
= 2ǫMcM(1) + 2ǫ2McM(2) +ǫ2[M(1)]2+... (5.145)
we de ine he con ol pa ame e ˜ǫin he usual way
(M2−M2
c) = M2
c˜ǫ(5.146)
Subs i u ing he se ies expansion o he ime de i a i e in e ms o ǫ(c . eq. 5.5)
ǫ˜
∂(1)
T+ǫ2˜
∂(2)
T−→ ∂T(5.147)
[ǫ1˜
∂(1)
T]2−→ ∂2
T(5.148)
and using he s anda d scaling
ǫkc√Aˆ
ξi−→ ξi(5.149)
he ampli ude equa ion can be w i en as4
∂Tξ1+δ
2∂2
Tξ1=1
2˜ǫξ1−1
2√Aξ∗
2ξ∗
3− |ξ1|2ξ1−B120
A(|ξ2|2+|ξ3|2)ξ1(5.150)
whe e we in oduce he dimensionless pa ame e δ=µ2kc(ρG +µ2kc)−1and whe e he
abb e ia ions Aand B120 a e gi en by
A=A′
32µ2k3
c≈5.750 and B120 =B′(θij =2π/3)
64µ2k3
c≈3.544 (5.151)
S a ing om eq. (5.141) ins ead o eq. (5.140) we ob ain he co esponding ampli ude
equa ion o he squa e pa e n
∂Tξ1+δ
2∂2
Tξ1=1
2˜ǫξ1− |ξ1|2ξ1−B90
A|ξ5|2ξ1(5.152)
whe e he cubic coe icien B90 is analogously gi en as
B90 =B′(θij =π/2)
64µ2k3
c≈4.021 (5.153)
4Recall ha ρGk−1
c=√ρGσT

5.5 Ampli ude equa ion 69
The ac ha he linea con ibu ion on he igh hand side o eqs. (5.150) and (5.152) is
only p opo ional o he con ol pa ame e ˜ǫjus i ies a pos e io i ou choice o he ypical
ime scale τ0.
Le us i s conside he s a ic solu ions o eq. (5.150). The quad a ic con ibu ion
gi es ise o a ansc i ical bi u ca ion om he la su ace o a hexagonal pa e n a
he linea h eshold. As discussed o he phenomenological ampli ude equa ion (5.9),
a bis able egime exis s o nega i e con ol pa ame e alues ˜ǫwi h i s lowe bounda y
gi en by
˜ǫA=−1
8(A+ 2B120)(5.154)
The solu ion o he hexagonal pa e n akes he o m ξi=−|ξi|eiΦi o i∈ {1,2,3},
whe e he magni ude o he ampli udes eads
|ξi|=1+p1+8(A+2B120)˜ǫ
4√A(1 + 2B120/A)(5.155)
and whe e he phases ha e o ul ill he condi ion PiΦi= 0.
In es iga ing he alues o he cubic coe icien s we ealize, ha B120/A < 1 indica ing
ha he hexagon solu ion is always s able wi h espec o s ipe solu ions a he linea
h eshold. S ipes and squa es a e mu ually exclusi e pa e n and since B90 + 2B30 <
A+ 2B120 and B90/A < 1, he hexagons a e loosing s abili y wi h espec o squa es a
he c i ical con ol pa ame e ˜ǫBgi en by (c . sec ion 5.1.2)
˜ǫB=B90 + 2B30
2(A+ 2B120 −B90 −2B30)2(5.156)
whe e he cubic coe icien B30 ≈4.188 desc ibes he nonlinea in e ac ion be ween he
hexagonal and he squa e pa e n.
The squa e pa e n is s able o con ol pa ame e s la ge han
˜ǫS=A+B90
2(A+B90 −B120 −B30)2(5.157)
Since ˜ǫS<˜ǫB, also a bis able egime be ween he hexagons and squa es exis s.
Le us now ocus on he dynamical beha io o he pa e ns beyond he linea h esh-
old. We assume ha he hexagonal pa e n wi h he ampli ude |ξi|, eq. (5.155), has de el-
oped and dis u b i homogeneously in space by a small excess ampli ude ,|ξi|→|ξi|+ .
The linea ized ampli ude equa ion (5.150) o he dis u bances hen eads
∂T +δ
2∂2
T =1
2˜ǫ−1
√A|ξi| −31 + 2B120
A|ξi|2 (5.158)
Subs i u ing he solu ion (5.155) in he igh hand side o eq. (5.158) i can be simpli ied
o −(˜ǫ/2+ |ξi|/(2√A)) , which is always nega i e abo e he linea h eshold. This
e lec s he ac ha he exponen ial g ow h o he in ini esimal dis u bances o he la
su ace abo e he linea h eshold ge s nonlinea ly sa u a ed by he cubic coe icien s and
70 The ampli ude equa ion
dimensionless ime T
Ampli ude |ξi|
543210
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Figu e 5.3: Quali a i e ime dependen beha io (no o scale) o he su ace
spikes acco ding o eq. (5.158). The ime Tas well as he ampli udes |ξi|a e
dimensionless a iables. I he con ol pa ame e ˜ǫis sligh ly beyond he c i ical
h eshold he plo may be conside ed as he quali a i e dynamics om he la
su ace |ξi|= 0 o he spike su ace |ξi|= 1.
a s able pa e n de elops. Eq. (5.158) he e o e akes he o m o a damped ha monic
oscilla o which can be sol ed by using he ansa z =| |eλT wi h he eigen alues
λ1/2=−1
δ±s1
δ2−˜ǫ√A+|ξi|
√Aδ (5.159)
whe e he eigen equency Ω o he oscilla o is gi en by Ω2=˜ǫ√A+|ξi
|
√Aδ .
These las esul s a e s ill in dimensionless uni s. I we choose he ime scale ν2/µ2 o
compa e dissipa i e and oscilla o y p ocesses as sugges ed by eq. (5.132), he eigen alues
ead
λ1/2=−(√ρGσT+µ2)
ν2±s(√ρGσT+µ2)2
ν2
2−µ2(˜ǫ√A+|ξi|)√ρGσT+µ2
√Aν2
2
(5.160)
This esul is in ui i e, since he damping a e is in e sely p opo ional o he dissipa i e
p ocesses, gi en by ν2, whe eas he eigen equency inc eases wi h inc easing shea modu-
lus. We also ealize ha he elaxa ion owa ds he equilib ium pa e n becomes as e in
a s onge g a i a ional ield as well as o la ge su ace ensions and elas ic highe shea
moduli o he medium.
The bi u ca ion om he la su ace owa ds hexagons is ansc i ical and he e o e
in ol es a non con inuous ansi ion. I he con ol pa ame e is sligh ly abo e i s c i ical
alue, he s ill la su ace (a T= 0) can be in e p e ed as a dis u bance o he s able
s a iona y solu ion (5.155). The dynamics owa ds hexagons om he la su ace is hen
desc ibed by equa ion (5.158) gi ing ise o an o e shoo and a damped oscilla ion owa ds
he equilib ium alue (c . ig. 5.3).
5.6 On he Newell-Ope a o 71
u(x, )
~
G
Figu e 5.4: Ske ch o a physical si ua ion which is desc ibed by he sine-Go don
equa ion (5.161). The line o pendulums connec ed by a o sion wi e is exposed o
he g a i a ional ield, which is di ec ed downwa ds. The angle om he no mal
is deno ed by u(x, ).
5.6 On he Newell-Ope a o
In he p e ious discussion on he nonlinea p ope ies o he Rosensweig ins abili y we
assumed spa ially homogeneous pa e ns wi h no long wa eleng h a ia ions. By addi-
ionally escaling he spa ial coo dina es in he same way as he ime coo dina e (5.4),
one could also implemen hese possible a ia ions in space as we men ioned al eady in
sec ion 5.1.1. As a consequence, he ampli ude equa ion addi ionally con ains de i a i es
o he ampli udes wi h espec o he scaled spa ial coo dina es. Fo ypical nonlinea
di e en ial equa ions hese linea con ibu ions o he ampli ude equa ion can be ob-
ained sys ema ically by a s anda d me hod exploi ing he linea p ope ies o he sys em
[81, 82]. This me hod is desc ibed in [83] and we summa ize some o he ideas be o e we
discuss a possible applica ion o he Rosensweig ins abili y.
We illus a e his s anda d me hod by assuming he ollowing nonlinea model equa-
ion [83], he so-called sine-Go don equa ion
∂2
u−c2∂2
xu+ω2
psin u= 0 (5.161)
A physical sys em ha is desc ibed by his equa ion is e.g. a line o pendulums ha a e
connec ed by a ho izon al o sion wi e, whe e he wis angle is deno ed by u(x, ) (c .
ig. 5.4). The second con ibu ion in eq. (5.161) is hen gi en by he wis o he wi e
while, he hi d con ibu ion is due o g a i y. Expanded in e ms o u, he sine-Go don
equa ion eads
∂2
u−c2∂2
xu+ω2
pu=1
6ω2
pu3+O(u5) (5.162)
The linea pa s can be sol ed by a sinusoidal ansa z u=ae−iω +ikx +c.c., whe e he
equency ωand he wa e ec o ka e no independen om each o he , bu ela ed by
he dispe sion ela ion
ω2=ω2
p+c2k2(5.163)
72 The ampli ude equa ion
No e, ha he dispe sion ela ion is he Fou ie ans o m o he linea ized equa ion
(5.161) and he e o e con ains only he linea p ope ies o he basic equa ion.
To accoun o he nonlinea con ibu ions o he model equa ion (5.161) we can apply
he same ideas as in he case o he Rosensweig ins abili y and expand he o sion angle
uin e ms o a small pa ame e ǫ
u=ǫ(u0+ǫu1+ǫ2u2+...) (5.164)
whe e, howe e , ǫhas a di e en meaning as in he p e ious sec ions. In pa icula i is no
connec ed o a con ol pa ame e , bu me ely models an expansion o small ampli udes o
he pendulums. In addi ion we can also escale ime acco ding o T1=ǫ and T2=ǫ2 and
ob ain by sol ing he di e en o de s in ǫsuccessi ely he ollowing nonlinea solu ion5
u=ǫaeikx−i „ω−ω2
p
4ωǫ2aa∗«−ǫ2a3
48e3(kx−ω )+c.c. (5.165)
whe e he sol abili y condi ions in he second and hi d o de in ǫ ead, espec i ely
∂T1a= 0 (5.166)
∂T2a=iω2
p
4ω|a|2a(5.167)
Un il now we ollowed he same app oach as p e iously in ou discussion o he
Rosensweig ins abili y. Addi ionally we will now escale space acco ding o X1=ǫx.
To a oid esonan g ow h in he second o de in ǫ, he ampli udes ha e o ul ill
∂T1a+c2k
ω∂Xa= 0 (5.168)
which s a es ha he ampli ude ahas o a el wi h he g oup eloci y
ω′=∂kω(k) = c2k
ω(5.169)
whe e one uses he dispe sion ela ion ω(k), eq. (5.163). In he hi d o de he sol abili y
condi ion eads
∂a
∂T2
=iω′′
2
∂2a
∂ζ2+iω2
p
ω|a|2a(5.170)
wi h ζ= (X−ω′T1), which is also known as he nonlinea Sch ¨
odinge equa ion. The linea
con ibu ions a ising in eq. (5.170) ha e been calcula ed by using he linea p ope ies o
he basic equa ion (5.161), in pa icula he dispe sion ela ion (5.163). The s uc u e o
hese linea con ibu ions is uni e sal and can be collec ed in o an ope a o
LN=∂
∂T2−iω′′
2
∂2
∂ζ2(5.171)
The ques ion ha a ises is, whe he we can simila ly calcula e he co esponding
coe icien s o he scaled spa ial de i a i es in he case o he Rosensweig ins abili y by
5No e, ha eq. (5.165) is al eady he nonlinea solu ion whe e he second o de co ec ion o he
equency (gi en by eq. (5.167)) has al eady been subs i u ed.
5.7 Discussion and compa ison 73
simply aking he de i a i e o he known dispe sion ela ion (3.20) wi h espec o he
wa e ec o k. Bu he e is a undamen al di e ence be ween he sys em o equa ions
we used o desc ibe he Rosensweig ins abili y (c . sec ion 2.2.1) and he sine-Go den
equa ion (5.161). We ealize, ha o he la e case he dispe sion ela ion is solely
de e mined by he sine-Go don equa ion i sel , in pa icula i is no hing bu he Fou ie
ans o m o he linea ized sine-Go don equa ion. In he case o he Rosensweig ins abili y
we addi ionally ha e o sa is y bounda y condi ions. I hese we e only de e mined by he
s ess balance a a ixed su ace, he de e mina ion o co esponding con ibu ions o he
ampli ude equa ion could be done by jus using he ope a o LN.
In he case o he Rosensweig ins abili y, howe e , we addi ionally ha e o ake in o
accoun he de o mabili y o he su ace and along wi h i he kinema ic bounda y con-
di ion. I he su ace de o ms, also i s no mal ec o nchanges in he cou se o ime,
which we ook in o accoun in ou p e ious discussions by explici ly expanding he la e
in e ms o he su ace de lec ion ξ. All he di e en o de s o nin ol e g adien s o he
su ace de lec ion ξas can be seen in eqs. (B.17-B.19). Upon escaling he spa ial coo -
dina es we also mus expand he g adien s appea ing in nin e ms o ǫ, which leads o
addi ional con ibu ions o he highe o de bounda y condi ions solely due o he la ge
scale a ia ions o he no mal ec o . These con ibu ions a e no con ained in he linea
dispe sion ela ion and, o cou se, canno be implemen ed in o i by any means, since
he dispe sion ela ion only conside s he linea p ope ies o he sys em o equa ions
and he e o e assumes a s ill la su ace. One a he has o expand he se o bounda y
condi ions wi h he scaled spa ial coo dina es om he beginning. The con ibu ions o
he second spa ial de i a i e in he ampli ude equa ion may hen be sepa a ed in o hose
due o g adien s in he s ess enso ( o example ∂j i), which a e he ones ha ollow
di ec ly om he dispe sion ela ion, and hose solely due o he de o mabili y o he su -
ace. Fu he mo e we ha e o e alua e he bounda y condi ions a he physical bounda y,
z=ξ. In ou calcula ions we accoun ed o his ac by expanding he eigen ec o s in
e m o ξa ound z= 0. This again in ol es g adien s wi h espec o z, which ha e o
be escaled as well and which a e no con ained in he dispe sion ela ion. Addi ionally,
one has o expec con ibu ions o he second o de spa ial de i a i es in he ampli ude
equa ion ha a e due o he bulk equa ions. In he case o he scaled ime de i a i e we
showed ha possible con ibu ions due o he bulk scale ou in he s a ic limi , bu i is
a om ob ious ha his is also he case o he spa ial de i a i es.
The p e ious discussion sugges s ha also in he case o a de o mable su ace he linea
con ibu ions o he ampli ude equa ion a e o a common s uc u e and can be collec ed
in o an ex ended ope a o simila o LN. The de e mina ion o he la e is, howe e ,
beyond he scope o his hesis and will be le o u u e wo k.
5.7 Discussion and compa ison
In his chap e we succeeded in de i ing an ampli ude equa ion o he Rosensweig ins a-
bili y in iso opic magne ic gels based on he undamen al hyd odynamic equa ions. An
impo an s ep was o ind he adjoin linea sys em o equa ions oge he wi h i s co e-
sponding bounda y condi ions in he p esence o a de o mable su ace. Two assump ions
u ned ou o be e y impo an in o de o ind he adjoin sys em. Besides he dynamic
ea men o he Rosensweig ins abili y, he medium has o be conside ed comp essible o

74 The ampli ude equa ion
he adjoining p ocess. The eason o he la e assump ion is o main ain he symme y
o he s ess enso du ing he adjoining p ocess. While we can assume an incomp essible
medium a e he adjoining p ocess, he dynamic ea men o he sys em o equa ions
u ns ou o be also impo an in he discussion o he highe pe u ba i e o de s.
Wi h he help o he adjoin sys em we we e able o sa is y F edholm’s heo em and o
pe o m a weakly nonlinea analysis. I u ned ou ha due o he decoupling o he mag-
ne ic bulk equa ions om he hyd odynamic ones, F edholm’s heo em does no con ain
a con ol pa ame e . We sol ed his p oblem by obse ing ha he no mal s ess bound-
a y condi ion consis s o wo pa s. One is p opo ional o he highe ha monics o he
cha ac e is ic wa eleng h and me ely inc eases he hyd os a ic p essu e in he medium.
The o he one is p opo ional o he main cha ac e is ic wa e ec o and se es as an
addi ional sol abili y condi ion p o iding he dependence be ween he scaled g ow h a e
and he con ol pa ame e . Bo h sol abili y condi ions show quali a i e di e en beha -
io in he s a ic limi . While he sol abili y condi ion ob ained om he no mal s ess
bounda y emains ini e, he bulk con ibu ions scale wi h he linea g ow h a e. The
la e beha io is media ed by he kinema ic bounda y condi ion and has been aken in o
accoun while combining bo h sol abili y condi ions in o one. Fu he mo e i e eals he
ac ha bo h s a es, he ini ial la su ace and he inal spiked one, a e mo ionless s a es
whe e he eloci y ield anishes iden ically. While combining he bulk sol abili y condi-
ion wi h he no mal s ess bounda y one has some eedom o choose he ela i e weigh
o he bounda y wi h espec o he bulk ia he wo di e en ly scaled ime de i a i es.
I seems easonable o weigh hese single con ibu ions equally wi h espec o each o he ,
which is implici ly also done, o example, in he nonlinea discussions using an ex ended
scala p oduc [54, 56].
Upon combining he second and he hi d o de sol abili y condi ion ollowing he
s anda d p ocedu e, we ob ained a se o ampli ude equa ions o he special cases o
s ipes, squa es and hexagons. The la e con ains a quad a ic coe icien ha ende s he
bi u ca ion om he la su ace o he hexagonal pa e n ansc i ical. The calcula ed
cubic coe icien s addi ionally e eal ha a he linea onse hexagons a e he s able
su ace pa e n. Fo high magne ic ield s eng hs ins ead, hexagons become uns able and
a squa e pa e n de elops. Bo h ansi ions, om he la su ace o hexagons and om
hexagons o squa es, in ol e bis able egions. We ob ained quali a i ely he same esul s
in he case o e o luids, whe e he de i a ion o he co esponding ampli ude equa ion
and he de e mina ion o he nonlinea coe icien s has been discussed in appendix E.
The esul s o he s a ic pa e ns in his chap e a e in quali a i e acco dance wi h
he bi u ca ion scena io ob ained wi h he ene gy me hod (chap e 4). The cubic coe i-
cien s in his chap e , howe e , a e independen om he elas ic shea modulus and he
magne ic suscep ibili y. This is due o he assump ions o chap e 2, whe e we modeled
he magne ic gel as a linea elas ic and a linea ly magne izable medium and whe e we
neglec ed magne os ic i e e ec s. The esul s in his chap e a e he e o e alid o a
ini e magne ic suscep ibili y and o ini e shea moduli. As we ealized in chap e 4,
his was di e en o he ene gy me hod e en hough he same app oxima ions we e used.
Howe e , we minimized he ene gy densi y wi h espec o he highe ha monics and he
main cha ac e is ic modes independen ly. As a consequence, he o h o de coe icien s
o he ene gy me hod ( hese coe icien s quali a i ely co espond o he cubic coe icien s
in an ǫ−expansion) showed an in e se p opo ionali y on he con ol pa ame e ˜ǫ. This
dependence is omi ed in he subsequen discussions o he ene gy me hod o simplici y,
5.7 Discussion and compa ison 75
which ende s his app oach alid in he asymp o ic limi o a anishing magne ic suscep-
ibili y only. In e ospec his minimiza ion p ocedu e and he simpli ica ion a e wa ds
seems o be unsys ema ic.
In addi ion o he s a ic p ope ies o he su ace pa e ns, he analysis in his chap e
p o ides us wi h nonlinea dynamical p ocesses. We ob ain he ypical i s o de ime
de i a i e ha desc ibes he g ow h o he su ace spikes beyond he linea h eshold bu
ha also accoun s o he dissipa i e p ocesses in he medium. The ypical ime scale
o he g ow h (o elaxa ion) p ocesses inc eases o inc easing iscosi ies and becomes
smalle o inc easing su ace ension and shea moduli. Addi ionally, howe e , we ind a
second o de ime de i a i e in he case o magne ic gels.
The analysis in his chap e elucida ed he main aspec s o he unde lying mechanism
ha lead o he Rosensweig ins abili y. Bu i also un a eled ha o a be e quan i a i e
unde s anding addi ional phenomena ha e o be aken in o accoun . Two nonlinea p op-
e ies ha e been neglec ed. The nonlinea magne iza ion beha io , ha al eady e ec s
he linea h eshold, and nonlinea elas ic p ope ies. Addi ionally, he magne os ic i e
e ec migh in luence he bi u ca ion beha io in magne ic gels.
76 The ampli ude equa ion
Chap e 6
Rosensweig ins abili y in ilms and
memb anes
6.1 Mo i a ion
In he p e ious chap e s we ha e emphasized, ha in he scope o ou assump ions he
d i ing o ce o he Rosensweig ins abili y is mani es in he bounda y only, i.e. he
bulk equa ions o he hyd odynamic a iables and o he magne ic ield a e decoupled.
On he one hand his enabled us o ind he eigen ec o s o he magne ic ield and he
hyd odynamic a iables sepa a ely, bu on he o he hand we had o ind a solu ion on how
o implemen he d i ing o ce in o he ampli ude equa ion. Wi h he p e ious discussions
on he Rosensweig ins abili y in mind he ques ion a ises, how he cha ac e is ics o he
Rosensweig ins abili y will change i we educe he elas ic medium o be a bounda y laye
only, namely i we deal wi h hin ilms o memb anes made o a magne ic gel. Rannache
and Engel ocused in [84] on a hin bu s ill ini e ilm hickness allowing o pe is al ic
pe u ba ions o he ini ial s a e whe e bo h su aces we e pa allel. He e, howe e , we
wan o discuss he linea s abili y o he memb ane in he limi o a mac oscopically
anishing ilm hickness ea ing a quasi- wo dimensional elas ic magne ic medium. This
es ic s us o modes whe e bo h su aces a e dis o ed in phase keeping he ilm hickness
cons an 1.
Be o e we s a wi h he Rosensweig ins abili y in magne ic memb anes, we elabo a e
on he hin ilm limi in o de o ob ain he iscoelas ic p ope ies o he memb ane. This
pa b ie ly summa izes he wo k o Ha den and Pleine [86].
6.2 Film p ope ies in iscoelas ic media
I we discuss hin ilms o memb anes sandwiched be ween wo luids, i is easonable
o s a wi h h ee media, he memb ane mo hickness d, whose mid-plane is placed a
z= 0, he luid aabo e he memb ane (z > d/2) and he luid bbelow he memb ane (z <
−d/2) as depic ed schema ically in ig. 6.1. In ou discussion o he Rosensweig ins abili y,
we will allow he luids aand b o be e o luids wi h he magne ic pe meabili ies µaand
µb, espec i ely. Besides hei supe pa amagne ic p ope y, we assume ha hey beha e
as usual New onian liquids. We will i s concen a e on he hyd odynamic deg ees o
1The discussions in his chap e ha e been published in [85].
77
84 Rosensweig ins abili y in ilms and memb anes
6.6 Rosensweig ins abili y
As done in sec ion 3.4, eqs. (6.19-6.20), and (6.27) can be sligh ly ein e p e ed: These
a e condi ions o an ex e nal ield s eng h B, a which a su ace pe u ba ion ξwi h
wa e ec o kand ( eal) equency ω0 elaxes o ze o o g ows exponen ially o σneg-
a i e o posi i e, espec i ely (ω=ω0−iσ). Fo σ= 0 such a su ace pe u ba ion
is ma ginally s able (o uns able) agains in ini esimal dis u bances, since eq. (6.19) has
been ob ained by linea izing he dynamic equa ions and he bounda y condi ions abou
he g ound s a e. The unc ions ω0and Bs ill depend on kand he la e has o be mini-
mized wi h espec o kin o de o ge he ue linea ins abili y h eshold. The e is no
gua an ee ha a h eshold exis s o a ini e equency due o he addi ional equi emen
ω2
0>0. We he e o e discuss i s he s a iona y case. Assuming ω0= 0 he h eshold
condi ion σ= 0 leads o ˜
C(z)(k, ω=0) = 0. We will u he analyze his condi ion o he
special cases, whe e he su ace magne ism can be ei he neglec ed o has only a small
in luence in sec ion 6.6.1, a pe manen ly magne ized ilm wi h no magne ic con as o
he su ounding luids in sec ion 6.6.2, while he gene al case, when bo h des abilizing
magne ic ield e ec s a e p esen , is discussed in sec ion 6.6.3. The possibili y o an oscil-
la o y ins abili y and he case o hyd odynamically symme ic con igu a ions is discussed
in he inal subsec ion 6.6.4.
6.6.1 S a iona y, asymme ic case wi hou su ace magne ism
Dealing wi h he case o a s ong magne ic con as be ween he uppe and lowe bulk
luid (e.g. acuum and a e o luid, espec i ely), we neglec he su ace magne ic e ec .
Expe imen ally, his case can be ealized by a e ogel (o a non-magne ic gel) on op o a
e o luid and apo o acuum abo e he ilm. In ha case he h eshold magne ic ield
is
κ1B2(k) = ˜γk +ρ(b)G
k+cbk3(6.28)
and is ini e o a non-ze o magne ic con as , χa6=χb, o he bulk luids, only. Minimizing
wi h espec o kleads o he c i ical wa e ec o
k2
c=1
6cbp˜γ2+ 12ρ(b)Gcb−˜γ(6.29)
and he c i ical magne ic ield Bc=B(k=kc). Sligh ly abo e he minimum, he cu a u e
o he ma ginal s abili y cu e is gi en by
κ1(B(k)2−B2
c) = (1/kc)p˜γ2+ 12ρ(b)Gcb(k−kc)2(6.30)
The linea h eshold condi ions o his s a iona y ins abili y a e comple ely indepen-
den o he iscosi ies o bo h, he unde lying luid as well as he ilm i sel , esembling
he case o bulk ee su ace Rosensweig ins abili ies in e o luids and e ogels (c . chap-
e 3). In con as o he la e case, he e he c i ical wa e ec o does depend on he
ans e se elas ic p ope ies (c⊥) o he e ogel ( h ough ˜γ) as well as on he bending
elas ic modulus cb. The eason is ha bo h e ec s en e he no mal s ess bounda y con-
di ion wi h a k-dependence di e en om ha o he magne ic ield (c . eq. (6.27)), o o

6.6 Rosensweig ins abili y 85
ph ase i di e en ly, he magne ic ield de o ma ions do no in oduce a speci ic in e nal
leng h scale compa ed o o dina y 3 D elas ici y, bu hey do in ela ion wi h su ace
elas ici y.
On he o he hand, he linea g ow h a e σo he mos uns able mode is comple ely
de e mined by he ( ans e se) iscous p ope ies o he ilm and he bulk luid
σ=κ1(B2−B2
c)
ν⊥kc+νbk3
c+ 2ν(b)
2
(6.31)
whe e he wa e ec o o he mos uns able mode ku=kc(1 −˜
δ) wi h
˜
δ=κ1(B2−B2
c)
2˜γ+ 12cbk2
c
ν⊥+ 3νbk2
c
ν⊥kc+νbk3
c+ 2ν(b)
2
(6.32)
is sligh ly smalle han he c i ical one. I he dissipa ion in he ilm o memb ane can be
neglec ed, he g ow h a e, σ=κ1(B2−B2
c)/(2ν(b)
2), is gi en by he bulk luid iscosi y
as in he case o a bulk e o luid o e ogel (c . sec ion 5.3.4), and he mos uns able
mode is he c i ical one, ku=kcin linea o de [27].
The linea h eshold condi ions o he s a iona y ins abili y a e also independen o
he longi udinal ma e ial p ope ies (ǫ,ck) o he ilm and he e o e indis inguishable om
hose o an incomp essible ilm.
Since we a e ope a ing in he long wa eleng h limi , usually he bending elas ici y is
less impo an han o dina y elas ici y, excep o e y hin ilms, whe e c⊥and γa e
ze o o can be neglec ed. In he o me case, in pa icula o ρ(b)Gcb≪˜γ2 he c i ical
quan i ies can be simpli ied o
k2
c=ρ(b)G
˜γ1−3ρ(b)Gcb
˜γ2(6.33)
κ1B2
c= 2pρ(b)G˜γ1 + 1
2
ρ(b)Gcb
˜γ2(6.34)
O cou se, he c i ical wa eleng h and ield inc ease wi h inc easing elas ici y and scale
a onse wi h he ele an elas ic modulus o he e ogel c⊥wi h exponen s 1/2 and 1/4,
espec i ely. In he pu e e o luid case, c⊥= 0 = cb, he c i ical alues a e iden ical o
hose o he usual Rosensweig ins abili y, i.e. he e is no di e ence be ween a bulk ee
su ace and a ilm, excep o a possible di e ence in he su ace ension σTin he wo
cases.
In he opposi e, bending domina ed egime, ρ(b)Gcb≫˜γ2 he c i ical alues a e
k4
c=ρ(b)G
3cb 1−˜γ
p3ρ(b)Gcb!(6.35)
κ2
1B4
c=16
9ρ(b)Gp3ρ(b)Gcb 1 + 3
2
˜γ
p3ρ(b)Gcb!(6.36)
He e, he c i ical wa eleng h and ield scale a onse wi h he bending elas ic modulus o
he e ogel ilm cbwi h exponen s 1/4 and 1/8, espec i ely.
86 Rosensweig ins abili y in ilms and memb anes
6.6.2 Pe manen -magne ic, symme ic case
We now conside a ilm consis ing o a pe manen -magne ic gel wi h he in insic (su ace)
magne iza ion M′
0 o be igidly ancho ed o he elas ic deg ees o eedom. In pa icula
we choose i o be always an ipa allel o he ex e nal ield B. In his sec ion we jus
discuss he case o a magne ic symme y be ween he bulk luids aand b, being ei he
bo h non-magne ic o ha ing he same magne ic suscep ibili y. Fo his case he magne ic
con ibu ion s emming om he le hand side o eq. (6.22) cancels (κ1in eq. (6.25) is ze o)
and only he di e gence o he magne ic memb ane s ess enso gi es a ield dependen
con ibu ion o he h eshold condi ion o a s a iona y ins abili y
˜
C(z)(k) = ∆ρ Gk−2+ ˜γ−M′
0B+cbk2= 0 (6.37)
He e, ∆ρis he densi y di e ence be ween he medium abo e and below he ilm o
memb ane. Eq. (6.37) leads o an ins abili y wi h a cha ac e is ic mode
k4
c=∆ρ G
cb
(6.38)
when he applied c i ical ield eaches he h eshold alue
Bc=1
M′
0˜γ+ 2pcb∆ρ G.(6.39)
No e ha he c i ical wa e ec o is independen o M′
0, domina ed by he bending elas ic
coe icien , and a he simila o eq. (6.35). The h eshold ield is in e sely p opo ional
o he magni ude o he in insic pe manen magne iza ion.
6.6.3 The gene al case
We now discuss he gene al case, whe e bo h des abilizing magne ic ield e ec s a e
p esen , i.e. a uniaxial ilm wi h he pe manen magne iza ion opposi e o he ield
and a magne ic con as be ween he wo su ounding luids. The condi ion o ma ginal
s abili y agains s a iona y con ec ion, eq. (6.27),
˜
C(z)(k) = ∆ρ Gk−2+ ˜γ+cbk2−M′
0B−κ1B2k−1= 0 (6.40)
leads o he neu al cu e B=B(k). In p inciple, one could expec a compe i ion be ween
he wo di e en ins abili ies desc ibed in he wo p eceding subchap e s, i.e. a ansi ion
om a s a iona y ins abili y wi h a wa e ec o like ha o eq. (6.29) o one like ha o
eq. (6.38).
The minimum h eshold condi ion dB/dk = 0 allows us he calcula e he c i ical wa e
ec o kcas a eal oo o
κ1(3cbk4
c+ ˜γk2
c−∆ρ G)2+ 2M′2
0(cbk4
c−∆ρ G)k3
c= 0 (6.41)
In dimensionless o m eq. (6.41) con ains h ee ele an numbe s RB=cb/(∆ρ Gd4), RE=
˜γ/(∆ρ Gd2), and RM=M′2
0/(∆ρ Gκ1d3), i he wa e ec o is scaled by he ilm hickness
d. Fo RM> RB, RE he e a e wo di e en minimum solu ions, kc1and kc2, possible.
Howe e , he c i ical ields associa ed wi h hese wa e ec o s, Bc1=B(kc1) and Bc2=
6.7 Discussion 87
B(kc2), a e ne e equal, excep in he limi RM→ ∞, whe e kc1=−kc2and he case
o sec ion 6.6.2 is eached. Fo RM.RB, RE he e is only one minimum solu ion o
eq. (6.41), which ends o smalle RM o he solu ion o sec ion 6.6.1. Thus, o a gi en
se o ma e ial pa ame e s he e is always one de ini e ins abili y a a minimum Bc, and
ne e a compe i ion be ween ins abili ies o di e en kc.
6.6.4 Addi ional ema ks
Finally we will explo e he possibili y o an oscilla o y ins abili y. I we assume ha he
ilm comp essional modulus, ˜ε, and he longi udinal elas ic modulus ckand iscosi y νk
can be neglec ed (incomp essible ilm), one can show ha he cu e o ma ginal s abili y,
B=B(k, ω) has i s minimum a ω0= 0, and hus any oscilla o y s a e would ha e
a highe h eshold han he s a iona y one. In he gene al case, he p oo o he non-
exis ence o an oscilla o y ins abili y is much mo e in ol ed. One can show (unde he
p o iso ha ν⊥+νbk2and νka e o he same o de o magni ude) ha he e is no ini e
equency possible i ˜ǫk2≤√3(˜γk2+ ∆ρ G +cbk4). In he opposi e case he h eshold o
an oscilla o y ins abili y (i i exis s) is highe han ha o he s a iona y one.
I he densi ies o he wo bulk luids abo e and below he ilm o memb ane a e
iden ical, hei g a i a ional in luence on he in e ace undula ions cancels. The hin ilm
i sel is no sensi i e o g a i y, since i s olume is going o ze o in he wo-dimensional
limi . The e o e, he g a i y e m is absen in he no mal s ess bounda y condi ion and
he linea ins abili y c i e ion in he s a iona y case is C(z)= 0 (ins ead o ˜
C(z)= 0).
The gene al ma ginal s abili y cu e B=B(k) hen has a minimum o a anishing
k2
c∼∆ρ G →0 leading o a anishing h eshold B4
c∼∆ρ G →03. The lowes wa e ec o
o a ini e expe imen al se -up o ho izon al dimension L,kc= 2π/L gi es κ1B2
c≈2π˜γ/L,
since e ec s o bending and su ace magne iza ion a e negligible o la ge L. This means
he e is only one su ace exci a ion (spike) in he whole sample, go e ned by he (e ec i e)
su ace ension. This is a e y well known scena io, heo e ically and expe imen ally [92],
o o dina y e o luid ee su aces unde s ongly educed g a i y condi ions.
6.7 Discussion
The d i ing o ce o he Rosensweig ins abili y mani es s i sel in he bounda y condi ions,
only, o e o luids as well as e ogels (i magne os ic ion is neglec ed). The ques ion
a ises, how will he cha ac e is ics o he onse o he ins abili y change, i he elas ic
medium i sel is e y hin so ha i can be conside ed as a ilm o a memb ane. In
he p esen chap e we ha e add essed his ques ion by ex ending p e iously ob ained
dispe sion ela ions o su ace wa es a a hal -space e ogel bounda y o hose o he
memb ane su aces. The e y hin memb ane is su ounded by wo New onian luids
ha can be e o luids wi h di e en magne ic p ope ies. Possible gene aliza ions o
iscoelas ic su ounding luids and o iscoelas ic ( a he han elas ic) memb anes ha e
been ske ched. The magne ic ilm i sel can be ei he a supe pa amagne ic iso opic
magne ic gel, o an aniso opic e omagne ic one ha ing a ini e in insic magne iza ion.
3Since he limi s k→0 and κ1→0 a e no in e changeable, he o mulas o sec ion 6.6.2 a e no
applicable o he case o anishing g a i y; a he , one has o es ablish ela ions be ween he smallness
o ∆ρ G, he smallness o kc, and he smallness o κ1, in o de o ge a de ini e esul o Bcin ha case.
88 Rosensweig ins abili y in ilms and memb anes
Apa om he ma e ial p ope ies o he su ounding luids, he de i a ion o dis-
pe sion ela ions in hin ilms makes use o ce ain e ec i e ( equency and wa e ec o
dependen ) su ace ma e ial pa ame e s ha desc ibe he in e nal ilm p ope ies. Fo
su ace wa es an e ec i e elas ic su ace modulus is in oduced ha con ains he in a-
laye elas ic and iscous p ope ies. In he same manne we in oduce in ou discussion an
e ec i e su ace pe meabili y o he magne ic ilm desc ibing he induced o pe manen
magne ic ilm p ope ies, which gene ally a e di e en om he bulk quan i ies. In ecen
expe imen s [93] his kind o di e ence be ween bulk and su ace beha io in he magne ic
p ope ies has been seen when spin coa ing a e o luid.
In ou discussion we ha e es ic ed ou sel es o modes whe e he uppe and he lowe
su ace o he memb ane mo e in phase, esul ing in an undula ed memb ane o cons an
hickness (in linea app oxima ion). This is complemen a y o a p e ious discussion o
ilms o ini e hickness, whe e jus pe is al ic mo ions whe e aken in o accoun [84]. Fo
supe pa amagne ic ilms we ge wo di e en addi ional con ibu ions o he dispe sion
ela ion. One is due o he magne ic asymme y be ween he su ounding liquids. This
con ibu ion is o he same cha ac e as he magne ic pa o su ace wa es in he hal -
space case and anishes in he symme ic case (no magne ic con as be ween he wo
su ounding luids). The second con ibu ion comes om he magne izabili y o he hin
ilm i sel . This las con ibu ion, howe e , ac s always s abilizing and e ec i ely s i ens
he memb ane. Thus, a (symme ic) supe pa amagne ic memb ane in ai , o ins ance,
will ne e become uns able o undula ions o he ype desc ibed he e. An in ui i e eason
o his is he ac ha in he symme ic case he magne ic ield is no dis o ed in
he limi o an in ini ely hin memb ane e en i he memb ane i sel is subjec o small
pe u ba ions. As a esul , no des abilizing o ce ac s on he magne ic dipoles in he
ilm. In he p esen discussion we he e o e ocus on he case o high magne ic con as
be ween he su ounding luids discussing he in luence o he su ace elas ic p ope ies o
he cha ac e is ics o he Rosensweig ins abili y. Due o he elas ic and bending elas ic
su ace p ope ies, he cha ac e is ic mode a onse is shi ed o highe wa eleng hs and
he c i ical magne ic ield owa ds highe ield s eng hs. We can dis inguish he limi ing
cases o a bending domina ed egime and he egime whe e su ace elas ici y plays he
impo an ole.
Fo an aniso opic magne ic hin ilm o memb ane, i s pe manen magne iza ion
can lead o he Rosensweig ins abili y, i he applied ield is s ong enough and o ien ed
an ipa allel o i . In his case he magne ic asymme y be ween he su ounding liquids
is no needed and such a magne ic ilm su ounded by ai can become uns able.
Finally, he gene al case o an aniso opic magne ic memb ane sepa a ing wo liquids
o di e en magne ic p ope ies has been discussed. In p inciple, he e is a compe i ion
be ween he p e iously discussed ins abili y mechanisms (ei he based on he magne ic
con as o on he pe manen ilm magne iza ion), which gene ally occu a a di e en
wa eleng h. Howe e , i u ns ou ha such a pa e n compe i ion does no occu in he
sys em unde conside a ion, because he c i ical magne ic ield acco ding o one o he
mechanisms is always smalle han he o he one. Only in he limi ing case o in ini ely
high in insic magne iza ion (in ini ely low magne ic con as ) bo h c i ical ields can be
equal. In his case, howe e , he di e en cha ac e is ic modes a onse a e o he same
magni ude, bu o opposi e sign, and no compe i ion o wo di e en spa ial modes a ises.
Chap e 7
The adjoin sys em o he
Ma angoni con ec ion
In his chap e we will apply he me hod ha we in oduced in chap e 5 o de i e he
adjoin sys em o he Rosensweig ins abili y o he case o he Ma angoni ins abili y.
Also in he case o he Ma angoni ins abili y he adjoin sys em aking in o accoun
he de o mabili y o he su ace was unknown and nonlinea discussions whe e he e o e
es ic ed o la su aces1.
7.1 In oduc ion o Ma angoni con ec ion
The Ma angoni ins abili y is a p ominen example o a su ace ension d i en ins abili y.
I a empe a u e g adien is applied o a laye o a luid wi h a ee su ace, he conduc ing
s a e becomes uns able beyond a ce ain c i ical empe a u e g adien when hea ing is
done om below and con ec ion s a s. Fo hick laye s he ins abili y is d i en by buoy-
ancy (classical Rayleigh-B´ena d con ec ion), bu i he laye is smalle han abou 1 mm,
Pea son [94] p oposed luc ua ions o he su ace ension, ha a ise due o empe a u e
luc ua ions a he ee su ace, being he mechanism d i ing he con ec ion.
The Ma angoni ins abili y was in es iga ed ex ensi ely heo e ically. Nield [95] i s
compa ed linea ly he compe i ion be ween he buoyancy and he su ace ension d i en
ins abili y mechanism, bu bo h, Pea son and Nield, s ill conside ed a la , unde o mable
su ace. Sc i en and S e nling [96] and la e on Smi h [97] accoun ed o a ee de o mable
su ace. In Re . [96] only capilla y e ec s ha e been conside ed and an always uns able
conduc ing egime was ob ained due o missing s abilizing g a i a ional con ibu ions o
he long wa eleng h limi . Smi h discussed a laye model, a ligh luid abo e a hea ie
one. A comp ehensi e linea s udy was i s gi en by Takashima [50, 51] in 1981, who also
discussed he possibili y o an oscilla o y b anch ha could a ise o nega i e Ma angoni
numbe s. P´e ez-Ga c´ıa and Ca nei o [98] gene alized his app oach o he combina ion
o bo h, su ace d i en and buoyancy d i en con ec ion, which ma ches he esul s o
Takashima in he limi o negligible buoyancy o ces. All nonlinea heo e ical discussion
up o now assumed a la , unde o mable su ace. Rosenbla e al., o ins ance, discussed
he nonlinea egime in a cylind ical con aine [52] in e ms o an ex ended Gale kin
me hod. This discussion was la e on ex ended o ec angula essels [53, 54], bu o his
1This chap e is based on [74].
89

90 The adjoin sys em o he Ma angoni con ec ion
z
~n
x, y
z= 0
z=dξ
∇T
~
G
Figu e 7.1: Quali a i e ske ch o he geome y unde conside a ion in he case
o pu e Ma angoni con ec ion. The luid is con ined be ween he igid su ace
a z= 0 and he de o mable su ace ini ially a z=d. The de lec ion o he
de o mable su ace wi h espec o he la su ace is deno ed by ξwi h i s uni
no mal ec o npoin ing upwa ds. The applied empe a u e g adien is always
pa allel, he accele a ion due o g a i y Galways an ipa allel o he z−axis.
app oach no adjoin sys em is needed. The case o a ho izon ally in ini e laye o luid
was s udied in Re s. [55] and [99]. In Re . [56] a wo laye model was conside ed, whe e
he adjoin sys em was de i ed using he ansa z o [99] p o ided he su ace is la .
Inspi ed by he esul o he case o he Rosensweig ins abili y, we apply he same
o malism (sec ion 5.2) o he case o s a iona y Ma angoni con ec ion o ind he ad-
join sys em o equa ions o his case as well. Howe e , he e exis s a c ucial di e ence
be ween hese wo ins abili ies. While in he case o magne ic luids he ex e nal o ce
ac s no mal o he ee su ace, in he case o Ma angoni con ec ion he ex e nal o ce
is ac ing angen ially o he su ace. We can he e o e e i y ou o malism o any a -
bi a y di ec ion o he d i ing o ce. This ex e nal o ce o he Ma angoni ins abili y
is, as men ioned al eady, media ed by empe a u e luc ua ions. The su ace ension σT
is he e o e assumed o be empe a u e dependen and eads in a se ies expansion up o
linea o de in T
σT(T) = σT(TR)−γ(T−TR) (7.1)
wi h he change in su ace ension due o empe a u e luc ua ions γ=−(∂σT(T)/∂T)T=TR
and whe e TR ep esen s an a bi a y e e ence empe a u e. Fo he ollowing discussion
we will e e o σT(TR) as σT.
7.2 Basic equa ions and he adjoin sys em
To ind he adjoin sys em o he pu ely su ace d i en con ec ion we assume a iscous
New onian luid. As done in he case o he Rosensweig ins abili y, we assume i o be
comp essible wi h a ba o opic equa ion o s a e a he beginning, bu in he end we will
again use he limi o an incomp essible luid. Addi ionally we ha e o inco po a e he
equa ion o hea anspo wi h he empe a u e Tand he he mal di usi i y ˜χ. All he
o he a iables a e deno ed in he same way as in he p e ious discussion. As we wan
o discuss he pu ely su ace d i en con ibu ion o con ec ion, all con ibu ions due o
7.2 Basic equa ions and he adjoin sys em 91
buoyancy a e neglec ed. The sys em o equa ions hus eads
∂ ρ+∂k(ρ k) = 0 (7.2)
∂ gi+∂jTij =ρGi(7.3)
∂ T+ j∂jT= ˜χ∂j∂jT(7.4)
The s ess enso Tij o he luid unde conside a ion akes he o m
Tij = jgi+pδij −ν2(∂j i+∂i j)−ˆν(∂k k)δij (7.5)
We equi e he no mal as well as he angen ial s ess a he ee su ace be ween he
New onian luid and he acuum o be balanced, leading o he no mal and angen ial
bounda y condi ions, espec i ely
p−ρ0Gξ −2ν2∂z z−ˆν(∂k k) = −σT(∂2
x+∂2
y)ξ(7.6)
ν2(∂y z+∂z y) = −γ∂yT+γβ∂yξ(7.7)
ν2(∂x z+∂z x) = −γ∂xT+γβ∂xξ(7.8)
whe e βdeno es he applied empe a u e g adien ac oss he luid.
Addi ionally we ha e o speci y he phenomenological bounda y condi ions a he
su ace. Again he kinema ic bounda y condi ion (2.44) o a ee de o mable su ace is
assumed o hold. Second, we assume he hea lux Q h ough he su ace o be p opo -
ional o he local empe a u e g adien , whe e κdeno es he coe icien o (su ace) hea
conduc ion.
Q(T) = −κ∂zT(7.9)
A he bo om (z= 0) o he con aine we assume he usual igid bounda y condi ions
i=∂z z=T= 0 (7.10)
The s a e ec o |φinow becomes six dimensional and is de ined by
|φi= ( x, y, z, p, T, ρ) (7.11)
so ha he sys em o equa ions eads again in he gene al o m
L0|φi= 0 (7.12)
We use he usual scala p oduc , howe e , now he z−in eg a ion is bounded be ween he
bo om pla e (z= 0) and he ee su ace (z=ξ).
h¯
φ|φi= lim
L→∞
1
4L
L
Z
−L
dx
L
Z
−L
dy
ξ
Z
0
dz
Z
0
d ¯
φφ (7.13)
The adjoin linea ope a o hen u ns ou o be
L†
0= A C
B D !
92 The adjoin sys em o he Ma angoni con ec ion
A=


−ρ∂ −ν2∂2
i−(ˆν+ν2)∂2
x−ˆν∂x∂y−ν2∂y∂x−ˆν∂x∂z−ν2∂z∂x
−ˆν∂y∂x−ν2∂x∂y−ρ∂ −ν2∂2
i−(ˆν+ν2)∂2
y−ˆν∂y∂z−ν2∂z∂y
−ˆν∂z∂x−ν2∂x∂z−ˆν∂z∂y−ν2∂y∂z−ρ∂ −ν2∂2
i−(ˆν+ν2)∂2
z



(7.14)
B=


−∂x−∂y−∂z
0 0 0
0 0 0


C=


−∂x0 0
−∂y0 0
−∂z−β0


(7.15)
D=


0 0 −1
ρ0∂
0−∂ −˜χ∂2
i0
−1
ρ0∂ 0−c2
ρ0∂


(7.16)
The su ace con ibu ions o he in eg a ion by pa s should anish o ul ill eq. (5.22)
leading o he co esponding bounda y condi ions in he adjoin case.
iω2ν2∂z¯ z+iωˆν(∂k¯ k) + iω¯p+ρG¯ z+σTk2¯ z= 0 (7.17)
¯ x(−ikx)ˆ
T(z)−¯ xγβ(−ikx) + ˆ x(z)ν2(∂z¯ x+∂x¯ z) = 0 (7.18)
¯ y(−iky)ˆ
T(z)−¯ yγβ(−iky) + ˆ y(z)ν2(∂z¯ y+∂y¯ z) = 0 (7.19)
−˜χ¯
T∂zT+ ˜χT∂z¯
T= 0 (7.20)
In he las se o equa ions we ha e used he ac , ha e e y a iable o he o iginal sys em
is modula ed by ξ, in pa icula we used T(z) = ˆ
T(z)ξand x,y(z) = ˆ x,y(z)ξ. Ac ually
eq. (7.20) jus s a es, ha he adjoin empe a u e may di e om he o iginal one by
jus a cons an . Fo he phenomenological bounda y condi ions we ake he same o m
as o he o iginal case, namely
¯ z=i¯ω¯
ξ(7.21)
¯
Q(¯
T) = −κ ∂z¯
T(7.22)
The bounda y condi ions a he igid bo om u n ou o be sel -adjoin , bu a e epea ed
he e
¯ i=∂z¯ z= 0 (7.23)
¯
T= 0 (7.24)
7.3 The dimensionless ep esen a ion
Fo he u he discussion we gi e he dimensionless e sion o he p oblem discussed in
he p e ious sec ion, because i is common in all he o he discussion ega ding con ec ion.
Following he usual s eps [100], he linea ized dynamical equa ions o he de ia ions om
he conduc ing s a e o he empe a u e Θ and he e ical componen o he eloci y z
ead
(D2−k2)(D2−k2−iω) z(z) = 0 (7.25)
(D2−k2−iωP)Θ(z) = − z(z) (7.26)
7.3 The dimensionless ep esen a ion 93
The bounda y condi ions a he ee su ace using he s ess balance hen ead
(D2+k2) z(z) = −Mk2Θ(z)−1
Pξ(7.27)
CP(iω −D2+ 3k2)D z(z) = −(B−k2)k2ξ(7.28)
And o he phenomenological bounda y condi ions we ha e
z(z) = iωξ (7.29)
P(D+F)Θ(z) = Fξ(7.30)
A he bo om, he equa ions educe o
z=D z= Θ = 0 (7.31)
While escaling he a iables we ha e in oduced dimensionless numbe s such as he
P and l numbe P=ν2/˜χ, he Ma angoni numbe M=γβd2/(ρ˜χν2), he C ispa-
ion numbe C=ρν2˜χ/(σTd), he Bond numbe B=ρGd2/σTand he Bio numbe
F= (∂Q/∂T )d/κ as well as he dimensionless de i a i e wi h espec o z,D=d/dz.
Using he same a gumen s o he adjoin se o equa ions we ind
(D2−k2)(D2−k2+i¯ω)¯ z(z) = −A¯
Θ(z) (7.32)
(D2−k2+i¯ωP)¯
Θ(z) = 0 (7.33)
I is wo h men ioning he e ha in eq. (7.32) an addi ional numbe , A=β2d4/(˜χν2),
a ises. This is, howe e , consis en wi h condi ion (7.20), which allows he empe a u e
in he adjoin case o di e om he o iginal empe a u e by a cons an ac o . One
could escale he dimensionless adjoin empe a u e by exac ly his numbe A, esul ing
in a dimensionalized adjoin empe a u e. This, howe e , is no su p ising since also in
he discussion o he adjoin sys em o he Rosensweig p oblem, he adjoin s ain ield
acqui ed a di e en physical uni due o he dynamic coupling be ween eloci y ield and
he s ain ield. The adjoin bounda y condi ions s emming om he adjoining p ocess
u n ou o be
−M(D¯ z(z))k2ˆ
Θ−1
P= (Dˆ z)(D2+k2)¯ z(z) (7.34)
CP(ω¯ω−iωD2+3iωk2)D¯ z(z) = −(B−k2)k2¯ z(z) (7.35)
While he ones desc ibing he ee su ace a e
¯ z(z) = i¯ω¯
ξ(7.36)
P(D+F)¯
Θ(z) = F¯
ξ(7.37)
The sel -adjoin bounda y condi ions a he bo om a e epea ed he e in dimensionless
o m
¯ z=D¯ z= 0 (7.38)
¯
Θ = 0 (7.39)
100 Conclusions
we ound ha he s ipe pa e n is ne e s able wi h espec o ei he o he o he wo
pa e ns. A he linea onse , he hexagonal con igu a ion o su ace spikes u ns ou o
be he ene ge ically a o ed pa e n. Upon u he inc ease o he con ol pa ame e he
hexagonal pa e n in u n becomes ene ge ically uns able and a squa e pa e n de elops.
Bo h ansi ions, om he la su ace o hexagons and om hexagons o squa es, a e
accompanied by hys e e ic egions ha become smalle o inc easing elas ic shea moduli.
The ene gy me hod compa es he ene gy o he possible su ace pa e ns bu does nei-
he p edic , which o hese pa e ns can be dynamically a ained, no akes in o accoun
he dissipa i e p ocesses in he medium ha become impo an du ing he g ow h o he
su ace spikes. Fu he mo e, i is s ic ly alid only in he unphysical limi o a anishing
magne ic suscep ibili y. These d awbacks mo i a ed us o discuss in he i h chap e
he nonlinea egime o he Rosensweig ins abili y using an expansion o he undamen-
al hyd odynamic equa ions in e ms o he no malized di e ence ǫbe ween he applied
magne ic ield and he c i ical one.
When expanding he undamen al hyd odynamic equa ions in e ms o ǫ, he non-
linea i ies gi e ise o inhomogenei ies in he second and highe o de equa ions. To
sys ema ically gua an ee he sol abili y o hese equa ions using F edholm’s heo em, he
adjoin linea eigen ec o s a e needed. Fo sys ems in ol ing a de o mable su ace in
gene al and o he Rosensweig ins abili y in pa icula , he se o adjoin linea equa-
ions wi h hei co esponding bounda y condi ions we e no known. Fo he de i a ion
o he la e wo assump ions u ned ou o be c ucial. Fi s , one has o ea he sys em
dynamically and he s a ic limi should only be used a he e y end, and second, one
has o s a wi h he hyd odynamic se o equa ions desc ibing a comp essible medium.
The incomp essibili y assump ion can hen be used once he sys em o equa ions is ad-
join . The i s assump ion es s on he indings o he linea discussion, namely ha he
s a ic cha ac e o he Rosensweig ins abili y should be in e p e ed as a limi ing p ocess
a he han as a s a ic p ocess om he beginning. The second assump ion gua an ees he
symme y o he s ess enso ha is needed o consis en ly de ine he adjoin angen ial
bounda y condi ions. Wi h he se o adjoin equa ions and he co esponding bounda y
condi ions, he adjoin linea eigen ec o s we e ob ained. The eby igh a eling wa es
in he o iginal sys em ans o m in o le a eling wa es in he adjoin sys em and ice
e sa. Fu he mo e hey show he same s a ic p ope ies as he o iginal eigen ec o s,
namely ha he adjoin eloci y ield anishes in he s a ic limi whe eas he s ain ield
acqui es a ini e s a ic alue.
Wi h he adjoin linea sys em o he Rosensweig ins abili y a hand, we ul illed
he sol abili y condi ions in he second and in he hi d pe u ba i e o de in e ms o ǫ
and inally ob ained he ampli ude equa ion o he Rosensweig ins abili y. Wi hin he
scope o ou assump ions, in pa icula because we neglec ed magne os ic i e e ec s, he
hyd odynamic and he magne ic bulk equa ions decouple. Since we assumed a as elax-
ing magne ic ield which is go e ned by linea s a ic Maxwell equa ions, he sol abili y
condi ion o he magne ic equa ions is ul illed i ially. As a consequence, F edholm’s
heo em applied o he hyd odynamic bulk equa ions does no con ain he magne ic ield
a iables, which ac as he con ol pa ame e . Besides he bulk equa ions he Rosensweig
ins abili y c ucially depends on he bounda y condi ions and in pa icula on he no mal
s ess bounda y condi ion. We showed in ou analysis ha he no mal s ess bounda y
condi ion canno be ul illed i ially in he highe pe u ba i e o de s, bu a he ac s as
a supplemen o F edholm’s heo em, in o de o de e mine he highe o de co ec ions

101
o he con ol pa ame e .
In he de i ed ampli ude equa ion wo con ibu ions a e impo an . We succeeded o
he i s ime in de i ing he quad a ic coe icien in he ampli ude equa ion om he un-
damen al hyd odynamic equa ions. The quad a ic coe icien implies ha hexagons a e
he s able con igu a ion o su ace spikes a he linea h eshold and in addi ion ha he
bi u ca ion om he la su ace o he su ace spikes is ansc i ical in ol ing a bis able
egion be ween hexagonally o de ed spikes and he la su ace below he linea h eshold.
Bo h esul s a e expe imen ally e i ied p ope ies o he Rosensweig ins abili y. Addi-
ionally we de i ed a second o de ime de i a i e in he case o magne ic gels. The
linea ized ampli ude equa ion he e o e acqui es he o m o a damped ha monic oscilla-
o . I a ini e ampli ude su ace pa e n is dis u bed, his dis u bance will elax in he
o m o a damped oscilla ion. In he case o he Rosensweig ins abili y in e o luids, whose
ampli ude equa ion has also been de i ed in his hesis, his second o de ime de i a i e
is no p esen . The ampli udes o he s able s a ic su ace pa e ns a e de e mined by
he cubic coe icien s in he ampli ude equa ion. They show ha o high magne ic ield
s eng hs he hexagons become uns able and ans o m in o a squa e pa e n. A esul
ha is in acco dance wi h he ene gy me hod. The cubic coe icien s calcula ed in his
hesis a e independen o he elas ic shea modulus and he magne ic suscep ibili y, due
o he assump ion o a linea elas ic as well as a linea magne ic medium.
Ou discussions e ealed ha he Rosensweig ins abili y is a pu ely su ace d i en
ins abili y wi hin he scope o ou assump ions. A na u al ques ion is hen o ask: Wha
happens, i he supe pa amagne ic medium is jus a su ace, namely a hin ilm o a mem-
b ane. In chap e 6 we discussed his ques ion assuming a memb ane, ei he made o an
iso opic o an aniso opic magne ic gel, loa ing on a New onian liquid o on a e o luid.
In he i s case we ealized, ha he ilm does no become uns able, i we assume an
iso opic magne ic gel. An in ui i e eason o his is gi en by he cha ac e o he d i ing
o ce i sel . Small su ace luc ua ions ende he applied homogeneous magne ic ield lo-
cally inhomogeneous, which causes a Kel in o ce. In he case o memb anes we showed,
ha in he limi o anishing ilm hickness he magne ic ield emains undis o ed causing
no o ce ac ing o he magne ic ilm. This changes i we assume an aniso opic magne ic
gel, whe e he ozen-in magne iza ion is igidly ancho ed o he elas ic medium. In his
case he ilm becomes uns able wi h espec o pe iodical dis u bances i he ozen-in
magne iza ion is o ien ed opposi e o he applied magne ic ield. A ypical p ope y o
he Rosensweig ins abili y in iso opic magne ic gels is ha i s cha ac e is ic wa eleng h
is he same as o usual e o luids. I we assume a non-magne ic ilm loa ing on op o a
usual e o luid, his changes and he cha ac e is ic mode depends on he elas ic p ope ies
o he memb ane.
We ealized in his hesis ha he cha ac e o he Rosensweig ins abili y is owed o
he de o mabili y o he su ace be ween he magne ic medium and he acuum abo e.
Ano he e y p ominen example o an ins abili y which in ol es a de o mable su ace is
gi en by he pu e Ma angoni ins abili y. In his case, empe a u e luc ua ions a he
ee su ace o a luid cause luc ua ions o he su ace ension ha in u n de o m he
su ace and d i e con ec ion. In he case o he Ma angoni con ec ion, he adjoin sys em
o linea equa ions ha akes in o accoun a de o mable su ace also was unknown and
nonlinea discussions he e o e had o assume la and unde o mable su aces. Using he
same a gumen s as o he Rosensweig ins abili y, we we e able o de i e he adjoin sys em
and he co esponding bounda y condi ions o he Ma angoni ins abili y. As shown in
102 Conclusions
chap e 7, he adjoin bounda y condi ions in ol e he linea eigen ec o s o he o iginal
sys em. The la e p ope y is due o a bulk coupling be ween he empe a u e ield and
he eloci y ield, al hough his coupling does no d i e he ins abili y. As a consequence,
he solu ion o he adjoin sys em o he Ma angoni con ec ion is a he in ol ed and
an easy in e p e a ion in e ms o a ansla ion be ween igh and le a eling wa es,
as i was he case o he Rosensweig ins abili y, is no possible. The d i ing o ce in
he case o he Ma angoni ins abili y ac s pu ely angen ially, while o he Rosensweig
ins abili y he o ce is o hogonal. Thus he me hod used o de i e he adjoin sys ems
o bo h ins abili ies should also wo k o any a bi a y o ien a ion o he d i ing o ce a
he su ace.
Appendix A
Decoupling o he dynamic sys em
In his appendix we show explici ly he decoupling o he Maxwell equa ions and he
bulk equa ions o he magne ic medium unde he assump ions o linea magne os a ics
and negligence o magne os ic i e e ec s. Sepa a ing he ac ual magne ic ield in o he
applied magne ic ield H0( espec i ely B0 o he lux densi y) and he pe u ba ions due
o he de o med su ace deno ed as h espec i ely b
H=H0+h(A.1)
B=B0+b(A.2)
The la e can be exp essed as he g adien o a scala po en ial Φ
h=−∇Φ (A.3)
b=−µ∇Φ (A.4)
wi h µdeno ing he magne ic pe meabili y o he medium.
The magne ic ield en e s he dynamic bulk equa ions o he medium only h ough
he s ess enso Tij gi en by (2.35). Concen a ing on he magne ic con ibu ions o he
momen um conse a ion equa ion (2.33), we ob ain, since he applied magne ic ield is
assumed be cons an , he con ibu ions
∂jBihj+Hjbi+bihj−1
2(Bkhk+Hkbk+bkhk)δij(A.5)
Subs i u ing he pe u bed magne ic ields in e ms o he scala po en ial one can simpli y
(A.5) as
∂j−Bi∂jΦ−Bj∂iΦ + µ(∂iΦ)(∂jΦ) + Bk(∂kΦ)δij −1
2µ(∂kΦ)(∂kΦ)δij(A.6)
E alua ing he pa ial de i a i e ∂jlea es us wi h
−Bi∂j∂jΦ−Bj∂j∂iΦ + µ(∂iΦ)(∂j∂jΦ) + µ(∂jΦ)∂j∂iΦ + Bj∂j∂iΦ−µ(∂jΦ)∂j∂iΦ
(A.7)
Ob iously he second e m cancels he i h as well as he hi d e m cancels he six h.
The emaining wo con ibu ions cancel by ealizing ha he magne ic scala po en ial
has o sa is y he Laplace equa ion. In o al all magne ic con ibu ions cancel in he bulk
equa ions o he magne ic medium.
103
104 Decoupling o he dynamic sys em
Appendix B
Magne ic ields
Wi hin he scope o ou assump ions he hyd odynamic bulk equa ions comple ely de-
couple om he magne ic bulk equa ions. This enables us o ind solu ions o he bulk
sys em sepa a ely. Fu he mo e, since he magne ic sys em is en i ely independen o
any hyd odynamic a iable (excep ha he dis o ions o he magne ic ield should be
p opo ional o he su ace de o ma ion) i is wo h de e mining he magne ic ield o
a gi en su ace de o ma ion ξ(x, y, ) i s and subs i u e a e wa ds in o he sys em o
hyd odynamic equa ions. In his sec ion we gi e a de ailed de i a ion o all he magne ic
ield exp essions used in he main ex .
B.1 The Hea iside-Lo en z sys em o
elec omagne ic uni s
Th oughou his hesis he Hea iside-Lo en z o he a ionalized Gauss sys em o elec o-
magne ic uni s has been used o desc ibe he magne ic phenomena [69]. The choice o his
sys em is se by he undamen al hyd odynamic equa ions in chap e 2 and in pa icula
by he choice o he ene gy densi y (2.7). We will he e o e in oduce his sys em o uni s
in his sec ion based on he book o Jackson [69] and al hough his hesis only conside s
magne ic ields we will, o comple eness, include he elec ic deg ees o eedom in his
in oduc o y sec ion as well.
The undamen al laws in he elec odynamic heo y, Coulomb’s law o elec os a ics
and he Amp`e e law, only gi e p opo ionali ies be ween measu ed o ces, FCand FA e-
spec i ely, and he dis ance and he magni ude o wo elec ic cha ges o elec ic cu en s.
Coulomb’s law eads
FC=k1
q1q2
2(B.1)
whe e q1and q2a e elec ic cha ges, is he dis ance be ween hem and k1is a p opo -
ionali y cons an and Amp`e e’s law is gi en by
dFA
dl = 2k2
I1I2
(B.2)
ela ing he o ce pe uni leng h o he elec ic cu en s I1and I2 ha a e ca ied by
wo pa allel, in ini ely long conduc ing wi es o negligible c oss-sec ion sepa a ed by he
dis ance .
105

106 Magne ic ields
Quan i y Hea iside-Lo en z SI
Speed o ligh c(µ0ǫ0)−1/2
Magne ic induc ion B B/õ0
Magne ic ield Hõ0H
Magne ic scala po en ial Φ √µ0Φ
Magne iza ion Mõ0M
Magne ic pe meabili y µ µ/µ0
Table B.1: This able shows he con e sion ules be ween he Hea iside-Lo en z
sys em o uni s used in his hesis and he SI uni s ( aken om [69]) o he mac o-
scopic a iables ele an in his hesis. The magne ic pe meabili y o acuum is
gi en by µ0= 4π·10−7H/m and he speed o ligh by c= 2.99792458 ·108m/s.
The p opo ionali y cons an s k1and k2a e ei he gi en by he eqs. (B.1) and (B.2) i
he uni cha ge has been chosen independen ly o one can choose hem a bi a ily wi h he
consequence o de ining uni cha ge. Due o he common de ini ion o he elec ic cu en
as he ime a e o change o cha ge, one can gi e he ela i e dimension o k1wi h
espec o k2as k1=c2k21, whe e cdeno es he speed o ligh . In he Hea iside-Lo en z
sys em o uni s hese p opo ionali y cons an s a e a bi a ily chosen as k1= 1/(4π) and
k2= 1/(4πc2) and he Hea iside-Lo en z sys em he e o e di e s om he usual Gaussian
sys em o uni s by a ac o 4π.
Measu ed in he Hea iside-Lo en z uni s, he magne ic and he elec ic ield a iables
acqui e he same physical uni s and he cons i u i e equa ions ead
D=E+P(B.3)
H=B−M(B.4)
whe e Ddeno es he elec ic displacemen ield, E he elec ic ield and P he elec ic
pola iza ion. The las equa ion is exac ly he ela ion be ween he magne ic ield H, he
magne ic lux densi y Band he magne iza ion Mas we ob ain by he modynamic means
in eq. (2.9). In he Hea iside-Lo en z sys em o uni s he Maxwell equa ions acqui e he
o m
∇·D=ρel (B.5)
∇×H=J
c+∂D
c∂ (B.6)
∇×E=−∂B
c∂ (B.7)
∇·B= 0 (B.8)
whe e ρel and Jdeno e he elec ical cha ge densi y and i s co esponding cu en , e-
spec i ely. In he absence o he la e and o ime independen magne ic ields, which is
1A his poin one can only claim ha he ela i e dimensions o k1and k2a e ha o eloci y squa ed,
whe eas he magni ude is s ill a bi a y. De i ing he wa e equa ion om his, howe e , ixes he s ill
unde e mined magni ude o his eloci y o ha o he speed o ligh in acuum (c . [69]).
B.2 Expansion o highe pe u ba i e o de s 107
one o ou assump ions, hese equa ions become he s a ic Maxwell equa ions (2.36) and
(2.37) as used in his hesis. To con e he equa ions in his hesis o he SI sys em, he
ules gi en in able B.1 can be applied.
B.2 Expansion o highe pe u ba i e o de s
B.2.1 The Maxwell equa ions
Fo he nonlinea discussion o an ins abili y in ol ing a de o mable su ace, i is con-
enien o dis inguish he ex e nally applied magne ic ield Hex in all o de s om he
dis o ion ield hdue o he de o med su ace and co espondingly o he magne ic lux
densi ies
H=Hex +hand B=Bex +b(B.9)
The ex e nal magne ic ield is, in ou geome y, always di ec ed pa allel o he z−axis
(c . ig. 2.1 on p. 18), howe e , he magni ude emains unable. We he e o e expand he
applied ex e nal ield as well as he lux densi y acco ding o
Hex =Hc+ǫH(1) +ǫ2H(2) +... (B.10)
Bex =Bc+ǫB(1) +ǫ2B(2) +... (B.11)
whe e ǫdeno es he no malized di e ence be ween he applied magne ic ield and he
c i ical one ( his is he expansion we used in eq. (5.1)).
In addi ion, he de o med su ace will cause he magne ic ield o be dis o ed. These
de ia ions om he applied magne ic ield a e aken in o accoun by he ield hand b
ha a e also expanded simila ly as
h=ǫh(1) +ǫ2h(2) +... (B.12)
b=ǫb(1) +ǫ2b(2) +... (B.13)
The same expansion applies o he co esponding ields in he acuum.
The de ia ions om he applied ield s ill ha e o sa is y he linea magne os a ic
equa ions, b=µhand ∇·b= 0 = ∇×h, which allows o he in oduc ion o a magne ic
scala po en ial [69] h=−∇Φ ha is hen go e ned by he Laplace equa ion
∆Φ = 0 and ∆Φ ac = 0 (B.14)
The scala magne ic po en ials a ain he expansion in e ms o ǫ om he ields and
co espondingly a e w i en as
Φ = ǫΦ(1) +ǫ2Φ(2) +ǫ3Φ(3) +... (B.15)
This one o one co espondence be ween dis o ion ield hand he scala po en ial Φ
wi hin he di e en o de s is only ue wi hin he scope o ou assump ions o chap e 5
whe e long wa eleng h a ia ions o he a ising pa e n a e disca ded.
Since he Laplace equa ion is linea and homogeneous, i s expansion o he highe
o de s is i ial. The bulk solu ions can be ound in each o de independen ly and in
pa icula , F edholm’s heo em will be ul illed in each o de .
108 Magne ic ields
B.2.2 The bounda y condi ions
Since he su ace no mal nis no cons an bu depends on he su ace de lec ion (as do he
dis o ed ield con ibu ions), a highe ha monic coupling o p e ious o de s is possible
(in con as o he sys em o bulk equa ions). Fo he upcoming calcula ion i is use ul
o de e mine i s he ields a he bounda y z=ξ
H=Hc+ǫH(1) −(∇Φ(1))z=0+ǫ2H(2) −(∇Φ(2))−ξ(1)(∂z∇Φ(1))z=0
+ǫ3H(3) −(∇Φ(3))z=0−ξ(1)(∂z∇Φ(2))z=0−1
2[(ξ(1)2∂2
z+2ξ(2)∂z)∇Φ(1)]z=0(B.16)
and acco dingly o he magne ic ield H ac and he magne ic lux densi ies Band B ac.
The con ibu ions in (B.16) ha a e explici ly p opo ional o ξ(1) o ξ(2) a e due o he
de o mable su ace.
As men ioned, he su ace no mal n, ini ially di ec ed pa allel o he z−axis, changes
i s o ien a ion in he cou se o ime as he su ace pe u ba ion g ows (c . ig. 2.1 on
p. 18). To gi e a p ope expansion o he bounda y condi ions, we addi ionally ha e o
expand he su ace no mal as a unc ion o he su ace de lec ion ξ(x, y, )
n=n0+ǫn(1) +ǫ2n(2) +ǫ3n(3) (B.17)
wi h he di e en pe u ba i e con ibu ions gi en by
n(1) =


−∂xξ(1)
−∂yξ(1)
0


,n(2) =


−∂xξ(2)
−∂yξ(2)
1
2(∂xξ(1))2+1
2(∂yξ(1))2


(B.18)
and n(3) =


−∂xξ(3) −1
2(∂yξ(1))2(∂xξ(1))−1
2(∂xξ(1))3
−∂yξ(3) −1
2(∂xξ(1))2(∂yξ(1))−1
2(∂yξ(1))3
(∂yξ(1))(∂yξ(2)) + (∂xξ(1))(∂xξ(2))


(B.19)
Wi h he p e ious conside a ions on hand, we a e able o expand he bounda y con-
di ions in e ms o ǫ. The ac ha he no mal componen o he magne ic lux densi y is
con inuous a he bounda y gi es he ollowing condi ion
n·H ac −H=n·B ac −B+M
=n·M(B.20)
Conside he linea pe u ba i e o de o he las equa ion
n(1) ·H ac
c−Hc+n(0) ·H(1) ac −H(1)=n(1) ·M(0) +n(0) ·M(1) (B.21)
Fo he cons an con ibu ions (cons an wi h espec o xand y), we ind
H(1) ac
z−H(1)
z=M(1)
z(B.22)
while he con ibu ions p opo ional o n(1) cancel iden ically. The co esponding exp es-
sion o he second o de con ibu ion o he applied ield, H(2) ac
z−H(2)
z=M(2)
z, can be
ob ained s aigh o wa dly.
B.2 Expansion o highe pe u ba i e o de s 109
The bounda y condi ion o he angen ial componen s o he magne ic ield (2.38) is
gi en in linea o de
n(1) ×(H ac
c−Hc) + n(0) ×H(1) ac −H(1)= 0 (B.23)
which can be simpli ied subs i u ing eq. (B.22) o (wi h a∈ {x, y})
h(1) ac
a−h(1)
a=−(∂aξ(1))M0(B.24)
In he second pe u ba i e o de we ind
n(2) ×H ac
c−Hc+n(1) ×H(1) ac −H(1) −∇Φ(1) ac +∇Φ(1)
+n(0) ×H(2) ac −∇Φ(2) ac +kcξ(1)∇Φ(1) ac −H(2) +∇Φ(2) +kcξ(1)∇Φ(1)= 0
(B.25)
which is simpli ied in he same manne (by exploi ing he esul s o he p e ious o de )
o
∂aΦ(2) ac −∂aΦ(2)−(∂aξ(2))Mc−(∂aξ(1))M(1)
+(∂aξ(1))∂zΦ(1) ac −∂zΦ(1)−kcξ(1)∂aΦ(1) ac +∂aΦ(1)= 0 (B.26)
wi h a∈ {x, y}. Upon subs i u ing he linea solu ions (B.34) and (B.35) his immedia ely
leads o exp essions (B.39) and (B.40) used in sec ion B.4 o ind he magne ic eigen ec o s
in he second pe u ba i e o de . Finally we deduce o he angen ial bounda y condi ion
in he hi d pe u ba i e o de
n(3) ×(H ac
c−Hc) + n(2) ×H(1) ac −H(1) −(∇Φ(1) ac) + (∇Φ(1))
+n(1) ×H(2) ac −H(2) −(∇Φ(2) ac) + (∇Φ(2)) + kcξ(1)(∇Φ(1) ac) + kcξ(1)(∇Φ(1))
+n(0) ×H(3) ac −H(3) −(∇Φ(3) ac) + (∇Φ(3))−ξ(1)(∂z∇Φ(2) ac) + ξ(1)(∂z∇Φ(2))
−1
2(k2
cξ(1)2 −2kcξ(2))(∇Φ(1) ac) + 1
2(k2
cξ(1)2 + 2kcξ(2))(∇Φ(1))
= 0 (B.27)
whe e i will be su icien o ou discussion o conside only he con ibu ions p opo ional
o he main cha ac e is ic modes ξ(1) as discussed in sec ion 5.3.2.
Along he same lines he bounda y condi ion ha gua an ees he con inui y o he
no mal componen (2.39) o he magne ic lux densi y is de i ed. In i s pe u ba i e
o de we ge
n(1) ·(B ac
c−Bc) + n(0) ·B(1) ac −B(1)= 0 (B.28)
which is s aigh o wa dly simpli ied o
b(1) ac
z−b(1)
z= 0 (B.29)
Fo he co esponding condi ion in he second pe u ba i e o de we ob ain
n(2) ·B ac
c−Bc+n(1) ·B(1) ac −B(1) −∇Φ(1) ac +µ∇Φ(1)
+n(0) ·B(2) ac −B(2) −∇Φ(2) ac +µ∇Φ(2) +kcξ(1)∇Φ(1) ac +µkcξ(1)∇Φ(1)= 0
(B.30)
116 Magne ic ields

Appendix C
The hyd odynamic bounda y
condi ions
C.1 Expansion o he bounda y condi ions
In his sec ion we discuss he expansion o he hyd odynamic bounda y condi ions in
e ms o ǫ. Recall i s , ha we equi e he angen ial s ess a he ee su ace o anish
whe eas he no mal s ess is balanced by su ace ension and g a i y
n×T·n=n×T ac ·n(C.1)
n·T·n−n·T ac ·n=σT∇·n−ρGξ (C.2)
The con ibu ions o he s ess enso o he di e en pe u ba i e o de s a e de ined by
he expansions o he mac oscopic a iables, eqs. (5.1,5.2), and by he expansion o he
su ace no mal n, eq. (B.17). The linea eigen ec o s o he hyd odynamic se o equa ions
a e ei he p opo ional o ekz o eqz (c . sec ion 3.5). Fo he bounda y condi ions one
has o e alua e hem a z=ξand he e o e an expansion simila o (B.16) is needed ha
explici ly accoun s o he de o mabili y o he su ace.
C.2 The linea pe u ba i e o de
The bounda y condi ions in linea o de a e s aigh o wa dly calcula ed, bu a e epea ed
he e o comple eness. Fo he angen ial s ess bounda y condi ion we ob ain
2µ2ǫ(1)
yz +ν2∂z (1)
y+∂y (1)
z= 0 (C.3)
2µ2ǫ(1)
xz +ν2∂z (1)
x+∂x (1)
z= 0 (C.4)
whe eas he no mal s ess bounda y condi ion eads
2µ2ǫ(1)
zz + 2ν2∂z (1)
z−p(1) +Gρξ(1) −µH0∂zΦ(1) −H ac
0∂zΦ(1) ac=σT∇·n(1)
(C.5)
In sec ion 3.2 we in oduced po en ials o he i o a ional and he o a ional low
con ibu ions. To sol e he sys em o equa ions o he po en ials we ha e o ansla e
he bounda y condi ions abo e o he co esponding ones alid o he po en ials. We
117
118 The hyd odynamic bounda y condi ions
s a wi h he angen ial bounda y condi ions. To be able o exp ess he s ain ield in
e ms o he g adien o he eloci y ield (3.9), we i s ha e o ake he de i a i e o
eqs. (C.3,C.4) wi h espec o ime. Upon subs i u ion o eqs. (3.11) we end up wi h
˜µ2(∂2
z−∂2
y)Ψ(1)
x+ ˜µ2(∂y∂x)Ψ(1)
y+ 2˜µ2∂y∂zϕ(1) = 0 (C.6)
˜µ2(∂x∂y)Ψ(1)
x+ ˜µ2(∂2
z−∂2
x)Ψ(1)
y−2˜µ2∂x∂zϕ(1) = 0 (C.7)
whe e he coe icien (µ2+ν2∂(0)
) has been abb e ia ed by ˜µ2.
Fo he no mal s ess bounda y condi ion we addi ionally ha e o discuss he con ibu-
ions ha explici ly con ain he su ace de lec ion ξ(1). A a i s glance hese con ibu ions
may be conside ed as inhomogenei ies. Howe e , he de lec ion ξ(1) is ela ed o he local
eloci y. Upon aking he ime de i a i e o eq. (C.5) we can subs i u e he linea ized
kinema ic bounda y condi ion ∂ ξ(1) = (1)
z. Following he same lines as in he case o he
angen ial bounda y condi ions, we ind
−(2˜µ2∂z∂y+Gρ∂y+σTk2∂y)Ψ(1)
x+ (2˜µ2∂z∂x+Gρ∂x+σTk2∂x)Ψ(1)
y
+(2˜µ2∂2
z+Gρ∂z+σTk2∂z)ϕ(1) −∂ p(1) −∂ H0µ∂zΦ(1) −H ac
0∂zΦ(1) ac= 0
(C.8)
Addi ionally, we can subs i u e he solu ion o he p essu e p(1), ob ained in sec ion
3.2, and he solu ions o he magne ic ields, ob ained in appendix B, o inally a i e a
−(2˜µ2∂z∂y+Gρ∂y+σTk2∂y−µ
1 + µM2
0∂z∂y)Ψ(1)
x
+(2˜µ2∂z∂x+Gρ∂x+σTk2∂x−µ
1 + µM2
0∂z∂x)Ψ(1)
y
+(2˜µ2∂2
z+Gρ∂z+σTk2∂z−ρω2−µ
1 + µM2
0∂2
z)ϕ(1) = 0 (C.9)
whe e again he explici su ace de lec ion ξ(1), a ising om he magne ic solu ions, has
been eplaced by he local eloci y.
The s eps leading o eq. (C.8), whe e we i s ook he ime de i a i e o eq. (C.5) o
implemen a e wa ds he kinema ic bounda y condi ion, a e e y c ucial. By inspec ion
o eq. (C.5) we ealize ha i is ini e in he s a iona y limi e ealing he o ce balance
be ween elas ic, g a i a ion, magne ic and su ace ension o ces. Eq. (C.8), howe e , is
a leas linea in he ime de i a i e and so a e he eigen ec o s de i ed om i . The
la e p ope y is in acco dance wi h he unde s anding ha he e is no mo ion in he
medium, i he su ace pa e n is ully de eloped. These conside a ions show ha , due o
he kinema ic bounda y condi ion, he bulk equa ions inhe en ly scale one o de highe
wi h espec o he ime de i a i e, when compa ed wi h he no mal s ess bounda y
condi ion. This di e en scaling beha io has o be aken in o accoun , i we combine he
sol abili y condi ions om he bulk wi h hose om he no mal s ess bounda y condi ion
(chap e 5).
C.3 The second pe u ba i e o de 119
C.3 The second pe u ba i e o de
In he second o de we ind he angen ial bounda y condi ions in ol ing he hyd ody-
namic ields as
2µ2ǫ(2)
xz +ν2∂z (2)
x+∂x (2)
z
=−ξ(1)∂z2µ2ǫ(1)
xz +ν2(∂x (1)
z+∂z (1)
x)+ (∂yξ(1))2µ2ǫ(1)
yz +ν2(∂y (1)
x+∂x (1)
y)
−2(∂xξ(1))µ2(ǫ(1)
zz +ǫ(1)
xx ) + ν2(∂z (1)
z+∂x (1)
x)+ρ (1)
x (1)
z≡Ω(2)
xz (C.10)
2µ2ǫ(2)
yz +ν2∂z (2)
y+∂y (2)
z
=−ξ(1)∂z2µ2ǫ(1)
yz +ν2(∂y (1)
z+∂z (1)
y)+ (∂xξ(1))2µ2ǫ(1)
xy +ν2(∂x (1)
y+∂y (1)
x)
−2(∂yξ(1))µ2(ǫ(1)
zz +ǫ(1)
yy ) + ν2(∂z (1)
z+∂y (1)
y)+ρ (1)
y (1)
z≡Ω(2)
yz (C.11)
In eqs. (C.11) and (C.10) he inhomogenei ies on he igh hand side ha e been abb e ia ed
by Ω(2)
xz and Ω(2)
yz , espec i ely. In pa icula hese inhomogenei ies a e p opo ional o
[ξ(1)]2.
The no mal s ess bounda y condi ion (C.2) eads in second o de
2µ2ǫ(2)
zz + 2ν2∂z (2)
z−p(2) +Gρξ(2) −µHc∂zΦ(2) +H ac
c∂zΦ(2) ac
=−2µ2[ǫ(1)
zz ]2+ρ[ (1)
z]2−McBc(∂yξ(1))2+ (∂xξ(1))2−1
2µ∂zΦ(1)2
+1
2∂zΦ(1) ac2−2µHc(∂yξ(1))(∂yΦ(1)) + 2H ac
c(∂yξ(1))(∂yΦ(1) ac)
+1
2µ(∂yΦ(1))2−1
2(∂yΦ(1) ac)2+1
2µ(∂xΦ(1))2−1
2(∂xΦ(1) ac)2
−2µHc(∂xξ(1))(∂xΦ(1)) + 2H ac
c(∂xξ(1))(∂xΦ(1) ac)
+ξ(1)∂z2µ2ǫ(1)
zz + 2ν2∂z (1)
z−p(1) −µ
1 + µM2
ckξ(1)+µ
1 + µM(1)Mckcξ(1)
−σT∆ξ(2) (C.12)
Fu he mo e, we ob ain o he kinema ic bounda y condi ion in second o de
∂(0)
ξ(2) +∂(1)
ξ(1) + ( (1) ·∇)ξ(1) = (2)
z+ξ(1)∂z (1)
z(C.13)
The physical bounda y is a z=ξ, gi ing ise o an addi ional dependence on ξ. In
eqs. (C.11-C.13) such e ms ha e al eady been made explici (e.g. he las one o (C.13)).
Thus hese bounda y condi ions a e e ec i e ones ha ha e o be aken a z= 0.
Inspec ing he exp essions (C.12) and (C.13) one immedia ely ealizes ha wo qual-
i a i ely di e en con ibu ions a e p esen . On he one hand we ob ain con ibu ions
p opo ional o he highe ha monic coupling [ξ(1)]2o he main cha ac e is ic mode. On
he o he hand, he e a e s ill con ibu ions p opo ional o he main cha ac e is ic mode
ξ(1) i sel . The la e will allow us o ind he linea con ibu ions in an ampli ude equa ion
e en hough he con ol pa ame e is no p esen in he bulk equa ions.
To sol e he co esponding hyd odynamic bulk equa ions, we in oduced a scala ,
ϕ(2), and a ec o po en ial, Ψ(2), in sec ion 5.3, o discuss po en ial and o a ional low
con ibu ions sepa a ely. Following he same lines as done in he linea o de (sec ion
C.2), we can ansla e he bounda y condi ions in o a co esponding se o equa ions o
120 The hyd odynamic bounda y condi ions
he ampli udes o he second o de po en ials ϕ(2) and Ψ(2). We ob ain o he angen ial
con ibu ions (C.11) and (C.10)
˜µ2(∂2
z−∂2
y)Ψ(2)
x+˜µ2(∂y∂x)Ψ(2)
y+2˜µ2∂y∂zϕ(2) = 2µ2 (1)
k∂kǫ(1)
yz +∂(0)
Ω(2)
yz (C.14)
−˜µ2(∂x∂y)Ψ(2)
x−˜µ2(∂2
z−∂2
x)Ψ(2)
y+2˜µ2∂x∂zϕ(2) = 2µ2 (1)
k∂kǫ(1)
xz +∂(0)
Ω(2)
xz (C.15)
using ǫ(1)
xz =ǫ(1)
yz ≡0 a he bounda y (c . eqs. (C.3,C.4)). The no mal s ess bounda y
condi ion (C.12) ansla es in o
−(2˜µ2∂y∂z+ρG∂y)Ψ(2)
x+ (2˜µ2∂z∂x+ρG∂x)Ψ(2)
y+ (2˜µ2∂2
z+ρG∂z)ϕ(2) −∂(0)
p(2)
=∂(0)
Hcµ∂zΦ(2) −H ac
c∂zΦ(2) ac+M(1)Mckc
µ
1 + µ∂(0)
ξ(1) +∂(0)
Ω(2)
zz
+2µ2 (1)
k∂kǫ(1)
zz −2ρGξ(1)∂z (1)
z+ρG∂(1)
ξ(1) + 2µ2∂(1)
ǫ(1)
zz −σT∂(0)
∆ξ(2) (C.16)
Eqs. (C.14-C.16) ollow om (C.11-C.12) by aking he ime de i a i e wi h espec o
(0) wi hou loss o gene ali y. This is why in eq. (C.16) only he con ibu ion ∂(0)
p(2) and
no con ibu ion ∂(1)
p(1) a ises, while ∂(0)
ǫ(2)
ij gi es ise o con ibu ions ∼ (2)
iand ∼∂(1)
ǫ(1)
ij
(c . eq. (5.58)).
C.4 The hi d pe u ba i e o de
We ake o e he p ocedu e o he p e ious sec ion o he hi d o de . I we use he solu-
ions (5.81,5.85,5.87) o he hyd odynamic bulk equa ions in second o de , he kinema ic
bounda y condi ion eads
∂(0)
ξ(3) +∂(1)
ξ(2) +∂(2)
ξ(1) + ( (1) ·∇)ξ(2) + ( (2) ·∇)ξ(1) (C.17)
= (3)
z+ξ(1)∂z (2,2)
z+ξ(1)∂z (2,1)hom
z−µ2+˜µ2
q˜µ2
k2
cξ(1)∂(1)
ξ(1) +ξ(2)∂z (1)
z+1
2ξ(1)2∂2
z (1)
z
The angen ial bounda y condi ions a e o he usual s uc u e and gi en as
2µ2ǫ(3)
yz +ν2∂z (3)
y+∂y (3)
z= Ω(3)
yx (C.18)
2µ2ǫ(3)
yz +ν2∂z (3)
y+∂y (3)
z= Ω(3)
xz (C.19)
whe e inhomogeneous con ibu ions, which a e a leas p opo ional o he highe ha -
monic couplings, a e collec ed in he abb e ia ion Ω(3)
ij , as done simila ly in second o de .
The only eason why we ha e o conside he hi d o de bounda y condi ions is o ob ain
he linea con ibu ions o he ampli ude equa ion. The gene al solu ion in ol ing he
highe ha monic couplings is no needed. Since Ω(3)
ij is a leas p opo ional o he highe
ha monic couplings i is he e o e unimpo an in his discussion and is no shown he e.
Taking he ime de i a i e o eqs. (C.18,C.19) oge he wi h (5.118) we ind
˜µ2(∂2
z−∂2
y)Ψ(3)
x+ ˜µ2(∂y∂x)Ψ(3)
y+ 2˜µ2∂y∂zϕ(3) =∂(0)
Ω(3)
yx + 2µ2(∂(1)
ǫ(2)
yz +∂(2)
ǫ(1)
yz )
+2µ2( (1)
k∂kǫ(2)
yz + (2)
k∂kǫ(1)
yz )(C.20)
−˜µ2(∂x∂y)Ψ(3)
x−˜µ2(∂2
z−∂2
x)Ψ(3)
y+ 2˜µ2∂x∂zϕ(3) =∂(0)
Ω(3)
xz + 2µ2(∂(1)
ǫ(2)
xz +∂(2)
ǫ(1)
xz )
+2µ2( (1)
k∂kǫ(2)
xz + (2)
k∂kǫ(1)
xz )(C.21)
C.4 The hi d pe u ba i e o de 121
Fo he no mal s ess bounda y condi ion we ob ain om eq. (C.2)
2µ2ǫ(3)
zz + 2ν2(∂z (3)
z)−p(3) +ρGξ(3) −(µHc∂zΦ(3) −H ac
c∂Φ(3) ac)
= (µH(2)∂zΦ(1) −H(2) ac∂zΦ(1) ac) + (µH(1)∂zΦ(2) −H(1) ac∂zΦ(2) ac)
+Ω(3)
zz +σT∇·n(3) (C.22)
which by a simila p ocedu e can be w i en as
−(2˜µ2∂z∂y+ρG∂y)Ψ(3)
x+ (2˜µ2∂z∂x+ρG∂x)Ψ(3)
y+ (2˜µ2∂2
z+ρG∂z)ϕ(3) −∂(0)
p(3)
= (µHc∂zΦ(3) −H ac
c∂zΦ(3) ac) + µ
1 + µ(McM(2) +M(1)2)kc∂(0)
ξ(1) +∂(0)
Ω(3)
zz
+ρG(∂(2)
ξ(1) +∂(1)
ξ(2) + (2)
k∂kξ(1) −ξ(1)∂z (2)
z−ξ(2)∂z (1)
z−1
2ξ(1)2∂2
z (1)
z)
+2µ2(∂(2)
ǫ(1)
zz +∂(1)
ǫ(2)
zz + ( (1)
k∂k)ǫ(2)
zz + ( (2)
k∂k)ǫ(1)
zz ) + σT∂(0)
∇·n(3) (C.23)

122 The hyd odynamic bounda y condi ions
Appendix D
Eigen ec o s in he second o de
In his appendix we gi e he con ibu ions o he eigen ec o s in he second pe u ba-
i e o de ha a e p opo ional o he highe ha monic couplings ξ(2). Due o he ac
ha we ha e o ea he sys em dynamically h oughou all o de s, he exp essions be-
come edious and ha e he e o e been calcula ed wi h Ma hema ica. In he ollowing he
solu ions o he hyd odynamic po en ials a e ep esen ed o he pa e ns unde consid-
e a ion, hexagons (θij = 2π/3), squa es (θij =π/2) and s ipes (θij = 0) as well as o he
in e ac ion be ween hexagons and squa es (θij =π/6) (c . ig. 5.2 on p. 59).
The inhomogeneous con ibu ions o he ec o po en ial, c . eq. (5.98), sepa a e in o
a con ibu ion ∼e(kc+q)zand ∼e2qz. Fo he hexagonal case (ij =ji = 12 = 23 = 31) we
ob ain
Ψinhom
NMij(z) = (k2
c+q2)(2µ2q2−5µ2qkc+ρ[D(0)
]2)
2q(k2
c−q2)(2kcq˜µ2+q2˜µ2−ρ[D(0)
]2)e(kc+q)zD(0)
+3k2
cq(µ2k2
c+ 2µ2q2−ρ[D(0)
]2)
(k4
c−5k2
cq2+ 4q4)(˜µ2k2
c−4˜µ2q2+ρ[D(0)
]2)e2qzD(0)
(D.1)
wi h {N, M} ∈ {R, L}. The abb e ia ion D(0)
s ands o 2iω(0) + 2σ(0), 2σ(0), and
−2iω(0) + 2σ(0) o N=M=R,N6=M, and N=M=L, espec i ely. The sec-
ond coe icien ˜
Ψinhom
NMij eads
˜
Ψinhom
NMij(z) = (k2
c+q2)(6k3
cµ2−10k2
cqµ2+kcq2µ2+2q3µ2+qρ[D(0)
]2)
2(2k4
c−2k3
cq−3k2
cq2+2kcq3+q4)(2k2
c˜µ2−2kcq˜µ2−q2˜µ2+ρ[D(0)
]2)e(kc+q)zD(0)
−k2
cq(k2
cµ2−2q2µ2+ρ[D(0)
]2)
(3k4
c−7k2
cq2+ 4q4)(3k2
c˜µ2−4q2˜µ2+ 2ρ[D(0)
]2)e2qzD(0)
(D.2)
Fo he squa e pa e n we ge (ij =ji = 15)
Ψinhom
NMij(z) = (k2
c+q2)(4µ2k3
c−10µ2qk2
c+2µ2q3+(kc+q)ρ[D(0)
]2)
2(k4
c−2qk3
c−2k2
cq2+2kcq3+q4)(˜µ2k2
c−2˜µ2qkc−˜µ2q2+ρ[D(0)
]2)e(q+kc)zD(0)
+qk2
c(2µ2q2−ρ[D(0)
]2)
4(k4
c−3q2k2
c+2q4)(2˜µ2k2
c−4˜µ2q2+ρ[D(0)
]2)e2qzD(0)
(D.3)
=˜
Ψinhom
NMij(z) (D.4)
123
124 Eigen ec o s in he second o de
Fo he s ipe geome y, i=j, we ob ain
Ψinhom
NMij(z) = (k2
c+q2)6k2
cµ2−4kcqµ2−2q2µ2−ρ[D(0)
]2
2(kc−q)(3k2
c+4kcq+q2)3k2
c˜µ2−2kcq˜µ2−q2˜µ2+2ρ[D(0)
]2e(kc+q)zD(0)
(D.5)
and
˜
Ψinhom
NMij(z) = k2
c4k4
cµ2−4q2(2q2µ2−ρ[D(0)
]2)−k2
c(20q2µ2+ 3ρ[D(0)
]2)
4(k2
c−q2)2(4q3˜µ2+qρ[D(0)
]2)e2qzD(0)
+Z1h4(kc−q)2(kc+q)3k2
c˜µ2+2kcq˜µ2+q2˜µ2−2ρ[D(0)
]2i−1e(kc+q)zD(0)
(D.6)
wi h he nume a o Z1being gi en by
Z1= (k2
c+q2)24k3
cqµ2−4q4µ2−2q2ρ[D(0)
]2+20k2
cq2µ2+3k2
cρ[D(0)
]2
+8kcq3µ2−4kcqρ[D(0)
]2(D.7)
In addi ion, o desc ibe he in e ac ion be ween he squa e and he hexagonal pa e n,
we need o conside also he case θij =π/6, (ij =ji = 14 = 36 = 25)
Ψinhom
NMij(z) = N−1
1n(k2
c+q2)(2q3µ2−10k2
cqµ2+k2
cµ2(1+3√3)+qρ[D(0)
]2(D.8)
+
√3kcq2µ2+kc(1−√3)ρ[D(0)
]2)oe(kc+q)zD(0)
−k2
cq(k2
cµ2(2√3−3) −2(2 −√3)(q2µ2−ρ[D(0)
]2))
((2 + √3)k2
c−4q2)(k2
c−q2)((2 + √3)k2
c˜µ2−(4q2˜µ2−2ρ[D(0)
]2))e2qzD(0)
whe e he denomina o N1is gi en by
N1= 2(k2
c−q2)n2(2+√3)k4
c˜µ2−4(1+√3)k3
cq˜µ2+q4˜µ2−q2ρ(D(0)
)2
+k2
c(1+√3)ρ[D(0)
]2+2(1−√3)q2˜µ2)+4kcq3˜µ2−2kcqρ[D(0)
]2o(D.9)
and
˜
Ψinhom
NMij(z) = (3 + 2√3)k2
cµ2+ (2 + √3)(2q2µ2−ρ[D(0)
]2)
(k2
c−q2)(√3−2)k2
c+ 4q2(√3−2)k2
c˜µ2+ 4q2˜µ2−ρ[D(0)
]2qk2
ce2qzD(0)
−N−1
2n(k2
c+q2)(3√3−1)k3
cµ2+10k2
cqµ2−2q3µ2
−2qρ[D(0)
]2+kc√3q2µ2−kc[1+√3]ρ[D(0)
]2oe(kc+q)zD(0)
(D.10)
wi h
N2= 2(k2
c−q2)h2(2−√3)k4
c˜µ2+4(√3−1)k3
cq˜µ2+q4˜µ2−q2ρ[D(0)
]2
+2k2
c[1+√3]q2˜µ2+k2
c[1−√3]ρ[D(0)
]2+4kcq3˜µ2−qkcρ[D(0)
]2i(D.11)
125
Fo he scala po en ial we ob ain om eq. (5.109) in he geome y o hexagons (ij =
ji = 12 = 23 = 31)
ˆϕNMij =h8ρ[D(0)
]2kcq(q+kc)(k2
c−4q2)i−1n2k5
cq(12˜µ2−14µ2−3ν2D(0)
)−12q4ρ[D(0)
]2
+2k4
c4˜µ2q2−16µ2q2+ν2q2D(0)
+2ρ[D(0)
]2
+k2
c19q2ρ[D(0)
]2+8q4[4˜µ2+4µ2+ν2]D(0)

−4kc9q3ρ[D(0)
]2−4q5[3µ2+ν2D(0)
]
+k3
c17qρ[D(0)
]2−96˜µ2q3+52µ2q3+20q3ν2D(0)
oD(0)
(D.12)
and o he case o squa es (ij =ji = 15)
ˆϕNMij =h4√2ρ[D(0)
]2kc(kc+q)(k2
c−2q2)(k2
c−2kcq−q2)i−1n4k7
c(6˜µ2−7µ2−2ν2D(0)
)
+2q5ρ[D(0)
]2−4k6
cq(10˜µ2−14µ2−5ν2D(0)
)
+k2
c16q5˜µ2−16q5µ2−8q5ν2D(0)
+9q3ρ[D(0)
]2
+2kc5q4ρ[D(0)
]2−4q6(3µ2+ν2D(0)
)+k5
c3ρ[D(0)
]2−4q2[22˜µ2−14µ2−3ν2D(0)
]
−k3
c35q2ρ[D(0)
]2−4q4[20˜µ2+15µ2+3ν2D(0)
]
−k4
c13qρ[D(0)
]2−4q3[18˜µ2−22µ2−9ν2D(0)
]oD(0)
(D.13)
Fo s ipes (i=j) we ob ain
ˆϕNMij =h(kc−q)(kc+q)(3kc+q)ρ[D(0)
]2i−1n2kc(2kc−q)ρ[D(0)
]2
+(kc−q)(3kc+q)k2
c(6˜µ2−7µ2)−3q2µ2+2kcq(˜µ2+2µ2)
−ν2(kc−q)(3kc+q)(3k2
c−2kcq+q2)D(0)
oD(0)
(D.14)
Fo θij =π/6, (ij =ji = 14 = 36 = 25) we ob ain
132 Usual e o luids
shown he e. Wi h he solu ions o he second o de , he hi d o de F edholm’s heo em
can be ul illed. The la e eads in he case o e o luids
h¯ i|ρ∂(2)
(1)
ii+h¯ i|ρ∂(1)
(2,1)
ii=−h¯ i|ρ∂(1)
(2,2)i−ρh¯ i|∂j( (1)
i (2,1)
j+ (2,1)
i (1)
j)i
−ρh¯ i|∂j( (1)
i (2,2)
j+ (2,2)
i (1)
j)i(E.5)
Since he analy ical exp essions o he eigen ec o s (2,2)
ia e bulky, he explici calcula ion
o he cubic coe icien s has been pe o med wi h Ma hema ica and he esul s a e shown
below. We should men ion, howe e , ha also in he hi d o de he igh hand side o
eq. (E.5) is p opo ional o [∂(0)
]3and he global ac o ([ω(0)]2−[σ(0)]2) can be canceled.
The discussion o he no mal s ess bounda y condi ion in he case o e o luids can
be aken om he sec ions 5.3.4 and 5.4. In he second o de , he addi ional condi ion o
he ampli udes (5.115) is alid o e ogels and e o luids, alike, and in he co esponding
hi d o de condi ion (5.133) we ha e o subs i u e µ2→0 wi h he consequence ha
he e is no second o de ime de i a i e. The ypical ime scale in he case o e o luids is
hen gi en by τ0=ν2kc/(ρG) which is in acco dance wi h p e ious heo e ical discussions
[39]. The inal ampli ude equa ion is de i ed in he same way as in sec ion 5.5 o magne ic
gels and inally esul s o he hexagonal pa e n in
∂Tξ1=1
2˜ǫ lξ1−2
3√Aξ∗
2ξ∗
3− |ξ1|2ξ1−B l
120
A l(|ξ2|2+|ξ3|2)ξ1(E.6)
Fo he squa e pa e n he quad a ic coe icien is absen and we ob ain
∂Tξ1= ˜ǫ lξ1− |ξ1|2ξ1−B l
90
A l|ξ5|2ξ1(E.7)
whe e he cubic coe icien s a e gi en by
A l≈8.625 (E.8)
B l
120 ≈3.150 (E.9)
B l
90 ≈4.266 (E.10)
The discussion o he di e en s able pa e ns ollows he same lines as in sec ion 5.5.
A he linea onse we ind hexagons o be he p e e ed pa e n, which emains subc i i-
cally s able o con ol pa ame e s la ge han
˜ǫA=−4
9(A l+ 2B l
120)(E.11)
Since B l
90 + 2B l
30 < A l+ 2B l
120 and B l
90/A l<1, whe e he cubic coe icien accoun ing
o he nonlinea in e ac ion be ween hexagons and squa es is gi en by B l
30 ≈4.545, he
hexagon pa e n ans o ms in o a squa e pa e n o con ol pa ame e s la ge han
˜ǫB=2(B l
90 + 2B l
30)
9(A l+ 2B l
120 −B l
90 −2B l
30)2(E.12)
The squa e pa e n in u n becomes uns able again o con ol pa ame e s lowe han
˜ǫS=2(A l+B l
90)
9(A l+B l
90 −B l
120 −B l
30)2(E.13)

Bibliog aphy
[1] F. Bi e , On inhomogenei ies in he magne iza ion o e omagne ic ma e ials,
Phys. Re . 38, 1903 (1931).
[2] W. C. Elmo e, Fe omagne ic colloid o s udying magne ic s uc u es, Phys. Re .
54, 309 (1938).
[3] R. E. Rosensweig, Fe ohyd odynamics, Camb idge Uni e si y P ess, Camb idge,
UK, 1985.
[4] C. Holm, A. I ano , S. Kan o o ich, E. Pyanzina and E. Rezniko , Equilib ium
p ope ies o a bidispe se e o luid wi h chain agg ega es: heo y and compu e
simula ions, J. Phys.: Condens. Ma e 18, S2737 (2006).
[5] A. Y. Zuba e and L. Y. Iskako a, Rheological p ope ies o e o luids wi h mi-
c os uc u es, J. Phys.: Condens. Ma e 18, S2771 (2006).
[6] L. M. Pop and S. Odenbach, In es iga ion o he mic oscopic eason o he mag-
ne o iscous e ec in e o luids s udied by small angle neu on sca e ing, J. Phys.:
Condens. Ma e 18, S2785 (2006).
[7] I. Hilge , E. Die ma , W. Linß, S. S eck and W. A. Kaise , De elopmen s o he
minimally in asi e ea men o umou s by a ge ed magne ic hea ing, J. Phys.:
Condens. Ma e 18, S2951 (2006).
[8] R. Ju gons, C. Selige , A. Hilpe , L. T ahms, S. Odenbach and C. Alexiou, D ug
loaded magne ic nanopa icles o cance he apy, J. Phys.: Condens. Ma e 18,
S2893 (2006).
[9] K. Au ich, G. Gl¨
ockl, E. Romanus, P. Webe , S. Nagel and W. Wei schies, Magne o-
op ical elaxa ion measu emen s o he cha ac e iza ion o biomolecula in e ac-
ions, J. Phys.: Condens. Ma e 18, S2847 (2006).
[10] T. Tanaka, Collapse o gels and he c i ical endpoin , Phys. Re . Le . 40, 820
(1978).
[11] T. Tanaka, I. Nishio, S. T. Sun and S. Ueno-Nishio, Collapse o gels in an elec ic
ield, Science 218, 467 (1982).
[12] M. Z ´ınyi, L. Ba si and A. B¨
uki, De o ma ion o e ogels induced by nonuni o m
magne ic ields, J. Chem. Phys. 104, 8750 (1996).
133
134 BIBLIOGRAPHY
[13] G. La e mann and M. K ekho a, The mo e e sible e ogels, Mac omol. Rapid
Commun. 27, 1373 (2006).
[14] Z. Va ga, J. Feh´e , G. Filipcsei and M. Z ´ınyi, Sma nanocomposi e polyme gels,
Mac omol. Symp. 200, 93 (2003).
[15] D. Collin, G. K. Aue nhamme , O. Ga a , P. Ma ino y and H. R. B and, F ozen-
in magne ic o de in uniaxial magne ic gels: P epa a ion and physical p ope ies,
Mac omol. Rapid Commun. 24, 737 (2003).
[16] T. Mi suma a, K. Juliac, K. Fu ukawa, K. Iwaku a, T. Taniguchi and K. Koyoma,
Aniso opy in longi udinal modulus o polyme gels con aining e i e, Mac omol.
Rapid Commun. 23, 175 (2002).
[17] J. Weilepp, J.-J. Zanna, N. Aß alg, P. S ein, L. Hilliou, M. Mauzac, H. Finkelmann,
H. R. B and and P. Ma ino y, Rheology o liquid c ys alline elas ome s in hei
iso opic and smec ic A s a e, Mac omolecules 32, 4566 (1999).
[18] J.-J. Zanna, P. S ein, J.-D. Ma y, M. Mauzac and P. Ma ino y, In luence o
molecula pa ame e s on he elas ic and iscoelas ic p ope ies o side-chain liquid
c ys alline elas ome s, Mac omolecules 35, 5459 (2002).
[19] M. D. Cowley and R. E. Rosensweig, The in e acial s abili y o a e omagne ic
luid, J. Fluid Mech. 30, 671 (1967).
[20] S. Bohlius, H. R. B and and H. Pleine , Su ace wa es and Rosensweig ins abili y
in iso opic e ogels, Z. Phys. Chem 200, 97 (2006).
[21] C. Gollwi ze , p i a e communica ion.
[22] A. T. Skjel o p, One- and wo-dimensional c ys alliza ion o magne ic holes, Phys.
Re . Le . 51, 2306 (1983).
[23] B. J. D. Gans, N. J. Duin, D. an den Ende and J. Mellema, The in luence o
pa icle size on he magne o heological p ope ies o an in e se e o luid, J. Chem.
Phys. 113, 2032 (2000).
[24] R. Saldi a -Gue e o, R. Rich e , I. Rehbe g, N. Aksel, L. Heymann and O. S.
Rod iguez-Fe n´andez, Viscoelas ici y o mono- and polydispe se in e se e o luids,
J. Chem. Phys. 125, 084907 (2006).
[25] C. Gollwi ze , Compu e ges ¨
u z e Radioskopie on Obe l¨
achenins abili ¨
a en in Fe -
o luiden, Diploma bei , Uni e si ¨
a Bay eu h, unpublished, 2005.
[26] A. Lange, B. Reimann and R. Rich e , Wa e numbe o maximal g ow h in iscous
magne ic luids o a bi a y dep h, Phys. Re . E 61, 5528 (2000).
[27] A. Lange, Scaling beha iou o he maximal g ow h a e in he Rosensweig ins abili y,
Eu ophys. Le . 55, 327 (2001).
[28] J.-C. Bac i and D. Salin, Fi s -o de ansi ion in he ins abili y o a magne ic luid
in e ace, J. Physique Le . 45, 559 (1984).
BIBLIOGRAPHY 135
[29] R. Rich e and J. Bl¨
asing, Measu ing su ace de o ma ions in magne ic luid by
adioscopy, J. Sci. Ins um. 72, 1729 (2001).
[30] C. Gollwi ze , G. Ma hies, R. Rich e , I. Rehbe g and L. Tobiska, The su ace
opog aphy o a magne ic luid: a quan i a i e compa ison be ween expe imen and
nume ical simula ion, J. Fluid Mech. 571, 455 (2007).
[31] B. Abou, J.-E. Wes eid and S. Roux, The no mal ield ins abili y in e o luids:
hexagon-squa e ansi ion mechanism and wa enumbe selec ion, J. Fluid Mech.
416, 217 (2000).
[32] C. Gollwi ze , I. Rehbe g and R. Rich e , Via hexagons o squa es: Expe imen s on
hys e e ic su ace ans o ma ions unde a ia ion o he no mal magne ic ield, J.
Phys.: Condens. Ma e 18, S2643 (2006).
[33] A. Gaili is, Fo ma ion o he hexagonal pa e n on he su ace o a e omagne ic
luid in an applied magne ic ield, J. Fluid Mech. 82, 401 (1977).
[34] R. F ied ichs and A. Engel, Pa e n and wa e numbe selec ion in magne ic luids,
Phys. Re . E 64, 021406 (2001).
[35] E. E. Twombly and J. W. Thomas, Ma hema ical heo y o non-linea wa es on he
su ace o a magne ic luid, IEEE T ans. Magn. 16, 214 (1980).
[36] E. E. Twombly and J. W. Thomas, Bi u ca ion ins abili ies o he ee su ace o a
e o luid, SIAM J. Ma h. Anal. 14, 736 (1983).
[37] M. Silbe and E. Knobloch, Pa e n selec ion in e o luids, Physica D 30, 83
(1988).
[38] A. Schl¨
u e , D. Lo z and F. Busse, On he s abili y o s eady ini e ampli ude
con ec ion, J. Fluid Mech. 23, 129 (1965).
[39] R. F ied ichs and A. Engel, Non-linea analysis o he Rosensweig ins abili y, Eu-
ophys. Le . 63, 826 (2003).
[40] R. F ied ichs, Low symme y pa e ns on magne ic luids, Phys. Re . E 66, 066215
(2002).
[41] S. K. Malik and M. Singh, Nonlinea dispe si e ins abili ies in magne ic luids, Q.
Appl. Ma h. 42, 359 (1984).
[42] S. K. Malik and M. Singh, Modula ional ins abili y in magne ic luids, Q. Appl.
Ma h. 43, 57 (1985).
[43] S. K. Malik and M. Singh, Nonlinea ocusing in magne ic luids, Q. Appl. Ma h.
44, 629 (1987).
[44] C. Kubs up, H. He e o and C. P´e ez-Ga c´ıa, F on s be ween hexagons and squa es
in a gene alized Swi -Hohenbe g equa ion, Phys. Re . E 54, 1560 (1996).
[45] H. He e o, C. P´e ez-Ga c´ıa and M. Bes eho n, S abili y o on s sepa a ing do-
mains wi h di e en symme ies in hyd odynamical ins abili ies, Chaos 4, 15 (1994).
136 BIBLIOGRAPHY
[46] O. La o a, G. Ma hies, T. Mi ko a, V. Pole iko and L. Tobiska, Nume ical
ea men o ee su ace p oblems in e ohyd odynamics, J. Phys.: Condens. Ma e
18, S2657 (2006).
[47] A. C. Newell and J. A. Whi ehead, Fini e bandwid h, ini e ampli ude con ec ion,
J. Fluid Mech. 38, 279 (1969).
[48] M. C. C oss and P. C. Hohenbe g, Pa e n o ma ion ou side o equilib ium, Re .
Mod. Phys. 65, 851 (1993).
[49] A. Lange, The adjoin p oblem in he p esence o a de o med su ace: The example
o he Rosensweig ins abili y on magne ic luids, In . J. Mod. Phys. B 16, 1155
(2002).
[50] M. Takashima, Su ace ension d i en ins abili y in a ho izon al liquid laye wi h
a de o mable ee su ace. I. S a iona y con ec ion, J. Phys. Soc. Japan 50, 2745
(1981).
[51] M. Takashima, Su ace ension d i en ins abili y in a ho izon al liquid laye wi h a
de o mable ee su ace. II. O e s abili y, J. Phys. Soc. Japan 50, 2751 (1981).
[52] S. Rosenbla , S. H. Da is and G. M. Homsy, Nonlinea Ma angoni con ec ion in
bounded laye s. Pa 1. Ci cula cylind ical con aine s, J. Fluid Mech. 120, 91
(1982).
[53] S. Rosenbla , G. M. Homsy and S. H. Da is, Nonlinea Ma angoni con ec ion in
bounded laye s. Pa 2. Rec angula cylind ical con aine s, J. Fluid Mech. 120, 123
(1982).
[54] P. C. Dauby, G. Lebon, P. Coline and J. C. Leg os, Hexagonal Ma angoni con-
ec ion in a ec angula box wi h slippe y walls, Q. J. Mech. Appl. Ma h. 46, 683
(1993).
[55] G. Lebon and A. Cloo , Buoyancy and su ace- ension d i en ins abili ies in p es-
ence o nega i e Rayleigh and Ma angoni numbe s, Ac a Mech. 43, 141 (1982).
[56] A. Engel and J. B. Swi , Plan o m selec ion in wo-laye B´ena d-Ma angoni con-
ec ion, Phys. Re . E 62, 6540 (2000).
[57] A. C. Skeldon and G. Guidoboni, Pa e n selec ion o Fa aday wa es in an incom-
p essible iscous luid, SIAM J. Appl. Ma h. 67, 1064 (2007).
[58] H. Pleine and H. R. B and, in Pa e n Fo ma ion in Liquid C ys als, edi ed by
A. Buka and L. K ame , Sp inge , New Yo k, 1996, chap e 2.
[59] D. Fo s e , Hyd odynamic Fluc ua ions, B oken Symme y and Co ela ion Func-
ions, Benjamin, Reading, Mass., 1975.
[60] E. Ja ko a, H. Pleine , H.-W. M¨
ulle and H. R. B and, Hyd odynamics o iso opic
e ogels, Phys. Re . E 68, 041706 (2003).
[61] L. Onsage , Recip ocal ela ion in i e e sible p ocesses I, Phys. Re . 37, 405 (1931).
BIBLIOGRAPHY 137
[62] L. Onsage , Recip ocal ela ion in i e e sible p ocesses II, Phys. Re . 38, 2265
(1931).
[63] S. R. deG oo and P. Mazu , G undlagen de The modynamik i e e sible P ozesse,
Hochschul aschenb¨
uche -Ve lag, Mannheim, 1969.
[64] L. D. Landau and E. M. Li schi z, Leh buch de Theo e ischen Physik, Band 7 –
Elas izi ¨
a s heo ie, Akademie-Ve lag, Be lin, 1974.
[65] P. M. Chaikin and T. C. Lubensky, P inciples o condensed ma e physics, Cam-
b idge Uni e si y P ess, Camb idge, UK, 1995.
[66] H. Temmen, H. Pleine , M. Liu and H. R. B and, Con ec i e nonlinea i y in non-
New onian luids, Phys. Re . Le . 84, 3228 (2000).
[67] H. Pleine , M. Liu and H. R. B and, The s uc u e o con ec i e nonlinea i ies in
polyme heology, Rheol. Ac a 39, 560 (2000).
[68] A. Menzel, H. Pleine and H. R. B and, Nonlinea ela i e o a ions in liquid
c ys alline elas ome s, J. Chem. Phys. 126, 234901 (2007).
[69] J. D. Jackson, Classical Elec odynamics, John Wiley & Sons, Inc., 1999.
[70] S. Bohlius, Mak oskopische Dynamik magne ische Gele, Diploma bei , Uni e si ¨
a
Bay eu h, unpublished, 2005.
[71] J. L. Ha den, H. Pleine and P. A. Pincus, Hyd odynamic su ace modes on con-
cen a ed polyme solu ions and gels, J. Chem. Phys. 94, 5208 (1991).
[72] J. Weilepp and H. R. B and, Compe i ion be ween he B´ena d-Ma angoni and he
Rosensweig ins abili y in magne ic luids, J. Phys. II 6, 419 (1996).
[73] S. Bohlius, H. Pleine and H. R. B and, Pa e n o ma ion in e ogels: Analysis o
he Rosensweig ins abili y using he ene gy me hod, J. Phys.: Condens. Ma e 18,
S2671 (2006).
[74] S. Bohlius, H. Pleine and H. R. B and, Solu ion o he adjoin p oblem o in-
s abili ies wi h a de o mable su ace: Rosensweig and Ma angoni ins abili y, Phys.
Fluids 19, 094103 (2007).
[75] S. Bohlius, H. Pleine and H. R. B and, The ampli ude equa ion o he Rosensweig
ins abili y, submi ed o Physica D .
[76] A. C. Newell, T. Passo and J. Lega, O de pa ame e equa ions o pa e ns, Annu.
Re . Fluid Mech. 25, 399 (1993).
[77] I. S. A anson and L. K ame , The wo ld o he complex Ginzbu g-Landau equa ion,
Re . Mod. Phys. 74, 99 (2002).
[78] S. Cilibe o, P. Coulle , J. Lega, E. Pampaloni and C. P´e ez-Ga c´ıa, De ec in
Roll-Hexagon Compe i ion, Phys. Re . Le . 65, 2370 (1990).

138 BIBLIOGRAPHY
[79] M. Diewald, Chemisch ge iebene Kon ek ion, Disse a ion, Uni e si ¨
a Bay eu h,
unpublished, 1997.
[80] J. B aga d and M. G. Vela de, B´ena d-Ma angoni con ec ion: plan o ms and ela ed
heo e ical p edic ions, J. Fluid Mech. 368, 165 (1998).
[81] H. R. B and, P. S. Lomdahl and A. C. Newell, E olu ion o he o de pa ame e in
si ua ions wi h b oken o a ional symme y, Phys. Le . A 118, 67 (1986).
[82] H. R. B and, P. S. Lomdahl and A. C. Newell, Benjamin-Fei u bulence in con-
ec i e bina y luid mix u es, Physica D 23, 345 (1986).
[83] A. C. Newell, Soli ons in Ma hema ics and Physics, SIAM Se ies, 1985.
[84] D. Rannache and A. Engel, Double Rosensweig ins abili y on a e o luid sandwich
s uc u e, Phys. Re . E 69, 066306 (2004).
[85] S. Bohlius, H. R. B and and H. Pleine , Rosensweig ins abili y o e ogel hin ilms
o memb anes, Eu . Phys. J. E 26, 275 (2008).
[86] J. L. Ha den and H. Pleine , Hyd odynamic modes o iscoelas ic polyme ilms,
Phys. Re . E 49, 1411 (1994).
[87] M. G. Hilge s and A. C. Pipkin, Bending ene gy o highly elas ic memb anes, Q.
Appl. Ma h. 50, 389 (1992).
[88] L. K ame , Theo y o ligh sca e ing om luc ua ions o memb anes and mono-
laye s, J. Chem. Phys. 55, 2097 (1971).
[89] E. H. Lucassen and J. Lucassen, P ope ies o capilla y wa es, Ad an. Colloid
In e ace Sci. 2, 347 (1969).
[90] D. Lange in, Ligh -sca e ing s udy o monolaye iscoelas ici y, J. Colloid and
In e ace Sci. 80, 412 (1981).
[91] S. Bohlius, H. R. B and and H. Pleine , Mac oscopic dynamics o uniaxial magne ic
gels, Phys. Re . E 70, 061411 (2004).
[92] D. R. Pe i and H. R. B and, On he in luence o nea ze o g a i y on he
Rosensweig ins abili y in magne ic luids, Phys. Le . A 159, 55 (1991).
[93] B. G a , p i a e communica ion.
[94] J. R. A. Pea son, On con ec ion cells induced by su ace ension, J. Fluid Mech. 4,
489 (1958).
[95] D. A. Nield, Su ace ension and buoyancy e ec s in cellula con ec ion, J. Fluid
Mech. 19, 341 (1964).
[96] L. E. Sc i en and C. V. S e nling, On cellula con ec ion d i en by su ace- ension
g adien s: E ec s o mean su ace ension and su ace iscosi y, J. Fluid Mech. 19,
321 (1964).
BIBLIOGRAPHY 139
[97] K. A. Smi h, On con ec i e ins abili y induced by su ace- ension g adien s, J. Fluid
Mech. 24, 401 (1966).
[98] C. P´e ez-Ga c´ıa and G. Ca nei o, Linea s abili y analysis o B´ena d-Ma angoni
con ec ion in luids wi h a de o mable su ace, Phys. Fluids A 3, 292 (1991).
[99] P. M. Pa men ie , V. C. Regnie , G. Lebon and J. C. Leg os, Nonlinea analysis
o coupled g a i a ional and capilla y he mocon ec ion in hin luid laye s, Phys.
Re . E 54, 411 (1996).
[100] S. Chand asekha , Hyd odynamic and Hyd omagne ic S abili y, Ox o d Uni e si y
P ess, Ox o d, 1961.
140 BIBLIOGRAPHY
Lis o Publica ions
This hesis is di ec ly connec ed o he ollowing publica ions in which he esul s o his
wo k ha e been published o will be published:
S. Bohlius, H. R. B and and H. Pleine ,
Su ace wa es and Rosensweig ins abili y in iso opic magne ic gels,
Z. Phys. Chem. 200, 97 (2006)
S. Bohlius, H. Pleine and H. R. B and,
Pa e n o ma ion in e ogels:
Analysis o he Rosensweig ins abili y using he ene gy me hod,
J. Phys.: Condens. Ma e 18, S2671 (2006)
S. Bohlius, H. Pleine and H. R. B and,
Solu ion o he adjoin p oblem o ins abili ies wi h a de o mable su ace,
Phys. Fluids 19, 094103 (2007)
S. Bohlius, H. R. B and and H. Pleine ,
Rosensweig ins abili y o e ogel hin ilms o memb anes,
Eu . Phys. J. E. 26, 275 (2008)
S. Bohlius, H. Pleine and H. R. B and,
The ampli ude equa ion o he Rosensweig ins abili y,
submi ed o Physica D
141