Influence of side walls and undulated topography on viscous gravity-driven film flow

In luence o side walls and undula ed opog aphy on iscous g a i y–d i en ilm low Von de Fakul ¨ a ¨ u Angewand e Na u wissenscha en de Uni e si ¨ a Bay eu h zu E langung de W¨ u de eines Dok o -Ingenieu s (D .-Ing.) genehmig e Disse a ion o geleg on Dipl.-Phys. Thilo Pollak aus Gumme sbach E s gu ach e : P o . D . e . na . N. Aksel Zwei gu ach e : P o . D . V. Bon ozoglou Tag de m¨ undlichen P ¨ u ung: 07. Augus 2012 Leh s uhl ¨ u Technische Mechanik und S ¨ omungsmechanik Uni e si ¨ a Bay eu h 2012 1 Lis o Jou nal publica ions - Pollak T. and K¨ ohle W.: C i ical assessmen o di usion coe icien s in semidilu e o concen a ed solu ions o polys y ene in oluene, The Jou nal o Chemical Physics, 130 (2009), 124905 - Wie schem A., Pollak T., Heining C. and Aksel N.: Supp ession o eddies in ilms o e opog aphy, Physics o Fluids, 22 (2010), 113603 - Haas A., Pollak T. and Aksel N.: Side wall e ec s in hin g a i y–d i en ilm low - s eady and d aining low, Physics o Fluids, 23 (2011), 062107 - Pollak T., Haas A. and Aksel N.: Side wall e ec s on he ins abili y o hin g a i y- d i en ilms - F om long–wa e o sho –wa e ins abili y, Physics o Fluids, 23 (2011), 094110 - Heining C., Pollak T. and Aksel N.: Pa e n o ma ion and mixing in h ee–dimensional ilm low, Physics o Fluids, 24 (2012), 042102 - Heining C., Sellie M. and Pollak T.,: Flow domain iden i ica ion om ee su ace eloci y in hin ine ial ilms, Jou nal o Fluid Mechanics, submi ed - Pollak T., Aksel A.: Expe imen al e idence o mul iple ins abili y b anches o g a i y– d i en ilms o e opog aphy,in p epa a ion 2 3 Abs ac While a g a i y–d i en iscous ilm low down an inclined la plane o in ini e ex en can be desc ibed by an easy analy ical solu ion, low p oblems in na u e, like glacie mo emen s o he liquid ilm on he human eye a e much mo e complex. Also o op imize a la ge numbe o echnical applica ions, like coa ing applica ions o hea exchange de ices, one has o in es iga e and unde s and how di e en in luencing ac o s, like opological ea u es on he subs a e o a ini e wid h o he sys em, in luence he low and i s s abili y isola ed om each o he . By in oducing a wa y s uc u e o he unde lying opog aphy, which could be o example a model o oughness, new e ec s eme ge in he low, which canno be obse ed in lows o e a la incline. Eddies can sepa a e om he main low a he lee side o he undula ion o kinema ic easons, o induced by ine ial e ec s. In biological sys ems hese eddies a e dead wa e a eas, which a e cu o om nu ien supply, in hea exchange applica ions hei appea ance has a s ong impac on he con ec i e hea anspo wi hin he liquid. Fu he mo e, he ampli ude o ee su ace o he liquid can be ampli ied immensely when he liquid is in esonance wi h he undula ion o he unde lying opog aphy. In his wo k we s udy expe imen ally as well as nume ically he complex in e ac ion o his esonance phenomenon wi h he appea ing o eddy s uc u es in he alleys o he undula ion and show, ha one can supp ess low sepa a ion selec i ely e en a a he high Reynolds numbe s when one exploi s he esonance phenomenon speci ically. Ano he pa o his wo k deals wi h he ques ion how he p esence o side walls and he con ac angle o he liquid he e in luences he ee su ace shape o he liquid, he eloci y ield and he globally anspo ed olume lux. While an addi ional no–slip condi ion a he wall causes addi ional ic ion and leads hus o a lowe olume lux, capilla y ele a ion a he side walls can gene a e a eloci y o e shoo in he icini y o he walls, depending on he ilm hickness and he we ing p ope ies o he liquid, which coun e ac s he addi ional ic ion coming om he walls. An ex ensi e heo e ical pa ame e s udy, which is supplemen ed wi h expe imen al da a, p o ides c i e ia o he i s onse o he eloci y o e shoo and gi es answe o he ques ion when he coun e ac ing in luences on he global olume lux jus cancel each o he . An expe imen al s udy o he ee su ace shape o a d aining low shows ha his con igu a ion canno be desc ibed by a se ies o quasi–s eady s a es, e en when a dynamic con ac angle is aken in o conside a ion, al hough he low changes only e y slowly in ime. Addi ional ime dependen nume ical simula ions o he d aining low e eal an inden a ion o he ee su ace in he icini y o he side wall, which could p omo e ilm up u e in echnical hin ilm applica ions. 4 Fu he mo e, side wall e ec s play an impo an ole o he physical s abili y o he low. Wa es de elop a he ee su ace o a g a i y–d i en low and g ow while hey a e a eling downs eam, when a c i ical olume lux is exceeded. I is shown by expe imen al a ia ion o he con ac angle, he ilm hickness and he side wall dis ance, ha he p esence o side walls gene a es di e en e ec s which ha e compe ing in luences on he s abili y o he low. Capilla y ele a ion leads o a p e ensioning o he ee su ace, which ends o s abilize he ee su ace, jus as he addi ional no–slip condi ion a he wall does. The eme ging o a eloci y o e shoo in he capilla y ele a ion on he o he hand leads o a des abiliza ion o he low. In he sys em s udied he e he s abilizing in luence o he side walls domina es o e he des abilizing in luence which is o compa a i ely sho ange, which means ha his low con igu a ion is mo e s able han he co esponding low o in ini e ex en . Howe e , he esul s sugges ha he des abilizing in luences should domina e o e he s abilizing in luences in simila low con igu a ions when he ilm would become e en hinne . While ee su ace ilm lows ypically o m long wa es a i s , we ind o his low con igu a ion, ha he ype o ins abili y changes om a long–wa e ype in he middle o he channel o a sho –wa e ype ins abili y, as i is well known o bounda y laye lows, as he side wall dis ance is educed. 5 Zusammen assung W¨ ah end sich eine schwe k a sge iebene iskose S ¨ omung, die eine unendlich ausge- dehn e und gla e Ebene hinab ließ , du ch eine ein ache analy ische L¨ osung besch eiben l¨ ass , sind die S ¨ omungsp obleme in de Na u , wie zum Beispiel eine Gle sche bewe- gung ode ein Fl¨ ussigkei s ilm au dem menschlichen Auge, wei aus komplizie e . Auch um zahl eiche echnische Anwendungen, wie beispielsweise Beschich ungs- ode W¨ a me- auschp ozesse, op imie en zu k¨ onnen, m¨ ussen Ein luss ak o en, wie das Vo handensein eine S uk u au de Obe l¨ ache des Bodens ode eine endliche B ei e des Sys ems und de en Ein l¨ usse au das S ¨ omungs eld und die physikalische S abili ¨ a de S ¨ omung iso- lie un e such und e s anden we den. Du ch das Vo handensein eines gewell en Un e g undes, de zum Beispiel ein Modell ¨ u Rauhei sein k¨ onn e, en s ehen neue E ek e in de S ¨ omung, die bei einem gla en Un e g und nich beobach e we den k¨ onnen. Sowohl aus ein kinema ischen G ¨ unden, abe auch du ch ¨ aghei sinduzie e E ek e kann die S ¨ omung au de Windscha ensei e de Bodens uk u om Boden abl¨ osen, so dass in den Bodenmulden Rezi kula ionsgebie- e en s ehen. In biologischen Sys emen s ellen diese Regionen To wasse gebie e da , die nich mi N¨ ah s o en e so g we den, in W¨ a me ausche anwendungen ha ih Au e- en einen s a ken Ein luss au den kon ek i en W¨ a me anspo . Neben dem En s ehen eine S ¨ omungsabl¨ osung kann du ch Au e en on Resonanz zwischen dem gewell en Bo- den und de Fl¨ ussigkei die Ampli ude de eien Fl¨ ussigkei sobe l¨ ache immens e s ¨ a k we den. In diese A bei wi d das komplizie e Zusammenspiel aus Resonanz und dem En s ehen on Rezi kula ionsgebie en in den Bodenmulden sowohl nume isch als auch expe imen ell un e such und es wi d gezeig , dass man du ch geschick e Ausnu zung de Resonanz das Au e en de Wi bels uk u en auch bei ela i hohen Reynoldszahlen geziel un e binden kann. Ein wei e e Teil diese Disse a ion be ass sich mi de F age, wie sich das Vo han- densein on Sei enw¨ anden und de Kon ak winkel de Fl¨ ussigkei do au die Fo m de eien Obe l¨ ache, das Geschwindigkei s eld und den globalen Volumens om auswi k . W¨ ah end eine zus¨ a zliche Ha bedingung an de Wand zu zus¨ a zliche Reibung und da- mi zu einem ge inge en Volumens om ¨ uh , kann in Abh¨ angigkei on Kon ak winkel und Filmdicke du ch kapilla e Anhebung ein Geschwindigkei s¨ ube schuss in de N¨ ahe de Sei enwand en s ehen, de dem Ein luss de Ha bedingung en gegenwi k . Eine um ang- eiche heo e ische Pa ame e s udie, die du ch expe imen elle E gebnisse e g¨ anz wi d, lie e K i e ien ¨ u das e s e Au e en eines Geschwindigkei s¨ ube schusses und bean wo - e die F age, wann sich die en gegenwi kenden Ein l¨ usse au den globalen Volumens om ge ade gegensei ig au heben. Die expe imen elle Un e suchung de Fo m de eien Obe l¨ ache eine D ainage- 6 s ¨ omung zeig , dass sich diese S ¨ omung auch du ch Ein ¨ uh ung eines dynamischen Kon- ak winkels nich du ch eine Folge quasis a ische Zus ¨ ande besch eiben l¨ ass , obwohl sie sich zei lich nu seh langsam e ¨ ande . Zus¨ a zliche zei abh¨ angige nume ische Simula i- onen de D ainages ¨ omung o enba en eine Ve ie ung de eien Obe l¨ ache in de N¨ ahe de Sei enwand, die in echnischen D¨ unn ilmanwendungen einen Ab iss des Fl¨ ussigkei s- ilms he o u en k¨ onn e. Da ¨ ube hinaus spielen Sei enwande ek e auch eine en scheidende Rolle ¨ u die physi- kalische S abili ¨ a de S ¨ omung. Au de eien Obe l¨ ache eine schwe k a sge iebenen Films ¨ omung bilden sich Wellen aus, die anwachsen, w¨ ah end sie die Ebene hinab lie- ßen, sobald ein k i ische Volumens om ¨ ube sch i en wi d. Es wi d du ch expe imen elle Va ia ion des Kon ak winkels, de Filmdicke und des Sei enwandabs andes gezeig , dass e schiedene E ek e, die du ch das Vo handensein on Sei enw¨ anden au e en, mi einan- de konku ie ende Ein l¨ usse au die S abili ¨ a de S ¨ omung haben. So ha die kapilla e Anhebung eine Vo k ¨ ummung de eien Obe l¨ ache zu Folge, welche zusammen mi de zus¨ a zlichen Ha bedingung an de Wand zu eine S abilisie ung de S ¨ omung ¨ uh . Das Au e en eines Geschwindigkei s¨ ube schusses in de kapilla en Anhebung ¨ uh hingegen zu eine Des abilisie ung de S ¨ omung. Bei de hie un e such en S ¨ omung ¨ ube wie- gen die lang eichwei igen s abilisie enden Ein l¨ usse den e gleichsweise ku z eichwei igen des abilisie enden Ein luss de Sei enwand, so dass dieses Sys em insgesam gegen¨ ube eine que zu Haup s ¨ omungs ich ung unendlich ausgedehn en S ¨ omung du ch die Sei- enw¨ ande s abilisie wi d. Die E gebnisse legen jedoch nahe, dass ¨ u ¨ ahnliche S ¨ omungs- kon igu a ionen, die eine noch ge inge e Filmdicke au weisen, de Ne oein luss de Sei en- wand au die S ¨ omung auch des abilisie end sein k¨ onn e. W¨ ah end sich bei Films ¨ omun- gen ypische weise zue s lange Wellen au de eien Obe l¨ ache ausbilden, inden wi ¨ u diese S ¨ omung, dass sich du ch eine Ve inge ung des Sei enwandabs andes ein ¨ Ube gang on eine Langwellenins abili ¨ a zu eine Ku zwellenins abili ¨ a , wie man sie ypische - weise on G enzschich s ¨ omungen kenn , ollzieh . 7 Con en s 1 In oduc ion 10 2 Expe imen al sys ems and se ups 16 2.1 Liquids ..................................... 16 2.2 Flow acili ies.................................. 17 2.3 T ace pa icles................................. 18 2.4 Expe imen alse ups .............................. 20 2.4.1 Flow a e ................................ 20 2.4.2 De ec ion o he ee su ace shape . . . . . . . . . . . . . . . . . . 21 2.4.3 S eamline de ec ion . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.4 Veloci y ield measu emen s . . . . . . . . . . . . . . . . . . . . . . 24 2.4.5 S abili y measu emen s . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Two–dimensional ilm low 32 3.1 Supp essiono eddies.............................. 32 3.1.1 P oblem o mula ion . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1.2 Expe imen al and nume ical indings . . . . . . . . . . . . . . . . . 34 3.1.3 Physical in e p e a ion and discussion . . . . . . . . . . . . . . . . 42 3.1.4 Conclusions............................... 44 4 Th ee–dimensional ilm low 46 4.1 Basic low.................................... 46 4.1.1 Go e ning equa ions . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1.2 Flow ype classi ica ion . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.3 Flow a es udy............................. 51 4.1.4 Veloci y ield .............................. 54 4.1.5 F eesu aceshape ........................... 57 4.1.6 Conclusions............................... 60 4.2 S abili y nea he side walls . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.1 Resul s ................................. 62 4.2.2 Conclusions............................... 65 5 Conclusions and ou look 68 8 Chap e 1. In oduc ion 15 Chap e 2 Expe imen al sys ems and se ups 2.1 Liquids Th ee di e en silicone oils om Basildon and Elbesil wi h dynamic iscosi ies η anging om app oxima ely 10 mPas o 1000 mPas, which all showed New onian beha io wi hin he conside ed shea a e and empe a u e ange ha e been in es iga ed. The main luid p ope ies a he mean measu emen empe a u e Ta e summa ized in Table 2.1. The densi y, he dynamic iscosi y, he kinema ic iscosi y and he su ace ension a e deno ed by ρ,η,ν=η/ρ and σ espec i ely. Densi y measu emen s ha e been ca ied ou wi h a Moh Wes phal balance om Go l. Ke n &Sohn GmbH wi h an absolu e accu acy o ±0.3 kg/m3. The empe a u e o he liquid in he Moh balance was con olled by a Lauda he mos a ype ecoline RE204. Measu emen s o he su ace ension σwe e done wi h a Lauda ing– ensiome e ype TE1CA-M whose luid empe a u e was con olled by a Lauda he mos a ype RC 6 CP. The esolu ion o he ing– ensiome e was 0.1 mN/m. The dynamic iscosi y ηo he liquids has been measu ed wi h di e en Ubbelohde iscosime e capilla ies ype 501 om Scho . The capilla ies we e plunged in o a wa- e ba h whose empe a u e was con olled by a Scho he mos a wi hin an accu acy o 0.05 ◦C. The p ecisions o he di e en iscosime e capilla ies we e speci ied o be be ween 0.65% and 0.8%. All luid p ope y measu emen s ha e been ca ied ou in a empe a u e in e al om 20 −30 ◦C in 1 ◦C s eps. The unce ain y o he liquid p ope ies du ing an expe imen al un is essen ially de e mined by he unce ain y o he liquid’s empe a u e and was hus calcula ed om he empe a u e dependence o he liquid p ope ies. The empe a u e o he liquid lowing in he channel was measu ed downs eam o he egion o in e es by Ahlbo n Mess- und Regelungs echnik GmbH PT-100 and NTC Manu ac u e Name T/ [◦C] ρ/ [kg/m3]η/ [mPas] ν/ [mm2/s] σ/ [mN/m] Basildon BC10cs 25 924.3 10.72 11.6 18.9 Basildon BC50 24 950.6 50.0 52.6 19.6 Elbesil B1000 24 969.0 1,076 1,110 20.4 Table 2.1: Liquid p ope ies o he used silicone oils. 16 Chap e 2. Expe imen al sys ems and se ups empe a u e senso s wi h an accu acy o 0.1◦C. The s a ic con ac angle θa he h ee–phase con ac lines liquid/ai /channel side wall has been measu ed wi h a con ac angle goniome e om Da aphysics ype OCA 20. All measu ed con ac angles we e ound o be independen o he empe a u e wi hin he measu emen e o o app oxima ely 2◦and he empe a u e ange in es iga ed. 2.2 Flow acili ies The expe imen s ha e been ca ied ou in wo di e en 170 ±1 mm b oad channels wi h la bo oms made o aluminum. The side walls o channel 1 we e made o Plexiglas®and channel 2 was ea u ed wi h side wall clamps which allowed o moun side walls made up o di e en ma e ials o a y he con ac angle θa he iple poin ai /liquid/side wall. In his wo k we ha e limi ed ou con ac angle s udy o he wo ex eme cases which a e echnically possible. Silicone oil, which was he only luid used h oughou all ex- pe imen al uns in o de o keep all ma e ial pa ame e s (see sec ion 2.1), in pa icula he su ace ension σ, cons an , shows nea ly pe ec we ing cha ac e is ics on he as majo i y o subs a es. Because o i s excellen plana i y we chose Plexiglass®as a side wall ma e ial o co e his case. The s a ic con ac angle θo silicone oil wi h plane Plexiglass®was measu ed wi h a sessile d op me hod using a s a ic con ac angle go- niome e ype OCA 20 om da aphysics o be 8◦±2◦. The second se o side walls was made up o glass which has been coa ed wi h p o.Glass®Clea 105 om nanoga e o enla ge θ o 52◦±2◦which was he la ges con ac angle we we e able o achie e wi h he u ilized luid. The o e all leng h o channel 1 was abou wo me e s and o channel 2 was abou hal a me e . The inclina ion angle αo bo h channels could ha e been a ied con inuously be ween 0◦and 90◦and has been de e mined by a digi al p o ac o wi h an accu acy o 0.1◦. The spanwise e enness o he channel has been checked by placing a wa e le el wi h a display accu acy o 0.1 mm/m pe pendicula o he side walls o he channel. Pe pendicula i y was assu ed wi h a 90 ◦–aluminum angle placed o he side wall. α g x z pump ib a ion isola ing able liquid ese oi in low ank ˙ V ˙ V ˙ V channel Figu e 2.1: Ske ch o he low ci cui including he channel which is moun ed on a ib a ion isola ing able and a pump which anspo s he liquid om a la ge liquid ese oi o a smalle in low ank on op o he channel. Depending on he desi ed low a e ˙qone o wo di e en eccen ic pumps om Johs ad p o ided a cons an adjus able olume lux ˙ V= ˙qB, wi h Bbeing he channel wid h, 17 2.3. T ace pa icles om a la ge liquid ese oi in o a smalle in low ank on op o he channel. Channel 1 was equipped wi h a pump ype AFJ 15.1B which p o ided a olume lux in he ange o 1 l/min o 10 l/min. Channel 2 was equipped wi h a smalle pump ype AFJ 06B which p o ided a olume lux in he ange o 0.025 l/min o 1 l/min. Smalle olume luxes ha e been ealized by an adjus able bypass in he ube sys em be ween he pump and he channel. F om he smalle in low ank on op o he channel he liquid lows g a i y–d i en, down he channel, back in o he liquid ese oi o close he low ci cui as illus a ed in in Figu e 2.1. A sinusoidally undula ed aluminum inlay, as illus a ed in Figu e 2.2, consis ing o 50 pe iods wi h a wa eleng h λ= 10 mm and an ampli ude a= 1 mm co e ing he whole wid h o he channel was inse ed di ec ly below he in low o he channel 1. The g a i a ional accele a ion is deno ed by g, which can be w i en as g= (gsin α, −gcos α) in he (x, z)–coo dina e sys em gi en in Figu e 2.2. α g x z a λ Figu e 2.2: Geome y o he wo–dimensional undula ed inlay. Du ing all expe imen al uns he empe a u e o he liquid was con olled by a TC300 he mos a om Haake ia a hea exchange coil si ing in he la ge liquid ese oi . 2.3 T ace pa icles We ha e used wo di e en ypes o ace pa icles. The mean diame e and he densi y o Red Fluo escen Polyme Mic osphe es om Duke Scien i ics, which will be called luo escen ace pa icles in he ollowing, ha e been speci ied by he manu ac u e o 7µm and 1050 kg/m3, espec i ely. Addi ionally we ha e de e mined he olume weigh ed pa icle size dis ibu ion wi h a Mas e size 2000 de ice om Mal e n, which is plo ed in Figu e 2.3. The median pa icle size x50 and he g ade o dispe si y ξd, which is de ined in[71] ξd=x84 −x16 2x50 ,(2.1) o he luo escen ace pa icles ha e been ound o be x50 = 7.122 µm and ξd= 0.3058. The quan i ies x16,x50 and x84 deno e he pa icle sizes, which a e g ea e han o equal o 16%, 50% and 84% o all pa icles, espec i ely. The sedimen a ion speed used o small sphe es alling in a iscous liquid can be calcu- la ed o be[72] used =2g 2 s 9η(ρs−ρ),(2.2) whe e sand ρsa e he adius and he densi y o he sphe e. 18 Chap e 2. Expe imen al sys ems and se ups 0.1 1 10 100 1000 Pa icle size / [µm] 0 5 10 15 20 Volume % Red Fluo escen Polyme Mic osphe es Z O2/MgO powde Figu e 2.3: Pa icle size dis ibu ions. The esul ing sedimen a ion speeds o he luo escen ace pa icles used in he h ee di e en silicone oils BC10cs, BC50 and B1000 (see Table 2.1) we e 3.2×10−4mm/s, 5.5×10−5mm/s and 2.1×10−6mm/s, espec i ely. All hese sedimen a ion eloci ies a e a each case o de s o magni udes smalle han he ypical low eloci ies measu ed. The e o e, he sedimen a ion dis ance du ing one expe imen al un did no exceed he pa icles diame e . Figu e 2.4 shows he emission spec um o he luo escen ace pa icles dissol ed in silicone oil which has been measu ed wi h a Ca y Eclipse luo escence spec opho ome e om Agilen Technologies a an exci a ion wa eleng h o 532 nm. 500 550 600 650 700 750 800 λ / [nm] 0 100 200 300 400 In ensi y / [a.u.] Figu e 2.4: Emission spec um o Red Fluo escen Polyme Mic osphe es om Duke Scien i ics in silicone oil. Wa eleng h o he exci a ion ligh was 532 nm. The second ype o ace pa icles was made up o a powde o Z O2/MgO om Good- ellow. The mean pa icle diame e is speci ied by he manu ac u e o be 0.8µm. The densi y is speci ied o be 5700 kg/m3. The measu ed pa icle size dis ibu ion is plo ed in Figu e 2.3. The median pa icle size and he g ade o dispe si y ha e been measu ed o be x50 = 0.76 µm and ξd= 0.4682. These pa icles, which will be called sca e ing ace pa icles in he ollowing, we e dissol ed in he B1000 silicone oil om Elbesil, which is desc ibed in sec ion 2.1, only. Acco ding o equa ion (2.2) he sedimen a ion speed used 19 2.4. Expe imen al se ups o he Z O2/MgO-pa icles in his oil was abou 1.4×10−6mm/s, which was o de s o magni udes smalle , han all ypical eloci ies measu ed. The e o e, he sedimen a ion dis ance du ing one expe imen al un did no exceed he pa icles diame e . 2.4 Expe imen al se ups 2.4.1 Flow a e De e mina ion o he low a e ˙q, o he Reynolds numbe Re espec i ely, was done ei he by de e mining he ilm hickness do he liquid lowing o e a su icien ly long la pa in he middle o he channel o by a low me e which measu ed he o e all olume lux ˙ V h ough he channel. Unde pe ec ly s able low condi ions, a low Reynolds numbe s, he ilm hickness has been measu ed by a mic ome e sc ew wi h a needle ip. The mic ome e sc ew was moun ed o he channel in a way, ha i poin ed pe pendicula o he ee su ace o he liquid o he bo om o he channel, espec i ely. By sc ewing he needle slowly owa ds he luid, he posi ion o he ee su ace can be de ec ed, when he ip o he needle con ac s he liquid and a capilla y ele a ion o ms ins an aneously (See Figu e 2.5). The posi ion o he subs a e has been de e mined in a simila ashion by sc ewing he needle u he down un il a small mechanical esis ance was sensible. The accu acy o he ilm hickness de e mina ion is es ima ed o be be e han 10 µm. (a) (b) Figu e 2.5: Illus a ion o he ip o a needle which is less han 6.5µm abo e he su ace o a lowing liquid ilm (a) and jus ouching i (b) wha causes a capilla y ele a ion o o m ins an aneously. The wid h o he needle illus a ed is 400 µm. To de e mine he low a e ˙qa in e media e Reynolds numbe s o unde weakly uns eady low condi ions, he o e all a e age olume lux ˙ V h ough he channel was measu ed by an analog low me e which was ins alled be ween he ou low o he channel and he la ge liquid ese oi . Fo each olume lux measu emen he a e aging ime was a leas 600 s o educe he s a is ical e o o he olume lux measu emen o less han 0.1 %. When side wall e ec s a e neglec ed and he low is s eady, he eloci y p o ile o a 20 Chap e 2. Expe imen al sys ems and se ups liquid lowing down a la incline is ound o be pa abolic[73] u(z) = ρg sin α 2η(2hn−z)z, (2.3) whe e zis he ca esian coo dina e pe pendicula o he bo om and hnis he ilm hick- ness o he well known Nussel solu ion[9]. In eg a ing he eloci y p o ile om he bo om (z= 0) o he ee su ace o he liquid (z=hn) yields he low a e o he Nussel ilm low ˙q=Zhn 0 u(z)dz=ρg sin αh3 n 3η.(2.4) In eg a ing he low a e ˙qo e he channel wid h Byields he olume lux ˙ V. ˙ V= ˙qB =ρg sin αh3 nB 3η.(2.5) When he in luence o side walls on he low is neglec ed, he ela ion be ween he Reynolds numbe Re, he ilm hickness o a ilm lowing down a la channel and he measu ed olume lux is in he ollowing gi en by Re = 2ushn 3ν=¯uhn ν=˙q ν=˙ V νB ,(2.6) whe e usis he ee su ace eloci y and ¯u= 2us/3 is he mean low eloci y. I has o be emphasized, ha equa ions (2.5) and (2.6) a e alid, only i he eloci y ield uis assumed o be independen o he spanwise y-coo dina e, which is only he case when he impac o he p esence o side walls on he low iled is neglec ed. When he in luence o side walls on he low is in es iga ed in chap e 4, his assump ion has o be d opped. A de ailed discussion o he ela ion be ween he ilm hickness d, he measu ed olume lux ˙ Vand he Reynolds numbe Re, when he in luence o he p esence o side walls is conside ed will ollow in sec ions 4.1.2 and 4.1.3. 2.4.2 De ec ion o he ee su ace shape In bo h low acili ies he shape o he ee su ace has been measu ed by illumina ing luo escen ace pa icles, which a e desc ibed in sec ion 2.3, in he bulk o he liquid wi h a lase shee . The luo escen ligh has been de ec ed wi h a cha ged–coupled–de ice (CCD) came a. The lase shee o channel 1 has been p oduced by a con inuous–wa e (cw) a gon– Ion (A +) Lase om Spec a Physics emi ing a a wa eleng h o 514.5 nm wi h an app oxima e ou pu powe o 100 mW (See Figu e 2.6). The ligh shee was aligned pa allel o he side walls igh in he middle o he channel o s udy he in luence o he undula ed bo om on he ee su ace shape o he liquid lowing abo e i . The luo escen ligh om he luo escen ace pa icles in he liquid was de ec ed wi h a JAI CV-M300 8-bi monoch ome CCD came a wi h a esolu ion o 768x494 pixels, which was inclined by abou 10◦wi h espec o he spanwise di ec ion o he channel (See came a (2) in Figu es 2.6 and 2.7). The much b igh e sca e ed ligh om he unde lying opog aphy was blocked by an op ical long pass il e wi h a 50% cu –o wa eleng h o 550 nm, which was 21 2.4. Expe imen al se ups moun ed on a 60 mm Nikko mic o lens. The came as ield o iew co e ed app oxima ely 2.5 pe iods o he bo om undula ion. This came a se up esul ed in a spa ial esolu ion o abou 30 µm/pixel. The image was calib a ed spa ially wi h a ce amic calib a ion scale wi h a poin pa e n o 4 p /mm2. Liquid wi h Lase shee 514.5 nm de ec ion came a Su ace Inclined channel pu e Com− A gon Ion Lase Field came a luo escen Fil e de ec ion ace s (2) (1) Figu e 2.6: Expe imen al se up o he su ace con ou de ec ion and he s eamline de ec ion in he oughs o he undula ed inlay in channel 1. The eddy size is de e mined by de ec ing he pa h lines o he luo escen ace pa icles wi h he ho izon al came a (1). The inclined came a (2) images he ligh shee om he ai side. The su ace con ou co esponds o he uppe bo de line o he b igh shee as seen by his came a. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i u e o Physics.    Su ace de ec ion came a (2) H Hj Field de ec ion came a (1) Figu e 2.7: Pho o o he expe imen al se up o he su ace con ou de ec ion and he s eamline de ec ion in he oughs o he undula ed inlay in channel 1. The expe imen al se up o he ee su ace de ec ion in channel 2 is ske ched in Figu e 2.8. The lase shee has been aligned pe pendicula o he side walls o he channel and was expanded o illumina e he liquid in a egion om he side wall o app oxima ely 50 mm apa om i , o de ec he shape o he capilla y ele a ion o he liquid in he p oximi y o he side walls. The shee was p oduced by a equency doubled, pulsed 22 Chap e 2. Expe imen al sys ems and se ups neodymium-doped y ium aluminum ga ne (Nd:Yag) lase om New wa e esea ch ype Solo II 15Hz emi ing a a wa eleng h o 532 nm. The pulse leng h and ene gy is speci ied o be 6 ns and 100 mJ. An op ical de ice om Cosmica /Pen ax was a ached di ec ly o he lase head o c ea e he ligh shee which had a wid h o app oxima ely 100 µm. The sca e ed p ima y ligh om he bo om was blocked in on o he came a’s lens using he same long pass il e as desc ibed abo e. The luo escen ligh om he luo escen ace pa icles was de ec ed by a monoch ome CCD came a HiSense om Dan ec wi h a esolu ion o 1280x1024 and a cap u ing a e o 8 Hz. The came a was inclined by abou 15 ◦wi h espec o he channel inclina ion as illus a ed in Figu e 2.8. The spa ial esolu ion o his came a se up was abou 8 µm/pixel. Calib a ion o he image has been ca ied ou a p io i by placing a millime e scale a he lase shee posi ion. Came a and lase ha e been synch onized by a igge ing uni om Dan ec. Nd:YAG T igge PC came a CCD lase uni Ligh shee Fil e g Inle Ou le Figu e 2.8: Measu emen se up o he ee su ace shape de ec ion. Rep in ed wi h pe mission om [66]. ©2011, Ame ican Ins i u e o Physics. Cap u ing images om he liquid as desc ibed abo e esul ed in g ainy single images, because he luo escen ace pa icles si a disc e e poin s when one image is aken. The e o e, i was necessa y o a e age o supe impose se e al images o ge a uni o mly b igh shee in he image whose uppe bo de co esponds o he ee su ace con ou o he liquid a he posi ion o he lase shee . Depending on he case he pos –p ocessing wo k low di e s sligh ly. The e o e, a mo e de ailed desc ip ion abou he me hod o how he a e aging o supe imposing o he images was pe o med will be gi en whe e he co esponding esul s a e p esen ed. 2.4.3 S eamline de ec ion A de ec ion o he s eamline pa e n o he liquid in he h oughs o he undula ed inlay in channel 1 has been done in a simila way as desc ibed by Wie schem e al.[33, 35, 44]. The luo escen ace pa icles and ligh shee a e iden ical wi h he ones desc ibed in subsec ion 2.3 and 2.4.2. The sca e ed ligh om he illumina ed pa icles was de ec ed wi h a ame a e o up o 500 Hz depending on he mean low eloci y wi h a monoch ome high–speed came a CamReco d 600 om Op onics. The came a was aligned pe pendic- 23 2.4. Expe imen al se ups ula o he channel side walls as illus a ed in Figu es 2.6 and 2.7 (came a (1)). The s eamline pa e ns ha e been econs uc ed by supe posing 2048 single images aken in one un wi h he came a’s ull esolu ion o 1280x1024 pixel by aking a each pixel posi- ion he b igh es pixel o all images (see Figu e 2.9(b)). The con ou and posi ion o he unde lying opog aphy could ha e been econs uc ed in a simila ashion, bu by aking no he b igh es bu he da kes pixel o all pic u es a each pixel posi ion (see Figu e 2.9(a)). The b igh lines below con ou line o he unde lying opog aphy in Figu e 2.9(b) came om e lec ions a he aluminum subs a e. (a) Con ou o he unde lying opog aphy econ- s uc ed by aking he da kes pixel o a 2048 se ies images a each pixel posi ion. (b) S eamline pa e n econs uc ed by aking he b igh es pixel o a 2048 images se ies a each pixel posi ion. Figu e 2.9: Illus a ion o he e alua ion me hod o he econs uc ion o he shape o he unde - lying opog aphy and he s eamline pa e n om expe imen al single image da a. The esul ing spa ial esolu ion was abou 12 µm/pixel. Spa ial calib a ion o he images has been ca ied ou wi h help o he ce amic calib a ion scale as desc ibed in subsec ion 2.4.2. 2.4.4 Veloci y ield measu emen s Veloci y measu emen s ha e been done wi h a Lase -Dopple -Velocime e (LDV) om Dan ec/In en . A de ailed assessmen o he gene al accu acy o he LDV-measu emen echnique when i is applied on ilm lows is p o ided by Aksel and Schmid chen[74]. As ace pa icles a powde o Z O2/MgO wi h a mean pa icle diame e o 0.8µm, which is desc ibed mo e de ailed in sec ion 2.3, was used. The ligh sou ce o he LDV–sys em was a Spec a Physics A gon–Ion (A +) Lase emi ing ligh a h ee main wa eleng hs o 476.5 nm, 488 nm and 514.5 nm. A Dan ec Fibe Flow beam spli e di ided he lase beam in o wo equally in ense beams and cou- pled he h ee colo s in o 6 glass ib e op ics. Addi ionally, a B agg cell shi ed one o he wo lase beams by 40 MHz o highe equencies be o e he beams a e spli ed in o hei di e en wa eleng hs o wo easons. One, o gene a e he e odyne de ec ion signals wi h a su icien ly high equency om slow sca e ing ace pa icles and wo, o ob ain in o ma ion abou he di ec ion o he low. Because he LDV sys em has been used in he one–dimensional (1D) mode only, all wa eleng hs excep o he mos in ense (514.5 nm) we e blocked by mechanical shu e s. An op ical de ice (LDV-head) om Dan ec wi h an ex a ocussing uni c ossed he wo emaining wo king equency lase beams in an ellip ical measu emen olume which was speci ied o be 25 µm×24 µm×126 µm in size. 24 Chap e 2. Expe imen al sys ems and se ups 31 Chap e 3 Two–dimensional ilm low 3.1 Supp ession o eddies 3.1.1 P oblem o mula ion We conside a s eady wo–dimensional g a i y–d i en low o a New onian liquid down an inclined opog aphy which is sinusoidally undula ed in he main low di ec ion. The p o ile o he subs a e undula ion which is desc ibed by b(x) = acos(2πx/λ),(3.1) is illus a ed in Figu e 3.1, whe e ais he ampli ude and λ he wa eleng h o he pe iodic undula ion. The g a i a ional accele a ion gis gi en by g=g(sin α, −cos α) in he gi en x z h(x) b(x) α g Figu e 3.1: Viscous ilm low down a wa y incline. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i u e o Physics. (x, z)-coo dina e sys em, wi h αbeeing he mean inclina ion angle o he channel. The posi ion o he liquid’s ee su ace is deno ed by h(x). The ilm hickness dcan easily be calcula ed by subs ac ing he bo om con ou b(x) om he ee su ace shape h(x): d(x) = h(x)−b(x).(3.2) The Na ie –S okes equa ions and he con inui y equa ion o incomp essible liquids ρ(u·∇)u=−∇p+η∆u+ρg,∇·u= 0 (3.3) 32 Chap e 3. Two–dimensional ilm low a e ew i en in a dimensionless o m by in oducing e e ence quan i ies. As a cha ac e - is ic leng h we use he ilm hickness dno he co esponding low o e a la incline wi h he same low a e ˙q. F om he well known Nussel solu ion[73] i can be calcula ed o dn=3 p3ν˙q/(gsin α).(3.4) Consequen ly, eloci ies a e escaled wi h he mean eloci y o he co esponding Nussel ilm low which eads ¯un= ˙q/dn= (gd2 nsin α)/(3ν).(3.5) The p essu e is escaled wi h he dynamic p essu e ρ¯u2 n. Inse ing hese scalings in o (3.3) yields a dimensionless o mula ion o he Na ie –S okes and he con inui y equa ions Re (˜ u·∇)˜ u=−Re∇˜p+ ∆˜ u+˜ g,∇· ˜ u= 0,(3.6) wi h he Reynolds numbe Re = ¯undn/ν = ˙q/ν (3.7) and he dimensionless g a i y ec o ˜ g= (3,−3 co α), whe e a ilde˜ ·deno es a dimension- less quan i y. The shape o he unde lying opog aphy is desc ibed by he dimensionless s eepness pa ame e ξ=a/d and he dimensionless wa e numbe k= 2πd/λ. A he bo om o he opog aphy ˜y=˜ b(˜x) = ξcos(k˜x) he no–slip condi ion ˜ u=0 holds. Because he liquids ee su ace con ou is a s eamline he kinema ic bounda y condi ion d˜ h d˜x=˜ ˜u(3.8) has o be ul illed a ˜y=˜ h, whe e ˜uand ˜ a e he eloci y componen s in ˜xand ˜y di ec ion, espec i ely. The dynamic bounda y condi ion, which akes ca e o he balance o s esses a he ee su ace, has o be ul illed and eads n·T=3Bo−1 k2Re κn(3.9) when he iscosi y o ai is neglec ed. The ou e no mal uni ec o o he ee su ace is deno ed by n,Tis he s ess enso T=−(˜p−˜p0)I+ (1/Re)[∇˜ u+ (∇˜ u)T], Iis he iden i y ma ix, p0is he ambien p essu e and κis he cu a u e o he ee su ace which is gi en by κ=1 R= d2h dx2 h1 + dh dx2i3/2,(3.10) wi h Rbeing he adius o cu a u e o he ee su ace shape. The in e se Bond numbe Bo−1= 4π2σ/(ρgλ2sin α) is a measu e o he a io be ween su ace ension s esses and g a i a ional s esses. We ha e now o mula ed he p oblem (3.3) wi h he gi en bound- a y condi ions in a dimensionless o m in a way ha i is go e ned by i e independen dimensionless pa ame e s namely ξ,k, Bo−1, Re and co α, only. Addi ionally, we claim he low o be pe iodic as he unde lying opog aphy is. The e- o e, we assume he ee su ace shape, he p essu e and he eloci y ield o be pe iodic 33 3.1. Supp ession o eddies in downs eam- (˜x-) di ec ion: ˜ h(˜x) = ˜ h(˜x+ 2π/k) (3.11) ˜p(˜x, ˜y) = ˜p(˜x+ 2π/k, ˜y) (3.12) ˜ u(˜x, ˜y) = ˜ u(˜x+ 2π/k, ˜y).(3.13) Wi h he kinema ic (3.8) and he dynamic (3.9) bounda y condi ions a he ee su ace and he no–slip bounda y condi ion a he bo om, he pe iodic bounda y condi ions (3.11)-(3.13) we comple e he se o dimensionless ield equa ions (3.6). The ac ha he posi ion o he ee su ace, and he e o e also he domain o solu ion, is no known a p io i in oduces a u he deg ee o eedom, which has o be cap u ed by a nume ical p ocedu e. This has been achie ed by implemen ing an i e a i e p ocedu e s a ing wi h an ini ial guess o he ee su ace shape. In each solu ion s ep he Na ie – S okes equa ions and he con inui y equa ion (3.6) ha e been sol ed oge he wi h he no–slip condi ion a he subs a e u=0and he dynamic bounda y condi ion a he ee su ace (3.9). The kinema ic bounda y condi ion (3.8) canno be ul illed ye and is o mally in e p e ed as a i s o de di e en ial equa ion o he unknown ee su ace posi ion ˜ h(˜x). Wi h he new ee su ace posi ion he i e a i e p ocedu e is epea ed un il he di e ence be ween he solu ion o he p e ious and he cu en s ep is below a h eshold alue. Since he low a e is s ill a bi a y we claimed a ce ain low a e and he e o e a ce ain Reynolds numbe o ob ain a unique solu ion. In each i e a ion s ep, he eloci ies and he p essu e a e app oxima ed using he Taylo –Hood elemen pai wi h piecewise quad a ic eloci y app oxima ion and piecewise linea p essu e. The esul ing nonlinea equa ion o he nodal eloci y and p essu e is sol ed wi h New on me hod, which ypically con e ges in ou o six i e a ions. The nume ical p ocedu e desc ibed abo e has been implemen ed and all nume ical calcula ions p esen ed in his chap e ha e been ca ied ou by Ch is ian Heining[44]. 3.1.2 Expe imen al and nume ical indings The expe imen s ha e been ca ied ou wi h Basildon silicone oil BC10cs, which is de- sc ibed in sec ion 2.1, lowing o e a sinusoidally undula ed inlay wi h an ampli ude a= 1 mm and a wa eleng h λ= 10 mm placed in channel 1 nea i s in low as desc ibed in sec ion 2.2. Measu emen s ha e been done a ou di e en inclina ion angles anging om 5 ◦ o 14 ◦. The Reynolds numbe has been a ied be ween 3 and 63. S eamline pa e ns ha e been eco ded expe imen ally as desc ibed in sec ion 2.4.3. Figu e 3.2 shows a compa ison o expe imen ally and nume ically obse ed s eamlines a di e en low a es o Reynolds numbe s, espec i ely. The b igh sinusoidal line which is o e laid by a ed one co esponds o he subs a e geome y. The lines abo e a e he nume ically (g een) and expe imen ally (black & whi e) de e mined s eamlines. The uppe mos ed line co esponds o he nume ically de e mined ee su ace con ou . The b igh lines below he bo om and he in e sely ben lines abo e he ee su ace a e e lec ions coming om he subs a e o he ee su ace o he liquid. We ind ha he low shows quali a i ely di e en beha io depending on he low a e o he Reynolds numbe , espec i ely. When he mean ilm hickness dis small compa ed o he wa eleng h λand he ampli ude a he low can locally be well desc ibed by he Nussel solu ion wi h he local inclina ion angle[21]. The e o e, he liquid lows 34 Chap e 3. Two–dimensional ilm low Figu e 3.2: Compa ison o expe imen al pa h lines o nume ical s eamlines. The images a e o a ed by he mean inclina ion angle o he channel. The olume lux is con inuously inc eased o m a) o d). a) Re = 9: no eddy a low Reynolds numbe s. b) Re = 16: inc easing ine ia esul s in he gene a ion o an eddy in he ough o he undula ion. c) Re = 31: inc easing ine ia u he , he eddy anishes. d) Re = 48: low sepa a ion eappea s a e en highe Reynolds numbe s. Bo om con ou : lowe b igh sinusoidal line; lines below and in e sely ben lines in he uppe pa o he images a e e lec ions o he pa h lines a he bo om and a he ee su ace. Channel inclina ion angle α= 8◦. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i u e o Physics. smoo hly along he subs a e con ou when he Reynolds numbe s a e a he small (See Figu e 3.2a)). When he low a e is inc eased he low begins o sepa a e in o a egion whe e he low eci cula es wi hin he ough o he undula ion and in o a main low abo e (See Figu e 3.2b)). In con as o he as majo i y o sys ems whe e an inc ease o he in luence o ine ia leads o a g ow h o eci cula ion a eas[77], inc easing he Reynolds numbe in his sys em leads o diminu ion o he eddies un il hey anish comple ely as shown in Figu e 3.2c). Howe e , inc easing he Reynolds numbe u he , we ind a c i ical Reynolds numbe a which he eddies eappea as depic ed in Figu e 3.2d). Abo e his c i ical Reynolds numbe he eddies g ow mono onously in size wi h inc easing Reynolds numbe . All s eamline pa e ns shown in Figu e 3.2 co espond o a mean channel inclina ion angle αo 8 ◦. Quali a i ely simila esul s ha e been ob ained o o he inclina ion angles. We ema k ha closed pa h lines, he lack o ji e and he excellen ag eemen be ween expe imen and nume ics indica e ha uns eady mo ion is negligible and ha he pe iodici y (3.11)-(3.13) and wo–dimensionali y assump ions made in sec ion 3.1.1 hold. De ec ion o he ee su ace shape o he liquid has been done as desc ibed in sec- ion 2.4.2. Because he luo escen ace pa icles desc ibed in sec ion 2.3 si a disc e e 35 3.1. Supp ession o eddies poin s when an image is aken, a single eco ding esul ed in a g ainy image o he liquid (see Figu e 3.3(a)). Thus, he e alua ion o he ee su ace shape has been done wi h an image, which has been a e aged o e 50 single images (See Figu e 3.3(b)). We no e, ha he a e age ilm hickness in Figu e 3.3 appea s much hinne as i is because he image o he liquid below he ee su ace (and also o he unde lying opog aphy) is s ongly dis o ed by he cu ed su ace o he liquid. (a) Single image om su ace de ec ion came a (2). (b) Image a e aged o e 50 single eco dings. Figu e 3.3: Illus a ion o he a e aging p ocess o he de ec ion o he ee su ace shape. The uppe bo de o he b igh shee in each image co esponds o he con ou o he ee su ace. As Figu e 3.2 indica es no only he low in he oughs o he undula ion shows a s ong Reynolds numbe dependence bu also he ee su ace shape o he liquid changes conside ably wi h he Reynolds numbe . Figu e 3.4 p o ides a compa ison o measu ed (symbols) and calcula ed (lines) ee su ace shapes a di e en Reynolds numbe s o a mean channel inclina ion angle αo 8 ◦. The cu es a e s agge ed in z-di ec ion o a oid o e lapping. The Reynolds numbe anges om 6.6 (lowe mos cu e) o 56.2 (uppe mos cu e). A low Reynolds numbe s we ind he ee su ace shape o be a he ha monic and o small ampli ude. When Re is inc eased he ee su ace apidly gains in ampli ude, is shi ed downs eam and exhibi s a sha p nonlinea inden a ion in he ough which becomes maximal a Re = 31.5 (See ed ’x’-symbols in Figu e 3.2). Inc easing he Reynolds numbe jus a li le u he , om Re = 31.5 o Re = 32.1 in Figu e 3.2, causes his inden a ion o anish e y b usquely esul ing in a smoo h sinusoidal shape again (See g een ’+’-symbols in Figu e 3.2). Fu he inc ease o Re shi s he ee su ace con ou u he downs eam while i s ampli ude dec eases con inuously. Quali a i ely simila esul s ha e been ob ained o o he inclina ion angles. Reci cula ion a eas Based on he expe imen ally and nume ically ob ained s eamline pa e ns we ha e e al- ua ed he size o he eci cula ion a eas in he oughs o he undula ion. Figu e 3.5 shows he eddy a ea as a unc ion o he Reynolds numbe a ou di e en mean in- clina ion angles α. Excep o he s eepes inclina ion angle o 14 ◦eddies appea wi h inc easing Reynolds numbe a a i s c i ical Reynolds numbe Re1≈11 which seems o be independen o he channel inclina ion angle α. Then he eddy size inc eases un il i eaches a local maximum and sh inks again un il i anishes comple ely a a second c i ical Reynolds numbe Re2. Only beyond a hi d c i ical Reynolds numbe Re3 he eddies eappea and g ow mono onously in size wi h inc easing Reynolds numbe in he in es iga ed ange. While he second c i ical Reynolds numbe o he disappea ance o 36 Chap e 3. Two–dimensional ilm low 0510 15 20 25 x / [mm] 0 1 2 3 4 ee su ace posi ion / [mm] 56.2 48.8 42.6 36.4 32.1 31.5 24.1 18.9 11.6 6.6 Reynolds numbe Figu e 3.4: Compa ison o expe imen al and nume ical ee su ace shapes a di e en Reynolds numbe s. Expe imen al da a a e ep esen ed by symbols; nume ical da a a e ep esen ed by lines. The ee su ace posi ions a e shi ed pe pendicula o he mean low di ec ion o a oid o e lapping. The e ical posi ion augmen s wi h Reynolds numbe . Channel inclina ion angle α= 8 ◦. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i u e o Physics. he eci cula ion a eas showed a s ong channel inclina ion angle dependence, he c i ical Reynolds numbe s Re1and Re3 o he eme ging o eddies seemed o be a he indepen- den o αin he in es iga ed inclina ion angle ange. 10 20 30 40 50 Reynolds numbe 0 0.2 0.4 0.6 0.8 Eddy a ea / aλ α = 5° α = 8° α = 11° α = 14° Figu e 3.5: C oss–sec ional a ea o he eddy as a unc ion o he Reynolds numbe a di e en inclina ion angles. Expe imen al and nume ical da a a e ep esen ed by open and solid symbols, espec i ely. A leas 40 measu emen s pe inclina ion angle ha e been ca ied ou om Re ≈6 o Re ≈62 in equidis an s eps. Whe e no eddy was obse ed, mos da a poin s ha e been blanked ou o cla i y. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i u e o Physics. Thus, we ind ha eddies which appea a no oo s eep inclina ion angles a a c i ical Reynolds numbe Re1disappea again in an eddy– ee window whose ex en g ows wi h inc easing α. The window whe e eddies can be obse ed be ween Re1and Re2sh inks acco dingly a he expense o he eddy– ee window un il i anishes comple ely o an inclina ion angle α= 14 ◦. 37 3.1. Supp ession o eddies F ee su ace shape Quan i a i e analysis o he ee su ace shape da a shown in Figu e 3.4 has been done by decomposing hem in o Fou ie se ies by disc e e Fou ie ans o ma ion (DFT). Figu es 3.6-3.8 show he Reynolds numbe dependence o he ampli udes o he ze o h, he i s and he second Fou ie modes o all inclina ion angles s udied. The ze o h Fou ie mode is illus a ed in Figu e 3.6 and co esponds o he ilm heigh 1h(x) a e aged o e one pe iod o he bo om con ou . A Reynolds numbe s below ≈25 and abo e ≈37 we ind a mono onous inc ease o he a e age ilm hickness wi h inc easing olume lux o Reynolds numbe , espec i ely o all in es iga ed channel inclina ion angles, as i is common o g a i y–d i en ilm lows[33, 34, 35]. In he egion a in e media e Reynolds numbe s all da a se s e eal a spon aneous d op in he a e age ilm hickness wi h inc easing Reynolds numbe . The posi ion whe e his d op akes place shi s wi h inc easing channel inclina ion o smalle Reynolds numbe s. 0 10 20 30 40 50 60 Reynolds numbe 1 1.5 2 2.5 3 3.5 4 A e age ilm hickness / a α = 5° α = 8° α = 11° α = 14° Figu e 3.6: Mean ilm hickness a e aged o e one bo om pe iod as a unc ion o he Reynolds numbe a di e en inclina ion angles. Expe imen al and nume ical da a a e ep esen ed by sym- bols and lines, espec i ely. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i u e o Physics. The i s ha monic o he Fou ie ans o med ee su ace shape co esponds o he wa eleng h o he bo om con ou . I s ampli ude shows a peak which g ows and shi s i s posi ion om Re ≈35 o Re ≈27 o smalle Reynolds numbe s when he channel inclina ion angle αbecomes s eepe (See Figu e 3.7). The p esence o his peak e lec s he ac ha he ee su ace is s ong undula ed whe e he i s ha monic peaks, bu a he la o low and o high Reynolds numbe s as al eady illus a ed in Figu e 3.4. The ampli ude o he second ha monic o he ee su ace shape cha ac e izes i s nonlinea i y. We ind ha i g ows wi h inc easing Reynolds numbe and eaches a pla eau be o e i d ops discon inuously o a much smalle alue and ends apidly agains ze o o la ge Reynolds numbe s, as isible om Figu e 3.8. The g ow h and apid d op o he second Fou ie mode co esponds o he eme ging o he sha p inden a ion in he oughs o he ee su ace shape, as can be seen mos clea ly om he ed line in Figu e 3.4, and he ab up shape ansi ion o a smoo h sinusoidal one as ep esen ed exempla ily by he g een line in Figu e 3.4. The heigh o he pla eau g ows wi h s eepe channel inclina ions. 1o ilm hickness d(x) = h(x)−b(x). 38 Chap e 3. Two–dimensional ilm low 0 10 20 30 40 50 60 Reynolds numbe 0 0.2 0.4 0.6 0.8 1 Ampli ude o 1s ha monic / a α = 5° α = 8° α = 11° α = 14° Figu e 3.7: Ampli ude o he i s ha monic o he ee su ace shape as a unc ion o he Reynolds numbe a di e en inclina ion angles. Expe imen al and nume ical da a a e ep esen ed by sym- bols and lines, espec i ely. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i u e o Physics. In conjunc ion o he g ow h, he posi ion o he sha p d op is shi ed o smalle Reynolds numbe s wi h s eepening he mean channel inclina ion. All highe ha monics, which a e no shown he e, showed quali a i ely he same beha io as he second one bu wi h much smalle ampli udes. 0 10 20 30 40 50 60 Reynolds numbe 0 0.05 0.1 0.15 0.2 0.25 Ampli ude o 2nd ha monic / a α = 5° α = 8° α = 11° α = 14° Figu e 3.8: Ampli ude o he second ha monic o he ee su ace shape as a unc ion o he Reynolds numbe a di e en inclina ion angles. Expe imen al and nume ical da a a e ep esen ed by symbols and lines, espec i ely. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i u e o Physics. A compa ison o Figu es 3.6, 3.7 and 3.8 e eals ha he d op o he a e age ilm hickness, he peak posi ion o he i s Fou ie mode and he d op o he ampli ude o he second Fou ie mode seem o coincide a he same Reynolds numbe o each channel inclina ion angle. Figu e 3.9 complies he i s ha monics o he ou in es iga ed inclina- ion angles. We ind ha o all in es iga ed channel inclina ions he i s ha monic peaks, whe e he a e age ilm hickness d ops and he shape o he ee su ace unde goes a sha p ansi ion om an anha monic shape o a smoo h sinusoidal one. This posi ion whe e a su ace shape ansi ion occu s is indica ed by he dashed line in each diag am. The Reynolds numbe whe e his ansi ion akes place shi s wi h s eepe channel inclina ions 39 3.1. Supp ession o eddies o smalle Reynolds numbe s. 10 20 30 40 50 60 Reynolds numbe 0 0.2 0.4 0.6 0.8 1 Ampli ude o 1s and 2nd ha monic / a 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 A e age ilm hickness / a A e age ilm hickness 1s Fou ie mode 2nd Fou ie mode (a) Channel inclina ion angle: 5 ◦. 10 20 30 40 50 60 Reynolds numbe 0 0.2 0.4 0.6 0.8 Ampli ude o 1s and 2nd ha monic / a 0 0.5 1.0 1.5 2.0 2.5 3.0 A e age ilm hickness / a A e age ilm hickness 1s Fou ie mode 2nd Fou ie mode (b) Channel inclina ion angle: 8 ◦. Rep in ed wi h pe mission om [44]. ©2010, Ame ican Ins i- u e o Physics. 10 20 30 40 50 60 Reynolds numbe 0 0.2 0.4 0.6 0.8 Ampli ude o 1s and 2nd ha monic / a 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 A e age ilm hickness / a A e age ilm hickness 1s Fou ie mode 2nd Fou ie mode (c) Channel inclina ion angle: 11 ◦. 10 20 30 40 50 60 Reynolds numbe 0 0.2 0.4 0.6 0.8 1 Ampli ude o 1s and 2nd ha monic / a 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 A e age ilm hickness / a A e age ilm hickness 1s Fou ie mode 2nd Fou ie mode (d) Channel inclina ion angle: 14 ◦. Figu e 3.9: A e age ilm hickness and ampli ude o he i s wo Fou ie componen s. The ansi ion Reynolds numbe is indica ed by he dashed line. The expe imen al and nume ical da a a e ep esen ed by symbols and lines, espec i ely. Figu e 3.10 shows a snapsho o a ideo2, which illus a es he ansi ion o he liquid’s ee su ace shape while he Reynolds numbe has been inc eased con inuously o e he ansi ion Reynolds numbe om Re ≈26 o Re ≈34 om wo di e en pe spec i es a a ixed channel inclina ion angle o α= 8 ◦. The main ame depic s a sligh ly de o med pic u e o wo and a hal pe iods o he ee su ace shape. The main low di ec ion in he main ame is om igh o le . S a ing om low Reynolds numbe s (Re ≈26) we ind a ee su ace shape wi h sha p inden a ions which g ow wi h Reynolds numbe . The posi ion o he sha p inden a ion shi s sligh ly downs eam while i is ge ing sha pe un il he ansi ion Reynolds numbe (Re ≈32) is eached. He e he sha p inden a ion disappea s and he ee su ace shape changes e y ab up ly in o a smoo h sinusoidal one. A u he inc ease o he Reynolds numbe causes he ampli ude o he ee su ace undula ion o diminish con inuously. The inse in he lowe igh co ne shows he low o e he whole channel wid h om abo e. The g een line in he middle o he inse co esponds o he lase shee p oduced by he A +-lase desc ibed in sec ion 2.4.2. The main low di ec ion he e is om up 2Video a ailable a h p://www. ms.uni-bay eu h.de/ ideos/su aces.mp4 40 Chap e 4. Th ee–dimensional ilm low x z α u(z) g (a) Side iew y z B θ ∆h H h(y) (b) C oss–sec ional iew Figu e 4.1: Channel geome y illus a ing side wall e ec s on he low and he o ien a ion and po- si ion o he (x, y, z)–coo dina e sys em. Rep in ed wi h pe mission om [66]. ©2011, Ame ican Ins i u e o Physics. ead u(y, z = 0) = 0,(4.5) u(y=±B/2, z)=0.(4.6) The no pene a ion condi ion a he igid walls is i ially ul illed because he low has been pos ula ed o be unidi ec ional. Addi ionally o he bounda y condi ions a he walls a kinema ic bounda y condi ion a he ee su ace n·uz=h(y)= 0 (4.7) basically demands he ee su ace con ou o be a s eamline, o in o he wo ds liquid pa icles mus no lea e he ee su ace. When he iscosi y o ai is neglec ed he dynamic bounda y condi ion, which akes ca e o he balance o s esses a he ee su ace, eads h(p−p0)−σ Rin=T·nz=h(y),(4.8) wi h he s ess enso T=Tij =η∂ui ∂xj +∂uj ∂xi(4.9) and he ou e uni no mal ec o n=∇h k∇hk=ˆez−∂h ∂y ˆey 1 + ∂h ∂y 2.(4.10) 47 4.1. Basic low The p essu e o he su ounding ai is deno ed by p0. The cu a u e o he ee su ace κ which is he in e se adius o cu a u e o he ee su ace Rcan be calcula ed o κ=1 R=∇·n=−d dy ∂h ∂y 1 + ∂h ∂y 2=− ∂2h ∂y2 1 + ∂h ∂y 23/2.(4.11) Inse ing (4.10) and (4.11) in o he dynamic bounda y condi ion (4.8) leads o wo com- ponen s o he dynamic bounda y condi ion no mal and angen ial o he ee su ace (p−p0)z=h(y)=−σd dy ∂h ∂y 1 + ∂h ∂y 2,(4.12) ∂u ∂z z=h(y)=∂u ∂y ∂h ∂y .(4.13) Dimensionless o mula ion To e o mula e he p oblem in a dimensionless o m we ha e o ind some e e ence quan- i ies o scaling. We ake he ee su ace eloci y u =ρg sin αH2 2η(4.14) om he well known Nussel solu ion [73] as a e e ence o all eloci ies. Hyd os a ic p essu e is aken as a e e ence o he p essu e p =ρg cos α. (4.15) We in oduce a gene alized capilla y leng h L L= 2σ ρg cos α,(4.16) which akes ca e o a educed g a i a ional accele a ion pe pendicula o he channel due o i s inclina ion αand se es as a e e ence o all leng hs o esol e e ec s wi hin he capilla y ele a ion. In he ollowing all quan i ies which a e labeled by a ˜ ·deno e dimensionless a iables which a e scaled wi h he abo e e e ence quan i ies. Fu he mo e, we de ine a dimensionless capilla y ange l l=L B/2=2L B.(4.17) Applying he abo e scalings o he Na ie –S okes equa ions and he bounda y condi ions lead o a dimensionless o mula ion o he sys em in he ollowing o m. The Na ie – S okes equa ions ead: 0 = ∂2˜u ∂˜y2+∂2˜u ∂˜z2+ 2,(4.18) 0 = −∂˜p ∂˜y,(4.19) 0 = −∂˜p ∂˜z−1.(4.20) 48 Chap e 4. Th ee–dimensional ilm low The no–slip bounda y condi ions a he bo om and he side walls changes o ˜u(˜y, ˜z= 0) = 0,(4.21) ˜u(˜y=±1/l, ˜z) = 0 (4.22) and he angen ial and no mal componen s o he dynamic bounda y condi ion now ead ∂˜u ∂˜z˜z=˜ h(˜y)=∂˜u ∂˜y d˜ h d˜y,(4.23) (˜p−˜p0)˜z=˜ h(˜y)=−1 2 d d˜y ∂˜ h ∂˜y 1 + ∂˜ h ∂˜y2.(4.24) F ee su ace shape We in oduce an addi ional bounda y condi ion aking ca e o he con ac angle θbe ween he liquid and he side wall ∂˜ h ∂˜y˜y=±1/l =±co θ(4.25) and a decomposi ion o he ee su ace shape ˜ h(˜y) in o a cons an pa which is equal o he ilm heigh in he middle o he channel ˜ Hand a pa ζdepending on ˜ywhich akes ca e o he capilla y ele a ion in he icini y o he side wall ˜ h(˜y) = ˜ H+ζ(˜y).(4.26) E alua ing ζa ˜y=±1/l by inse ing he bounda y condi ion (4.25) in o (4.24) leads o he capilla y ele a ion heigh depic ed in Figu e 4.1(b) [66] ∆˜ h=ζ(˜y=±1/l) = √1−sin θ. (4.27) The ee su ace shape can be ob ained by in eg a ion o equa ion (4.24) as desc ibed in de ail by Scholle and Aksel[63] o Haas e al.[66]. Wi h he abb e ia ion G(x) := x−1 2√2ln √2 + x √2−x!,(4.28) he unc ion o he ilm ele a ion ζ(˜y) can be w i en down in an implici o m ˜y(ζ) = ˜ h−1(ζ) =    −1/l +hG√1 + sin θ−Gp2−ζ2i ˜y∈[−1/l, 0], 1/l −hG√1 + sin θ−Gp2−ζ2i ˜y∈[0,1/l].(4.29) Veloci y ield The eloci y ield can be desc ibed by he ollowing ansa z which is a solu ion o equa ion (4.18) and al eady ul ills he no-slip bounda y condi ions a he side walls ˜u=4 l2X n∈N+Dnelcn˜z+Ene−lcn˜z−(−1)n c3 ncos(lcn˜y),(4.30) 49 4.1. Basic low cn=n−1 2π. (4.31) The addi ional no-slip condi ion a he subs a e (4.21) leads o nequa ions o he ec o elemen s Dnand En Dn+En=(−1)n c3 n .(4.32) Inse ing he ansa z o he eloci y ield (4.30) in o he angen ial pa o he dynamic bounda y condi ion (4.23) leads o an in ini e sys em o algeb aic equa ions o he coe - icien s DnX n∈N+ GnmDn=dm,(4.33) wi h he ma ix elemen s Gnm =Z1/l −1/l 2 cosh(lcn˜ h(˜y)) sin(lcn˜y) sin(lcm˜y)d˜y(4.34) and he ec o elemen s dm=X n∈N+Z1/l −1/l (−1)n c3 ne−lcn˜ h(˜y)+lcn˜ h(˜y)sin(lcn˜y) sin(lcm˜y)d˜y −l cmX n∈N+Z1/l −1/l (−1)n˜ h(˜y) cn [cos(lcn˜y) cos(lcm˜y)] d˜y. (4.35) The coe icien s Dnand Enha e been calcula ed using MATLAB®[83] unca ing he in ini e sys em o algeb aic equa ions o a ce ain o de N∈N+. The accu acy o he powe se ies expansion has been assu ed by he demand Dn, En<10−6.[66] The heo e ical de i a ion and he implemen a ion o he p ocedu e o solu ion de- sc ibed abo e has been done and all heo e ical esul s p esen ed in his chap e ha e been calcula ed by And ´e Haas[66]. 4.1.2 Flow ype classi ica ion Depending on he magni ude o he capilla y ele a ion compa ed o he ilm heigh H and he channel wid h Bi is use ul o dis inguish di e en low ypes as illus a ed in Figu e 4.2. Fo a channel o in ini e ex en he low con igu a ion is equi alen he wo–dimensional case and he solu ion o he eloci y ield equals he well known Nussel solu ion. In his case we de ine he Reynolds numbe as Re(a)= Re2D =usH ν.(4.36) Because he ee su ace eloci y o a Nussel ilm low can easily be calcula ed o us=gsin αH2 2ν,(4.37) he Reynolds numbe can also be exp essed as a unc ion o he ilm heigh H Re2D =gsin αH3 2ν2.(4.38) 50 Chap e 4. Th ee–dimensional ilm low (a) In ini e b oad channel o slip condi ion a he side walls (equals he 2D case). (b) No capilla y ele a ion a he side walls (θ= 90◦ o ∆hH) (c) (θ < 90◦and ∆h∼H) (d) Capilla y co ne low (θ < 90◦and ∆hH) Figu e 4.2: C oss sec ional eloci y p o iles o di e en channel low ypes. The eloci y is colo coded: blue co esponds o slow and ed co esponds o as . Rep in ed wi h pe mission om [66]. ©2011, Ame ican Ins i u e o Physics. As soon as side walls a e in oduced he liquid in he icini y o he side wall ge s de- cele a ed due o he addi ional no–slip condi ion a he bounda y. Now, he ee su ace eloci y is no independen o he c osswise coo dina e yanymo e and i is use ul o de ine he Reynolds numbe in e ms o he co esponding olume lux ˙ V Re(b)=3 2Bν Z (y) 0ZB/2 −B/2 u(y, z)dydz=3˙ V 2Bν <Re2D.(4.39) Compa ed o he scena io depic ed in Figu e 4.2(a) less liquid is anspo ed due o he addi ional d ag a he side walls. The e o e, he Reynolds numbe o a ilm in his case is alway smalle han he Reynolds numbe o a wo–dimensional ilm o he same ilm hickness H. When His dec eased, he con ac angle θbe ween he liquid and he side wall becomes an impo an ac o . The in luence o capilla i y leads o a capilla y ele a ion o he liquid, when θis smalle han 90◦. Due o his locally hicke ilm a eloci y o e shoo close o he side walls may show up when he addi ional ilm hickness wins o e he addi ional d ag coming om he no–slip condi ion a he side walls (see Figu e 4.2(c)). In he limi o anishingly hin ilms (H→0) he low degene a es o a capilla y co ne low as depic ed in Figu e 4.2(d). Now mos o he liquid is anspo ed close o he side walls in he capilla y ele a ion and he Reynolds numbe is ob iously la ge han in he wo–dimensional case Re(d)=3˙ V 2Bν >Re2D →0, H →0.(4.40) When he con ac angle be ween he liquid and he side wall is la ge han 90◦ he cap- illa y ele a ion ∆hbecomes nega i e and no eloci y o e shoo can be obse ed. The e- o e, we es ic ou s udies wi hou loss o gene ali y o liquids wi h we ing p ope ies only. 4.1.3 Flow a e s udy Since he p esence o side walls and he esul ing capilla y ele a ion has a signi ican impac on he low a e, a s udy on he impo an pa ame e s is o be ca ied ou . The 51 4.1. Basic low olume lux o a wo–dimensional case ˙ V2D which is illus a ed in Figu e 4.2(a) is aken as a e e ence. Figu e 4.3 illus a es he inequali y (4.39) o wo di e en capilla y anges, one ma ch- ing he expe imen al se up (l= 0.028) and one e e ing o he capilla y ange o a na - owe (and/o s eepe ) channel (l= 0.1). When no capilla y ele a ion is p esen , as depic ed in Figu e 4.2(b) he addi ional d ag a he side walls leads o an o e all dec ease o he anspo ed liquid depending on he dimensionless ilm heigh ˜ Hand he capilla y ange l. The amoun o missing olume lux inc eases wi h dec easing channel wid h B and inc easing dimensionless ilm heigh ˜ H. Bo h an inc easing dimensionless ilm heigh and a na owing o he channel lead o an inc ease o he ela i e pa o he side wall a ea which esul s in a s onge impac o he side wall p esence on he olume lux. The e- o e, especially o na ow channels and ilm hicknesses which a e la ge compa ed o he capilla y leng h L he in luence o he side walls on he olume lux canno be neglec ed. 0.9 1 0 0.2 0.4 0.6 0.8 1 ˜ H ˙ Vθ=90◦/˙ V2D l= 0.028 l= 0.1 Figu e 4.3: Dec ease o he olume lux due o he addi ional no–slip condi ion a he side walls (wi hou capilla i y), as i is depic ed in Figu e 4.2(b), compa ed o he wo–dimensional case, which is depic ed in Figu e 4.2(a). Rep in ed wi h pe mission om [66]. ©2011, Ame ican Ins i u e o Physics. In he case depic ed in Figu e 4.2(c) i is no possible o make a simila gene al s a emen on he Reynolds numbe o he olume lux like in he equa ions (4.39) o (4.40), because bo h e e s o one an addi ional d ag coming om he side wall and wo a eloci y o e shoo ha e a compe ing in luence o he same o de on he low a e as illus a ed in Figu e 4.4. Figu e 4.5 compa es he amoun o anspo ed liquid when a capilla y ele a ion due o he p esence o side walls is conside ed wi h he wo–dimensional case. The con ac angles θa e chosen o i he expe imen al se up as desc ibed in sec ion 2.2. Fo small ilm hickness ˜ H he in luence o he capilla y ele a ion and he esul ing eloci y o e shoo becomes he mos impo an anspo mechanism. The a io o ˙ V / ˙ V2D becomes la ge han one and e en di e ges o ˜ H→0 because ˙ V2D hen also ends o ze o. Fo cons an ilm heigh ˜ H he p esence o a capilla y ele a ion becomes ob iously mo e impo an o na owe channels o la ge capilla y anges l(see Figu e 4.5(a)). Remembe , ha he wo–dimensional case is equal o he case l→0. The in luence o he capilla y ange on he olume lux inc eases wi h dec easing ilm heigh . One inds an explici ansi ion ilm heigh deno ed by ˜ h , which is independen o he capilla y leng h, whe e he in luences o he eloci y o e shoo due o capilla y ele a ion and he in luence o he no–slip condi ion 52 Chap e 4. Th ee–dimensional ilm low u umax us,m Figu e 4.4: F ee su ace eloci y p o ile in low di ec ion showing eloci y o e shoo and de ec compa ed o he plane low wi h he same ilm heigh . Rep in ed wi h pe mission om [70]. ©2011, Ame ican Ins i u e o Physics.[70] a he wall on he no malized olume lux ˙ V / ˙ V2D jus cancel each o he . 1 1.5 2 0 0.2 0.4 0.6 0.8 1 ˜ H ˙ V / ˙ V2D l= 0.028 l= 0.1 ˜ h (a) Va ia ion o he capilla y ange la ixed con- ac angle θ= 8◦. 1 1.5 2 0 0.2 0.4 0.6 1 ˙ V / ˙ V2D ˜ H θ= 8◦ θ= 52◦ ˜ h ˜ h (b) Va ia ion o he con ac angle θa ixed capil- la y ange l= 0.1. Figu e 4.5: In luence o capilla y e ec s a he side walls on he no malized olume lux. Rep in ed wi h pe mission om [66]. ©2011, Ame ican Ins i u e o Physics. The impac o he con ac angle θon he olume lux is illus a ed in Figu e 4.5(b). Simila o he capilla y ange in Figu e 4.5(a) also he in luence o he con ac angle on he olume lux is small o ilm heigh s ˜ Ho he o de o one bu gains in impo ance he hinne he liquid ilm ge s. One inds ha smalle con ac angles lead o a la ge eloci y o e shoo and he e o e o a la ge olume lux. The ansi ion ilm heigh ˜ h is no independen o he con ac angle. Fo la ge ˜ H he a io o ˙ V / ˙ V2D becomes smalle han one because he in luence o capilla i y on he eloci y ield looses impo ance. The olume lux ˙ V ends o ˙ Vθ=90◦ o ˜ H→ ∞ which is always smalle han ˙ V2D as depic ed in Figu e 4.3. Figu e 4.6 shows he dependence o he ansi ion ilm heigh ˜ h on he con ac angle θ. Fo a con ac angle o θ= 90◦no capilla y ele a ion is p esen and hus no eloci y o e shoo can be obse ed. Equali y o he olume lux o he 2D case ˙ V2D and he olume lux ˙ Vcan only be eached, when also he d ag in luence o he side wall ends o ze o which is only he case in he limi ˜ h→0. Dec easing he con ac angle leads o a mono onous inc ease o he ansi ion ilm hickness o ini e alues below one. Unde pe ec we ing condi ions (θ= 0◦) he ansi ion ilm heigh eaches a alue o app oxima ely ˜ h ≈0.92. To summa ize: T ea ing a channel o ini e wid h as wo–dimensional always leads o 53 4.1. Basic low 0 0.2 0.4 0.6 0.8 1 0 30 60 90 ˜ h θ/[◦] Figu e 4.6: Dependence o he ansi ion ilm hickness ˜ h on he con ac angle θ. Rep in ed wi h pe mission om [66]. ©2011, Ame ican Ins i u e o Physics. an o e es ima ion o he olume lux (o he Reynolds numbe ) when he ilm hickness H is la ge han he gene alized capilla y leng h L(o abo e he ed cu e in Figu e 4.6) due o he addi ional d ag coming om he side walls. This disc epancy becomes la ge especially o na ow channels. When he ilm hickness is small o o he same o de as he gene alized capilla y leng h he p esence o a capilla y ele a ion gains in impo ance one has o ake he con ac angle θin o accoun . The e ec o a esul ing eloci y o e shoo compe es wi h he addi ional d ag a he side wall. Fo hin ilms and small con ac angles (below he ed cu e in Figu e 4.6) ea ing he ilm as wo–dimensional leads o an unde es ima ion o he olume lux. 4.1.4 Veloci y ield Figu e 4.7 shows a compa ison o he heo e ical and measu ed eloci y p o iles o h ee ilms o di e en heigh s Hand wo di e en s a ic con ac angles θ. As liquid Elbesil silicone oil B1000 which is desc ibed in sec ion 2.1 was used. Fo each low con igu a ion he side wall dis ance dsdependence o he eloci y p o iles was measu ed by a Lase Dopple Velocime e desc ibed in sec ion 2.4.4 a h ee di e en measu emen heigh s Hm. The e o ba s o he measu ed da a deno e he oo mean squa e e o o he mean alue o all de ec ed eloci y signals in each measu emen olume. Because he numbe o e aluable coun s pe ime dec eases wi h he speed o he liquid in he measu emen olume, he measu emen ime has been adop ed o he low eloci y o ge easonable signal o noise a ios especially in nea wall egions. Addi ionally, mo e poin s ha e been eco ded in he icini y o he wall o esol e he eloci y o e shoo . The o e all ag eemen be ween he measu ed da a and he calcula ed alues o he low eloci ies is excellen . The small de ia ions which ne e exceed he oo mean squa e e o ba s a e in he mos cases o s a is ically na u e. Sys ema ic disc epancies such as in Figu e 4.7(a) can be explained by e o s in de e mining he dis ance be ween he channel bo om and he measu emen olume Hmo in de e mining he ilm heigh in he middle o he channel H. Fo la ge side wall dis ances he measu ed and calcula ed eloci y p o ile co esponds 54 Chap e 4. Th ee–dimensional ilm low d u Hm= 0.7585 mm Hm= 0.5309 mm Hm= 0.3034 mm su ace 0 5 0 10 20 [mm/ s] [mm] s (a) H= 0.965 mm, θ= 8◦. d u Hm= 0.7585 mm Hm= 0.5309 mm Hm= 0.3034 mm su ace 0 5 0 10 20 [mm/ s] [mm] s (b) H= 0.960 mm, θ= 52◦. d u Hm= 1.0619 mm Hm= 0.7586 mm Hm= 0.3793 mm su ace 0 5 10 0 10 20 [mm/ s] [mm] s (c) H= 1.550 mm, θ= 8◦. d u Hm= 1.0619 mm Hm= 0.7585 mm Hm= 0.3793 mm su ace 0 5 10 0 10 20 [mm/ s] [mm] s (d) H= 1.550 mm, θ= 52◦. d u Hm= 1.8962 mm Hm= 1.5170 mm Hm= 1.1377 mm su ace 0 5 10 15 0 10 20 [mm/ s] [mm] s (e) H= 2.208 mm, θ= 8◦. u d Hm= 1.8962 mm Hm= 1.5170 mm Hm= 1.1377 mm su ace 0 5 10 15 0 10 20 [mm/ s] [mm] s ( ) H= 2.204 mm, θ= 52◦. Figu e 4.7: Compa ison o measu ed (poin s) and calcula ed (lines) eloci y p o iles o di e en ilm heigh s H, measu emen heigh s Hmand con ac angles θ. Rep in ed wi h pe mission om [66]. ©2011, Ame ican Ins i u e o Physics. o he Nussel solu ion. In he nea wall egion a eloci y o e shoo is obse ed whose magni ude, quan i ied by he a io o he highes eloci y umax and he su ace eloci y in he middle o he channel us,m, depends s ongly on he ilm hickness Hand he con ac angle θ. Figu e 4.8(a) shows he dependence o he magni ude o he eloci y o e shoo on he ilm hickness ˜ H o bo h con ac angles θmeasu ed. Fo hick ilms he a io o umax/us,m ends o one, meaning ha no eloci y o e shoo can be obse ed. When he ilm hickness ˜ His app oxima ely 0.5 o less he magni ude o he eloci y o e shoo becomes conside able and depends on he con ac angle θas also shown in Figu e 4.7. As he ilm hickness is dec eased u he he eloci y o e shoo di e ges, because he ee 55 4.1. Basic low su ace eloci y in he middle o he channel ends o ze o. The low hen degene a es o a capilla y co ne low as illus a ed in Figu e 4.2(d). 0 2 4 6 0 0.5 1 1.5 umax/us,m ˜ H θ= 8◦ θ= 52◦ (a) No malized maximal ee su ace eloci y. 0 0.5 1 1.5 ˜ymax 0 1/2l 1/l θ= 8◦ θ= 52◦ ˜ hc˜ hc ˜ H (b) Posi ion o he maximum ee su ace eloci y. Figu e 4.8: In luence o he ilm hickness on he eloci y o e shoo . Rep in ed wi h pe mission om [66]. ©2011, Ame ican Ins i u e o Physics. No only he magni ude o he eloci y o e shoo bu also i s loca ion depends on he ilm heigh ˜ has well as on he con ac angle θ. Figu e 4.8(b) shows ilm heigh dependence o he ˜y-posi ion ˜ymax whe e he la ges ee su ace eloci y umax is loca ed. An inc ease o he ilm heigh leads o a shi o he maximum ee su ace eloci y owa ds he middle o he channel un il he eloci y o e shoo disappea s and he maximal eloci y is loca ed in he middle o he channel a ˜y= 0. The c i ical ilm hickness a which he eloci y o e shoo disappea s, labeled in Figu e 4.8(b) by ˜ hc, shi s wi h inc easing con ac angle θ o smalle alues. Bo h he posi ion and he magni ude o he eloci y o e shoo depend on he con ac angle θand he ilm heigh H. The e o e, we in oduce a combined dimensionless pa am- e e which can be a ibu ed o a ce ain shape o he ee su ace eloci y p o ile and depends on he dimensionless capilla y ele a ion heigh ∆˜ h(θ) and he dimensionless ilm heigh ˜ Hin he o m o [66] =∆˜ h(θ)c1 ˜ Hc2.(4.41) The ee pa ame e s c1and c2we e ob ained by i ing he esul s o addi ional simula ions: c1= 0.0435 ±0.002 and c2= 0.9814 ±0.0176.[66] Wi h hese pa ame e s inse ed in equa ion (4.41) all expe imen al se ups wi h he same pa ame e show he same beha io o he eloci y o e shoo . Thus, we can now ind a c i ical a io c o he onse o a eloci y o e shoo o c= 0.733 ±0.002. Inse ing cin o he equa ion (4.27) o he capilla y ele a ion heigh leads o an empi ical h eshold o he onse o a eloci y o e shoo in e ms o a c i ical ilm hickness ˜ hc= −1/c2 c(1 −sin θ)c1/(2c2)(4.42) which is illus a ed in Figu e 4.9. The c i ical ilm hickness ˜ hcis la ge han ansi ion ilm hickness ˜ h (compa e Figu es 4.6 and 4.9), because ˜ hcdesc ibes he ilm hickness whe e a eloci y o e shoo jus eme ges. The magni ude o he eloci y will no become su icien ly s ong o balance he no–slip condi ion a he un il he ilm heigh is dec eased u he o he ansi ion ilm heigh ˜ h which, he e o e, has always o be smalle han ˜ hc. 56 Chap e 4. Th ee–dimensional ilm low s udied sys em o be mo e s able in he middle o he channel han he wo–dimensional case, especially a high exci a ion equencies. The neu al cu e in he middle o he channel shows no con ac angle dependence. Vlachogiannis e al. [68] and Geo gan aki e al. [69] s udied he in luence o a i- ni e channel wid h on he s abili y o he low a di e en inclina ion angles and luids p ope ies a e y low exci a ion equency ( e= 0.167 Hz). They also ound he low o be mo e s able unde he p esence o side walls, bu only i he Kapi za numbe Ka is su icien ly high (see Figu e 4.15). The Kapi za numbe is de ined as Ka = σ/(ρg1/3ν4/3) and ep esen s he a io o capilla y s esses o iscous s esses. I is a dimensionless ma e ial p ope y only and does no depend on low p ope ies. They ound he a io R∗= Rec/Rec,2D o depend only on he channel wid h and on he Kapi za numbe , bu no on he channel inclina ion o ma e ial p ope ies like o example he su ace ension σ. When he Kapi za numbe is o he o de o one o smalle hey ound no in luence o he channel wid h on he s abili y o he low when he channel was a leas 100 mm b oad. Al hough all ou measu emen s we e done a a Kapi za numbe o app oxima ely i e we ound a s ong s abilizing in luence o he side walls on he low which is no in line wi h he da a shown in Figu e 4.15. Ye , we suppose ha hese indings a e no in con lic wi h each o he , because an ex apola ion o ou da a shown in Figu e 4.14 yields ha i migh coincide wi h he wo–dimensional case a he limi o e y low equency e→0 as p oposed by he wo k o Geo gan aki e al. [69]. (a) Ra io R∗as a unc ion o Ka o se e al incli- na ion angles, luids and channel wid hs. The dimension o he gi en su ace ension is mN/m. The uppe cu e co esponds o a channel wid h o B= 100 mm. The lowe cu e co esponds o a channel wid h o B= 250 mm. (b) Ra io R∗as a unc ion o Ka o se e al channel wid hs B,α= 3◦. Figu e 4.15: Ra io R∗as a unc ion o Ka. Rep in ed wi h pe mission om [69]. ©2011, Ame ican Physical Socie y. URL: h p://p e.aps.o g/abs ac /PRE/ 84/i2/e026325 In he icini y o he side walls all ou measu emen s showed a u he s abiliza ion o he low compa ed o he da a om he cen e –line measu emen s. Fu he mo e, he con ac angle θ, which did no play a ole in he middle o he channel, gains in impo ance when he side wall dis ance dsis educed. Measu emen s done in he channel wi h he coa ed glass side walls (θ= 52 ◦) show neu al cu es which a e signi ican ly shi ed 63 4.2. S abili y nea he side walls o lowe Reynolds numbe s han he measu emen s done in he channel wi h un ea ed Plexiglas®side walls (θ= 8 ◦). We accoun wo e ec s o his phenomenon. One, he smalle con ac angle causes a la ge capilla y ele a ion heigh ∆h(see e.g. Figu e 4.10 o equa ion (4.27)) and hus o a la ge con ac a ea be ween he liquid and he e a ding side wall. Two, when a ini e su ace ension o he liquid is conside ed, he smalle con ac angle leads o a s onge cu a u e and he e o e o a s onge p e ensioning o he ee su ace in he icini y o he side wall which hinde s ee su ace wa es o de elop and hus ends o s abilize he low. We ind a ema kable ange o he con ac angle in luence on he s abili y o he low. A a side wall dis ance o 10 mm, which is abou ou imes he capilla y leng h Lo he ilm heigh H, he di e ence be ween he neu al cu es a θ= 8 ◦and θ= 52 ◦is up o 25%. E en up o a side wall dis ance o 40 mm, which is abou 17 imes he capilla y leng h Lo he ilm heigh H, he di e ence be ween he neu al cu es a θ= 8 ◦and θ= 52 ◦is s ill mo e han 7%. Addi ionally, he shape o he neu al cu es changes when he side wall dis ance is dec eased. In he middle o he channel we obse e he longes wa es o become ini ially uns able as p edic ed by Benjamin[46] and Yih[47] o he wo–dimensional case. Fo side wall dis ances o 10 mm in he case o uncoa ed side walls and 20 mm in he case o coa ed side walls he ype o he ins abili y changes om a long–wa e ype o a sho –wa e ype ins abili y in he in es iga ed equency ange. This ype o ins abili y is well known o bounda y laye lows as obse ed expe imen ally by Schubaue and Sk ams ad[86] o a pla e which is aligned pa allel o a plane low. This con igu a ion was la e desc ibed in de ail by Schlich ing and Ge sen[87]. The mechanisms o he ins abili y a e qui e di e en because in he p esen wo k an ins abili y o a ee su ace nea a side wall and no he ins abili y o a bulk is in es iga ed. Typical c i ical Reynolds numbe s ound o his bulk ins abili y in a bounda y laye a e abou wo o de s o magni ude la ge han o he ee su ace low in es iga ed he e[87]. Howe e he simila i y o he shape o he neu al cu es close o he side wall sugges s o ea he nea wall egion as a capilla y bounda y laye wi h a ange o ou o eigh imes he capilla y leng h L. O he g a i y–d i en ee su ace lows showing a sho –wa e ins abili y a e desc ibed in a wo–dimensional heo e ical amewo k by D’Alessio e al.[59] o New onian liquids a e y high in e se Bond numbe s, which means ha capilla y o ces domina e o e g a i y, o by Heining and Aksel[60] o powe –law liquids lowing down a sinusoidally undula ed incline. A in e media e side wall dis ances he neu al cu es nei he show he cha ac e o a ypical long–wa e ins abili y no he ypical sho –wa e ins abili y. In his ansi ion egion we obse e, ha he neu al cu es ha e an in lec ion poin in he in es iga ed equency ange. The size o he ansi ion egion seems o be la ge o smalle con ac angles θ. Figu e 4.16 shows he side wall dis ance dependence o he neu al poin s o bo h in es iga ed con ac angles and wo di e en exci a ion equencies. Because he mea- su emen s we e done a sligh ly di e en exci a ion equencies he da a shown in Figu e 4.14 ha e been in e pola ed linea ly o p o ide compa abili y. In he middle o he chan- nel we do no obse e a con ac angle dependence as a compa ison o he Figu es 4.14(a) and 4.14(b) al eady e ealed. Reducing dsleads a i s o a mono onous inc ease o he Reynolds numbe a which ee su ace wa es a e nei he damped, no ampli ied while 64 Chap e 4. Th ee–dimensional ilm low a elling downs eam, which means, ha he low is ge ing mo e and mo e s able due o he e a ding in luence o he side wall and he p e ensioning o he ee su ace coming in o play. Especially a low exci a ion equency ewe obse e he la ge ampli ude and ange o he con ac angle in luence. 0 10 20 30 40channel cen e ds/L 1 1.5 2 2.5 3 Re e = 2Hz; θ = 8° e = 2Hz; θ = 52° e = 6Hz; θ = 8° e = 6Hz; θ = 52° Figu e 4.16: Side wall dis ance dependence o he neu al poin s a wo di e en con ac angles θ and exci a ion equencies e. Rep in ed wi h pe mission om [70]. ©2011, Ame ican Ins i u e o Physics. Howe e , his mono onous beha io is b oken e y close o he side wall, al hough he s abilizing e ec s o he side wall should be s onges he e. This can also be seen in Figu e 4.14: Especially in he case o uncoa ed side walls he neu al cu e o ds= 5 mm is le o he neu al cu e o ds= 10 mm o e he whole in es iga ed exci a ion equency ange. Ob iously ano he (compe ing) e ec , namely he p esence o a eloci y o e shoo , gains in impo ance he e. When he ilm hickness is smalle han he c i ical ilm hickness hc, which is abou ∼1.3L o small con ac angles θ, no only he ilm hickness in he icini y o he side wall is la ge han he ilm hickness in he middle o he channel H, bu also a eloci y o e shoo is obse ed due o he capilla y ele a ion (see sec ion 4.1). Bo h, he highe ilm hickness as well as he highe eloci y a he ee su ace cause he local Reynolds numbe Reloc(ds) = h(ds)us(ds)/ν o exceed he (global) Reynolds numbe Re = 3 ˙ V /(2νB) a some ds(See Figu e 4.17). In hose egions he onse o wa es a he ee su ace is p omo ed and he low ends o be mo e uns able. Due o he mo e p onounced eloci y o e shoo and capilla y ele a ion o smalle con ac angles he magni ude o he local Reynolds numbe o e shoo is la ge o θ= 8 ◦. Tha explains why he peaks in he side wall dis ance dependence o he neu al poin s in Figu e 4.16 a e mo e p onounced o θ= 8 ◦ han o θ= 52 ◦. Compa ed o he s abilizing e ec s, he des abilizing in luence o he local Reynolds numbe o e shoo seems o be o a much sho e ange. 4.2.2 Conclusions We ha e shown ha he neu al cu e o he onse on a p ima y ins abili y in g a i y– d i en ee su ace lows depends s ongly on he dis ance o he side wall o he channel. In he s udied sys em he low in he icini y o he side wall was always mo e s able 65 4.2. S abili y nea he side walls Reloc ∼us·h us h ds Figu e 4.17: Ske ch o he side wall dis ance dependence o he local Reynolds numbe Reloc when a eloci y o e shoo is p esen . han he low in he middle o he channel. A di ec consequence is, ha a low may be uns able a some egions, bu shows o he egions whe e he ee su ace wa es coming om he uns able egions a e damped a he same ime. O cou se a low has o be ea ed as uns able as soon as i s su ace wa es appea somewhe e o mos applica ions. Ne e heless, we ound ha a coexis ence o s able and uns able egions is possible, which is impo an o unde s and he in luence o he side walls on he ins abili y o he low and he unde lying mechanisms esponsible.[70] In he middle o he channel we obse e a long–wa e ype ins abili y as ound by Benjamin[46] and Yih[47] o a pu ely wo–dimensional low. When he dis ance o he side wall is educed he ins abili y unde goes a ansi ion om he long–wa e ype o a sho –wa e ype as i is ypical o bounda y laye lows[86, 87]. Geo gan aki e al.[69] es ic ed hei s udies on he in luence o he channel wid h on he ins abili y o ilm low o he limi o e y long wa es. They ind he a io R∗, which is he c i ical Reynolds numbe no malized wi h he c i ical Reynolds numbe o a ilm lowing down a plane o in ini e ex en , o be a unc ion o he channel wid h and he Kapi za numbe . Since we ha e shown, ha he mos uns able wa e may also ha e a ini e wa eleng h in nea wall egions, we p opose ha a exci a ion equency a ia ion has o be ca ied ou o de e mine he a io R∗p ope ly o all channel wid hs and Kapi za numbe s. One has o conside di e en compe ing e ec s o he side walls on he ins abili y o he ee su ace, some end o s abilize and some end o des abilize he low. The addi ional no slip condi ion a he wall and he p e ensioning o he ee su ace due o capilla y ele a ion end o s abilize he low. These e ec s a e mo e p onounced when he con ac angle be ween he liquid and he side wall θis small. Fo hin ilms he capilla y ele a ion leads o a eloci y o e shoo a he ee su ace and hus o an o e shoo o he local Reynolds numbe which p omo es he onse o ee su ace wa es. Compa ed o he e ec s which end o s abilize he low, which a e s ill signi ican up o side wall dis ances o 17L, he in luence o he local Reynolds numbe o e shoo on he s abili y o he ee su ace seems o be o a he sho ange. Also he magni ude o he impac o he Reynolds numbe o e shoo in he icini y 66 Chap e 4. Th ee–dimensional ilm low o he side wall is clea ly o mino impo ance compa ed o he s abilizing e ec s o he in es iga ed low con igu a ion. Bu we ema k ha a u he educ ion o he ilm heigh will ampli y he magni ude o he Reynolds numbe o e shoo un il i s in luence migh o e come he s abilizing e ec s. The e o e, we specula e ha he e should exis a c i ical ilm heigh h0 ca which a low ini ially becomes uns able close o he side walls be o e he Reynolds numbe in he middle o he channel eaches he classical esul o Rec= (5/4) co α. This would imply ha a ilm o hickness h<h0 cwhich is con ined by side walls ini ially becomes uns able a a smalle Reynolds numbe han a channel low o in ini e ex en (B→ ∞) wi h he same ilm hickness h. To p o e his assump ion ha he p esence o side walls ha e an o e all des abilizing in luence on e y hin g a i y–d i en ilm lows u he expe imen s ha e o be ca ied ou . 67 Chap e 5 Conclusions and ou look In he p esen wo k we s udy iscous g a i y–d i en ilm low down an inclined channel nume ically, analy ically and expe imen ally. Pa icula ocus lies on he in luence o a pe iodic wo–dimensional sinusoidally undula ed opog aphy on he low and he appea - ance and disappea ance o eddies in he alleys o he undula ion. The co esponding esul s a e p esen ed in chap e 3. Chap e 4 deals wi h he ques ion how he p esence o side walls and di e en con ac angles be ween he liquid and he side wall in luences he low s uc u e, he o e all olume lux and he physical s abili y o he low. The main indings a e summa ized in he ollowing. Fi s , we conside nume ically and expe imen ally a s eady low down an inclined pe- iodic wo–dimensional sinusoidally undula ed opog aphy. We ind, ha he ee su ace unde goes a sha p ansi ion om a s ongly anha monic shape wi h a s ong inden a ion jus abo e he alleys o he undula ion o a smoo h ha monic shape as he Reynolds num- be is inc eased abo e a ansi ion Reynolds numbe . A his ansi ion Reynolds numbe a hyd aulic jump which has been es ablished con inuously wi h inc easing Reynolds num- be disappea s ab up ly because he low changes i s ype om a sub- o a supe c i ical low in e ms o he F oude numbe . Fu he mo e, we ind ha eddies, which a e o med wi h inc easing Reynolds numbe in he alleys o he unde lying undula ion, disappea again jus in he icini y o he ansi ion Reynolds numbe . Howe e , he eddies a e supp essed when he hyd aulic jump is p esen below he ansi ion Reynolds numbe as well as abo e i . Hence, we conclude, ha no he ansi ion i sel is esponsible o he supp ession o he eddy s uc u es, bu an ampli ica ion o he ee su ace ampli ude, which comes along wi h he ansi ion and is well known as a esonance phenomenon in li e a u e. Such sys ema ic supp ession o eddies is o pa icula ele ance o open up new op i- mized p ocess windows o many indus ial applica ions whe e hei o ma ion is desi ed o undesi ed, depending on he pu pose, since hey ha e signi ican in luence on mac o- scopic sys em p ope ies. In hea exchange applica ions o example he o ma ion o such eddy s uc u es has a majo impac on he con ec i e hea anspo in he liquid while hey lead o d ag educ ion, which migh be use ul o bea ings o any kind o ma e ial anspo . In en i onmen al sys ems pa icles cap u ed in he eci cula ing low a e cu o om subsequen deli e y o po en ially necessa y nu ien subs ances and so is he wall which is in con ac wi h he eddy. The possibili y o gene a e o des oy eddy o ma ions in he alleys o an undula ion jus by damping o exci ing esonance seems 68 Chap e 5. Conclusions and ou look o be a good amewo k in gene al, di e en om applying ex e nal o ces o he sys em. In he second pa o he p esen wo k he in luence o side walls and he con ac angle be ween he liquid he e on he eloci y p o ile and he low a es has been s udied expe imen ally and heo e ically. In he case o a we ing liquid a capilla y ele a ion is o med a he side walls, which may lead o a eloci y o e shoo in he icini y o he wall. While his eloci y o e shoo leads o an inc ease o he anspo ed liquid in he channel an addi ional no–slip condi ion a he side walls has a con a y in luence on he low a e. He e, a c i e ion o he i s onse o a eloci y o e shoo is p esen ed, as well as a c i e ion whe e he in luences o he eloci y o e shoo and o he addi ional no–slip condi ion on he o e all olume lux jus cancel each o he . As long as hey do no cancel each o he , neglec ing he h ee–dimensionali y o he low would lead o an o e – o unde es ima ion o he low a e, wha becomes especially impo an in he case o a e y hin ilm, whe e mos liquid is anspo ed in he capilla y co ne o he channel. In addi ion o he s eady low case indings o he case o a slowly d aining low a e p esen ed. Expe imen s show, ha he ee su ace shape o he liquid can no be modeled by a se ies o quasi–s eady s a es, e en i a dynamic con ac angle is aken in o conside a ion, al hough he se ling speed o he liquid is o de s o magni ude smalle han he mean low eloci y. Time dependen nume ical simula ions o he slowly d aining low case e ealed an inden a ion o he ee su ace nea he side walls, which migh p omo e ilm up u e in indus ial hin ilm applica ions. Finally, we in es iga e expe imen ally he in luence o he p esence o side walls on he p ima y ins abili y o he ee su ace o he low and ind, ha he neu al s abili y cu e shows a s ong dependence on he dis ance o he side wall and on he we ing p ope ies o he liquid. In he middle o he channel, a away om he side walls, we obse e a long–wa e ype ins abili y o he ee su ace, which is independen o he con ac angle be ween he liquid and he side wall. Howe e , when he side wall dis ance is educed he ype o he ins abili y changes o a sho –wa e ype ins abili y, as i is well known om bounda y laye lows a much highe Reynolds numbe s, and shows a s ong con ac angle dependence. We ind, ha one has o conside di e en compe ing e ec s o he side walls on he s abili y o he low. The p e ensioning o he ee su ace due o he capilla y ele a ion and he addi ional no–slip bounda y condi ion a he side wall end o s abilize he low and a e sensible up o la ge side wall dis ances o abou 17 imes he gene alized capilla y leng h. The o ma ion o a eloci y o e shoo in he capilla y ele a ion, on he con a y, ends o des abilize he low, because he local Reynolds numbe in he nea wall egion can exceed he (global) Reynolds numbe o he channel low. Howe e , compa ed o he s abilizing e ec s coming om he side wall he des abilizing e ec o he side walls seems o be o a he sho ange and plays a mino pa only o he in es iga ed low con igu a ion. Thus, he p esence o side walls leads o an o e all s abiliza ion o he sys em s udied he e. Ne e heless, we specula e ha o e en hinne ilms he inc easing magni ude o he eloci y o e shoo migh cause he des abilizing e ec o win o e he s abilizing e ec s, which would implica e ha e y hin ilms bounded by side walls would be mo e uns able han he co esponding ilms o in ini e ex en . In indus ial applica ions which deal wi h e y hin ilms like cu ain coa ing p ocesses o example, one would like o op imize he p ocess in e ms o a as p ocessing and an e ec i e exploi a ion o he coa ing ma e ial. Since side walls lead o a nonuni o m coa ing a he edges one usually cu s he edges o he coa ed ma e ial away o ge a 69 uni o mly coa ed esul . He e, we show ha especially in hin ilms mos o he ma e ial is anspo ed jus whe e edges a e loca ed when he con ac angle o he liquid a he side wall is below 90◦. The speed o p ocessing in indus ial applica ions is o en limi ed by he s abili y o he low, which is also s ongly a ec ed by he side walls and he we ing p ope ies o he liquid. The e o e, we sugges ha he con ac angle should be conside ed o he design o such de ices since i holds some ele an po en ial o op imiza ion in coa ing and o he hin ilm applica ions. To summa ize, he p esen wo k deals wi h he in luence o a co uga ed opog aphy and o side walls on ilm lows. Ne e heless, se e al aspec s conce ning he s abili y o g a i y–d i en ilm lows emain open ques ions. Since ou expe imen s on he in luence o he side wall on he s abili y o he low we e es ic ed o side wall dis ances o 5 mm, we sugges a ull nume ical s udy o he Na ie –S okes equa ions in ol ing ex ensi e pa- ame e s udies especially o he case o e y hin ilms o ge a close look in o he liquid nea he side wall and hus a be e unde s anding o he unde lying physics. Howe e , we ema k, ha he compu a ional e o o such simula ions will quickly become a key issue, since o he in es iga ion o con ec i e ins abili y i is necessa y o simula e a he la ge domains which a e ully h ee–dimensional and ime dependen on a a he p ecise mesh o esol e capilla y e ec s p ope ly. In addi ion o he ques ion how side walls in lu- ence he lows’ s abili y, he low o e subs a es wi h ini e co uga ions has been s udied by se e al au ho s o e he las yea s heo e ically as well as expe imen ally. Howe e , he ques ion how a o ma ion o eddies in s eep subs a e undula ions migh in luence he s abili y o he low is e y in e es ing bu s ill open and is wo h o be add essed in u u e wo ks. 70 Chap e 5. Conclusions and ou look 71 Lis o symbols (x, y, z) spa ial coo dina es (˜x, ˜y, ˜z) dimensionless spa ial coo dina es (ˆex,ˆey,ˆez) uni ec o s in x-, y- and z-di ec ion (X, Y, Z) spa ial coo dina es in he e e ence ame o he a e se (x0, y0, z0) spa ial coo dina es in he e e ence ame o he sc een (u, , w) eloci y componen s in x-, y- and z-di ec ion (˜u, ˜ , ˜w) dimensionless eloci y componen s in x-, y- and z-di ec ion u eloci y ec o ime ρ, ν, η, σ luid p ope ies: densi y, kinema ic iscosi y, dynamic iscosi y and su ace ension nou e no mal uni ec o Ts ess enso Iiden i y ma ix g, ˜gdimensional and dimensionless accele a ion o g a i y g,˜ gdimensional and dimensionless g a i y ec o p, ˜pdimensional and dimensionless p essu e p0,˜p0dimensional and dimensionless ambien p essu e a, λ ampli ude and wa eleng h o he channel undula ion αchannel inclina ion angle Bchannel wid h b,˜ bdimensional and dimensionless con ou o he channel opog aphy l1, l2lase beam a el dis ances h, ˜ hdimensional and dimensionless posi ion o he ee su ace d, ˜ ddimensional and dimensionless ilm hickness hn, dnNussel ilm hickness H, ˜ Hdimensional and dimensionless ilm heigh in he middle o he channel Hmmeasu emen heigh us ee su ace eloci y us,m ee su ace eloci y in he middle o he channel umax maximal ee su ace eloci y ¯umean low eloci y ¯unmean eloci y o a Nussel ilm low uloc local ee su ace eloci y ymax y-loca ion o he maximal ee su ace eloci y 72 Bibliog aphy [29] Heining, C. 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Dezembe 1982 in Gumme sbach (NRW) Familiens and: Ledig S aa sb¨ u ge scha : Deu schland Sp achen: Deu sch, Englisch Pa si als asse 27a 95445 Bay eu h Tel.: 0172 9569022 E-mail: hilo.pollak@uni-bay eu h.de Schulbildung 08/89 - 07/93 G undschule S einenb ¨ uck bei Gumme sbach 08/93 - 06/99 S ¨ ad . Gymnasium Mol kes asse in Gumme sbach 07/99 Umzug nach Hi schaid (Baye n) 09/99 - 06/02 Dien zenho e –Gymnasium Bambe g (Baye n) Abschluss: Abi u (Leis ungsku se: Ma hema ik und Physik) S udium 10/03 - 12/08 Physiks udium an de Uni e si ¨ a Bay eu h (Baye n) Abschluss Diplom Physik (Thema de Abschlussa bei : ”L¨ angenskalenabh¨ angigkei de Di usion im bin¨ a en Glasbildne Polys y ol/Toluol“) 05/09 - heu e P omo ions udium am Leh s uhl ¨ u Technische Mechanik und S ¨ omungsmechanik an de Uni e si ¨ a Bay eu h Sons ige T¨ a igkei en 07/99 - 08/99 Fe iena bei bei INA Mo o enwe ke in Hi schaid 08/01 - 09/01 Fe iena bei bei BOSCH in Bambe g 07/02 - 03/03 G undweh diens in Ro h (Baye n), Lengg ies (Baye n) und Leipheim (Baye n) 10/05 - 09/07 Gew¨ ahl e S uden en e e ung Fachbe eich Ma he/Physik/In o ma ik de Uni e si ¨ a Bay eu h 03/09 - 04/11 G ¨ undungsmi glied und 2. Vo si zende des Absol en en- und F¨ o de e eins Ma he/Physik/In o ma ik Uni Bay eu h e.V. 19. Ap il 2012, Thilo Pollak 83 Danksagung Mein Dank gil allen Mi a bei e n des Leh s uhls ¨ u Technische Mechanik und S ¨ omungs- mechanik de Uni e si ¨ a Bay eu h die mich bei de E s ellung diese A bei a k ¨ a ig un e s ¨ uz haben. Hie m¨ och e ich mich insbesonde e beim Leh s uhlinhabe He n P o . D . Nu i Aksel bedanken, de meine A bei ¨ ube den gesam en Zei aum seh ge adlinig be eu ha , mi s e s hil sbe ei zu Sei e s and und sich imme Zei ¨ u Diskussionen ¨ ube o ene F ages ellungen nahm. Regelm¨ assig ¨ aum e e mi auch die M¨ oglichkei ein, an di e sen in e na ionalen Fach agungen eilzunehmen bei denen ich seh in e essan e Kon ak e zu Fo schungskollegen aus de ganzen Wel au bauen und e ie en konn e. Auße dem he o zuheben sind meine Kollegen D . And ´e Haas, D . Ch is ian Heining und D . Ricca do Pu aglesi die mi du ch zahl eiche Fachdiskussionen h¨ au ig hel en konn- en neue Ideen und L¨ osungsans¨ a ze zu e a bei en. G oße Dank gil insbesonde e auch Ca ola Lepski und Ma ion M¨ a kl, da du ch ih e un e s ¨ u zende Labo a bei ein z¨ ugiges und e ek i es wissenscha liches A bei en ¨ ube haup e s e m¨ oglich wu de. Wei e hin m¨ och e ich D . Lu z Heymann und Ka ja Helm ich ¨ u ih e be ei willige Hil e nich nu bei o ganisa o ischen F ages ellungen einen g oßen Dank aussp echen. Selbs e s ¨ andlich haben da ¨ ube hinaus auch alle ande en Mi a bei e des Leh s uhls zu eine A bei sa - mosph¨ a e beige agen, die ich imme als seh angenehm und kollegial emp and. Nich zule z m¨ och e ich mich bei meinen El e n bedanken, die mich w¨ ah end meine gesam en Ausbildung uneingesch ¨ ank un e s ¨ u z haben und mi so jeden nu e denk- lichen F ei aum gescha en haben um mein S udium und das Ve assen diese A bei e olg eich abzuschließen. 84 Selbs s ¨ andigkei se kl¨ a ung Hie mi e kl¨ a e ich, dass die o liegende Disse a ion ohne emde Hil e eigens ¨ andig und ausschließlich un e Ve wendung de angegebenen Hil smi el und Quellen e ass wu de. Ich e siche e, dass ich die o liegenden A bei e s malig ein eiche und keine ¨ uhe en Ve suche eine P omo ion un e nommen habe. (Dipl.-Phys. Thilo Pollak) Bay eu h, 19. Ap il 2012 85