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磁性流體在有限厚度流體層之界面不穩定現象研究

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Instabilities of Ferrofluids Immersed in Fluid Layer with a

Finite Thickness



   























 





























































































































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Instabilities of Ferrofluids Immersed in Fluid Layer with a Finite Thickness      StudentJia-Fen Liu    AdvisorChing-Yao Chen    A Thesis

Submitted to Department of Mechanical Engineering College of Engineering

National Chiao Tung University in Partial Fulfillment of The Requirements

for the Degree of Master of Science

in

Mechanical Engineering July 2010

Hsinchu, Taiwan, Republic of China

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!1-1 67 8 9 ? @ A B C D 67 E F G pH w D F G xᑒ Р7 I

psurfactantx´J K 8 9 pbase fluidx ··· 10

!1-2I J K L M  L M ···11 !1-367 c d •D E 6F N H  Ò r t O  L M ÆÉ P Q o R c d [2] ···11 !1-4à S X T U V W › X 67 8 9 ‘’ H  I J K L M N O P Q R [3] ··· 12 !1-5o Y ë … ` . ž 5 S Q ' Z N O P ? [ \ ] ^ •[4]··· 12 !1-667 8 9 ´N ` a b c 9 •D E 6F N H ? _ ` u v w Q R [5] 13 !1-7 Hele-Shaw a ½ ![11]··· 13 !1-867 c 9 ´b c d _ e f G H  u v o N O P Q R [6]··· 14 !1-9r s t pÑ g x´u v o N O P Q R pN g xY q^ •? Q R [8]··· 14 !1-10 67 8 9 ´` a b c 9 •\ h 6F N H ? O w Q R [9]··· 15 !1-11 i j N ` a b k ’ N gc 9 ? c _ l m n O w Q R ? V O .Æl ä _ î o [10]··· 16 !1-12N Y 6w p  67 c d A q r 6F G H ? O w Q R [11]··· 16 !1-13 .s t i j •q r ? ™ N g` a b c 9  u Ò ~ ´v w ~ X o x ´y w H z m n[12]··· 17 !1-1467 c d d {  Y | _  @ 67 c 9  g} .N Y ‚  q r 6F

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qgX < q r ~ -   c d l 3 z ? m n[13] ··· 17 !1-15 € q r ` a b c d • Hele-ShawCell  A u Ò ~ ´v w ~  m nge È Q s  ‚ ƒ „  O w N O P Q R [14] ··· 18 !1-16q r 6F ÆD E 6F N G H  k ’ … y [15]··· 18 !1-17 A D E 6F Ɔ U 1 h 6F m nG H  _ ` 7 O w Q R [16] 19 !1-1867 c d = > •‡ ˆ U 5 N A D E 6F G H  î ‰ U V [17] ··· 19 !2-1 /0 å L !··· 26 !2-2 /0 a ½ !··· 26 !2-3 67 8 9 pEMG 901x ··· 27 !2-4 67 8 9 pEMG 901x 6w p Š U ! ··· 27 !2-5 ` ‹ w Œ  Ž  ë ··· 28 !2-6 U  ± Œ 8 H h Ñ ? 6F ··· 28 !2-7  ‘ ’ Ɠ ” ··· 29 !2-8 X•Y•Z – ’ 6F î o ··· 29 !2-9 — ž ˜ ™ – š Æ CCD › m  ··· 30 !3-1 67 c d (Dœ750 µm)A ‡ D E 6F (Hœ350 Oe)G H  I J K L M N O P Q R W··· 45 !3-2 67 c d (Dœ844 µm)A ‡ D E 6F (Hœ350 Oe)G H  I J K L M N O P Q R W··· 46

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!3-3 67 c d (Dœ939 µm)A ‡ D E 6F (Hœ350 Oe)G H  I J K L M N O P Q R W··· 47 !3-4 67 c d (Dœ1060 µm)A ‡ D E 6F (Hœ350 Oe)G H  I J K L M N O P Q R W··· 48 !3-5 67 c d (Dœ786 µm)A ‡ D E 6F (Hœ440 Oe)G H  I J K L M N O P Q R W··· 49 !3-6 67 c d (Dœ794 µm)A ‡ D E 6F (Hœ440 Oe)G H  I J K L M N O P Q R W··· 50 !3-7 67 c d (Dœ896 µm)A ‡ D E 6F (Hœ440 Oe)G H  I J K L M N O P Q R W··· 51 !3-8 67 c d (Dœ1051 µm)A ‡ D E 6F (Hœ440 Oe)G H  I J K L M N O P Q R W··· 52 !3-9 67 c d (Dœ930 µm) { N ` a b c 9 A ‡ D E 6F (Hœ510 Oe) G H  I J K L M N O P Q R W··· 53 !3-10 67 c d (Dœ1120 µm) { N ` a b c 9 A ‡ D E 6F (Hœ510 Oe) G H  I J K L M N O P Q R W··· 54 !3-11 67 c d (Dœ1140 µm) { N ` a b c 9 A ‡ D E 6F (Hœ510 Oe) G H  I J K L M N O P Q R W··· 55 !3-12 67 c d (Dœ1190 µm) { N ` a b c 9 A ‡ D E 6F (Hœ510 Oe)

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G H  I J K L M N O P Q R W··· 56 !3-13 67 c d (Dœ1080 µm) { N ` a b c 9 A ‡ D E 6F (Hœ350 Oe) G H  % ' ž I J K L M N O P Q R W··· 57 !3-14 67 c d (Dœ2560 µm) { N ` a b c 9 A ‡ D E 6F (Hœ350 Oe) G H  % . ž I J K L M N O P Q R W··· 58 !3-15 67 c d (Dœ3410 µm) { N ` a b c 9 A ‡ D E 6F (Hœ350 Oe) G H  % Ž ž I J K L M N O P Q R W··· 59 !3-16 67 c d (Dœ3980 µm) { N ` a b c 9 A ‡ D E 6F (Hœ350 Oe) G H  % < ž I J K L M N O P Q R W··· 60 !3-17 67 c d (Dœ1000 µm) { N ` a b c 9 A ‡ D E 6F (Hœ440 Oe) G H  % ' ž I J K L M N O P Q R W··· 61 !3-18 67 c d (Dœ2240 µm) { N ` a b c 9 A ‡ D E 6F (Hœ440 Oe) G H  % . ž I J K L M N O P Q R W··· 62 !3-19 67 c d (Dœ3460 µm) { N ` a b c 9 A ‡ D E 6F (Hœ440 Oe) G H  % Ž ž I J K L M N O P Q R W··· 63 !3-20 67 c d (Dœ4070 µm) { N ` a b c 9 A ‡ D E 6F (Hœ440 Oe) G H  % < ž I J K L M N O P Q R W··· 64 !3-21 67 c d (Dœ1100 µm) { N ` a b c 9 A ‡ D E 6F (Hœ510 Oe) G H  % ' ž I J K L M N O P Q R W··· 65

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!3-22 67 c d (Dœ2250 µm) { N ` a b c 9 A ‡ D E 6F (Hœ510 Oe) G H  % . ž I J K L M N O P Q R W··· 66 !3-23 67 c d (Dœ3610 µm) { N ` a b c 9 A ‡ D E 6F (Hœ510 Oe) G H  % . ž I J K L M N O P Q R W··· 67 !3-24 67 c d (Dœ3830 µm) { N ` a b c 9 A ‡ D E 6F (Hœ510 Oe) G H  % < ž I J K L M N O P Q R W··· 68







































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N Y g`67c d ? î ‰ i ‹ î Ì Ng <  ž o paxc d E \ ‚ < 400 µmg|o x c d Ò · 4 S T † O P r t O  L M gec d ¸ ¹ “ B CV † s  º » Ì Ä ¼ S T Q o R c d pbxc d E \ @ < 500´1000 µmg'= 3Ò r t O  L M ÆÉ P Q o R c d ? H  g•|ž o N3 à× R c d ? H  geu Ò > z ? R c d > N O P pcxc d E \ @ < 1100´1400µmg3à× ? R c d î ‰ H  g½ 1 R c d 9‹ ; à c d / [° ± ‚ qGH  ? R c d , Ïï ‚ geR c d ?  ‚ Ê­ Ò 9 ¸ ¹ · 4 y‚ pdxc d E \  < 1500µmg; à ° ± ‚ ? c d 3àN O P ? Q RgeR c d î ‰ ÊÌ ¾ ¿ áN © “ W •2007$gGollwitzer et al.[3]à SXT U V W › X 678 9‘’H   I J K L M N O P Q RgyÑ ± 3„ í Ž 2i j s678 9‘’À Á g• !1-4› œ ‡ •ý Â Ô  ? 6F gƒ Ì 6F ? N BCQ R ï I J K L M Q RÅY ë ¸ ¹ · 4 9 Ò S  W e” ' ! ‘7  Rither[4]  6 g• o ? J Ñ  ~  §? 6 78 9g 678 9à 3 ’‹ ÆÄ _ ? ¢ £ ÏÕ x / 0 g•}  D E 6F y ™ Ç  6F ^ _ g¿ 6F L ý • Y Å } 6F Ng• o 68 9 @ ’Npõ!1-5x`. ž 5 S Q 'Z N O P ? [ \ ] g  —  GÆ 3S Q ? W    67 8 9 GH  u v o N O P Q R•1975$¯ 3k¥ Ç g Romankiw et al.[5]i j 50´1000 µm ‚ ? 678 9´ N `a b c 9GV † c d •D E 6F ? é SNH  ? _ ` u v w Q Rpõ!1-6GaxW— • u v w Q RH   Å< 678 9~ < 3È É – Ê gà SÈ É Ñ 1 Ë '– Ê  ï 678 9A „ < ¨ – Ê } pÙHele-Shaw Cellgõ!1-7xg} ³ D E 6F £g/¬ 9 Ñ S T 678 9ƒ A ‡ È É Ñ 1– Ê F Ì š2Í

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7 1 h? ¡ ­ ‹ W   —   G• É g678 9.ƒ ˆ . L ?  L •6F NH  u v w Q R ? @ Ï ¨ 7? [ > h á6c d = > Ö ×  4 V † Ñ Ò _ ` ? i ‹ 'WDickstein et al. [7] su v w Q RX < Z [¢ £  ŠÓ ­ g ʞ a k© } 3Ô  D ß ? 5 1Á Ï ‚ g‚ D ß u v w Q R? Õ Ö ¡ ¢ ‘ Ö × Ï ‚ Á§Õ × Ã 3Y Á§W 

؏u v o N O P Q RGà SHele-Shaw CellÑ – Ê ¤ g 678

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(54)

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(59)

!3-2 67cd (Dœ840 µm){ N` ab c9A ‡ D E 6F(Hœ350 Oe)

GHI J K L M NO P QRW

(60)

!3-3 67cd (Dœ940 µm){ N` ab c9A ‡ D E 6F(Hœ350 Oe)

GHI J K L M NO P QRW

(61)

!3-4 67cd (Dœ1060 µm){ N` ab c9A ‡ D E 6F(Hœ350 Oe)

GHI J K L M NO P QRW

(62)

!3-5 67cd (Dœ780 µm){ N` ab c9A ‡ D E 6F(Hœ440 Oe)

GHI J K L M NO P QRW

(63)

!3-6 67cd (Dœ790 µm){ N` ab c9A ‡ D E 6F(Hœ440 Oe)

GHI J K L M NO P QRW

(64)

!3-7 67cd (Dœ900 µm){ N` ab c9A ‡ D E 6F(Hœ440 Oe)

GHI J K L M NO P QRW

(65)

!3-8 67cd (Dœ1050 µm){ N` ab c9A ‡ D E 6F(Hœ440 Oe)

GHI J K L M NO P QRW

(66)

!3-9 67cd (Dœ930 µm){ N` ab c9A ‡ D E 6F(Hœ510 Oe)

GHI J K L M NO P QRW

(67)

!3-10 67cd (Dœ1120 µm){ N` ab c9A ‡ D E 6F(Hœ510 Oe)

GHI J K L M NO P QRW

(68)

!3-11 67cd (Dœ1140 µm){ N` ab c9A ‡ D E 6F(Hœ510 Oe)

GHI J K L M NO P QRW

(69)

!3-12 67cd (Dœ1190 µm){ N` ab c9A ‡ D E 6F(Hœ510 Oe)

GHI J K L M NO P QRW

(70)

!3-13 67cd (Dœ1080 µm){ ` ab c9A ‡ D E 6F(Hœ350 Oe)

GH%'žI J K L M NO P QRW

(71)

!3-14 67cd (Dœ2560 µm){ ` ab c9A ‡ D E 6F(Hœ350 Oe)

GH%. žI J K L M NO P QRW

(72)

!3-15 67cd (Dœ3410 µm){ ` ab c9A ‡ D E 6F(Hœ350 Oe)

GH%Ž žI J K L M NO P QRW

(73)

!3-16 67cd (Dœ3980 µm){ ` ab c9A ‡ D E 6F(Hœ350 Oe)

GH%<žI J K L M NO P QRW

(74)

!3-17 67cd (Dœ1000 µm){ ` ab c9A ‡ D E 6F(Hœ440 Oe)

GH%'žI J K L M NO P QRW

(75)

!3-18 67cd (Dœ2240 µm){ ` ab c9A ‡ D E 6F(Hœ440 Oe)

GH%. žI J K L M NO P QRW

(76)

!3-19 67cd (Dœ3460 µm){ ` ab c9A ‡ D E 6F(Hœ440 Oe)

GH%Ž žI J K L M NO P QRW

(77)

!3-20 67cd (Dœ4070 µm){ ` ab c9A ‡ D E 6F(Hœ440 Oe)

GH%<žI J K L M NO P QRW

(78)

!3-21 67cd (Dœ1100 µm){ ` ab c9A ‡ D E 6F(Hœ510 Oe)

GH%'žI J K L M NO P QRW

(79)

!3-22 67cd (Dœ2250 µm){ ` ab c9A ‡ D E 6F(Hœ510 Oe)

GH%. žI J K L M NO P QRW

(80)

!3-23 67cd (Dœ3610 µm){ ` ab c9A ‡ D E 6F(Hœ510 Oe)

GH%. žI J K L M NO P QRW

(81)

!3-24 67cd (Dœ3830 µm){ ` ab c9A ‡ D E 6F(Hœ510 Oe)

GH%<žI J K L M NO P QRW

(82)

















 































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(84)













































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[2] C.-Y. Chen and L.-W. Lo, “Breakup of thin films of micro magnetic drops in perpendicular fields,” J. Magn. Magn. Mater. , 305, 2, 440(2006).

[3] C. Gollwitzer, G. Matthies, R. Rither, I. Rehberg and L. Tobiska, “Surface topography of a magnetic fluid-a quantitative comparison between experiment and numerical simulation”, J.Fluid Mech,571,455-474,(2007).

[4] R. Richter and I. V. Barashenkov‚“Two-dimensional solitons on the surface of magnetic fluids”‚Phys.Rev.Lett.‚94‚184503‚(2005).

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(2007).

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[8] C.-Y. Chen, W.-K. Tsai and Jose Miranda, “A hybrid ferrohydrodynamic instability: coexisting peak and labyrinthine patterns,” Phys. Review E, 77, 056306 (2008).

[9] C.-Y.Chen, Y. S. Yang and Jose Miranda, “Miscible Ferrofluid Patterns in a Radial Magnetic Field,” Submitted(2009).

[10] J. A. Miranda and E. Alvarez-Lacalle, “Viscosity contrast effects on fingering formation in rotating Hele-Shaw flows,Phys. Review E, 72, 026306 (2005).

[11] R. M. Oliveira and J. A. Miranda, “Ferrofluid patterns in a radial magnetic field: Linear stability, nonlinear dynamics, and exact solutions, Phys. Rev. E 77, 016304(2008).

[12] C-Y Chen, C.-W. Huang, H. Gadêlha, and J. A. Miranda,“Radial viscous fingering in miscible Hele-Shaw flows: A numerical study,Phys. Review E, 77,016306(2008).

[13] A. V. Lebedev, A. Engel, K. I. Morozov, and H. Bauke, “Ferrofluid drops in rotating magnetic fields, New Journal of Physics 5,57.1–57.20(2003).

[14] C.-H. Chen and C.-Y. Chen, “Numerical simulations of interfacial instabilities on a rotating miscible droplet in a time-dependent gap Hele–Shaw cell with significant Coriolis effects, Int. J. Numer. Meth. Fluids, 51:881–895(2006).

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continuous-to-discrete phase transitions under simultaneously applied DC axial and AC in-plane rotating magnetic fields ” J.Magn.Magn. Mater.289‚353‚(2005).

[16] C.-Y. Chen, S.-Y. Wu and J. Miranda, “Fingering Patterns in the Lifting Flow of a Confined Miscible Ferrofluid, Phys. Rev. E., 75, 036310 (2007).

[17]   , ”      ”,           , (2009)

[18]S.A.Langer‚R. E. Goldstein‚ and D.P. Jackson‚ “Dynamics of labyrinthine pattern formation in magneticfields ”‚Phys.Rev.A46‚4894‚ (1992).

[19] D. P. Jackson‚R. E. Goldstein‚and A. Cebers‚“Hydrodynamics of fingering instabilities in dipolar domains ”‚Phys.Rev.E50‚298‚(1994).

[20]G. Pacitto‚C. Flament‚J.-C. Bacir‚and M. Widom‚ “Rayleugh-Taylor Instability with magnetic fluids:Experiment and theory” Phys. Rev. E62‚7941(2000).

[21]Ching-Yao Chen‚ “Numerical simulations of fingering instabilities in miscible magnetic fluids in a Hele-Shaw cell and the effects of Korteweg stresses”, Phys. Fluids 15, 1086 (2003).

[22]R. Friedrichs and A. Engel‚ “Pattern and wave number selection in magnetic fluid s”‚Phys.Rev.E 64‚021406‚(2001).

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[25]S. Bohlius‚ H. Pleiner and H. Brand‚“Solution of the adjoint problem for instability with a deformable surface: Rosensweig and Marangoni instability”,Phys.Fluids,19,0941 03,(2007). [26] Ching-Yao Chen‚ Z.-Y. Cheng, “An experimental study on Rosensweig instability of a ferrofluid droplet”, PHYSICS OF FLUIDS 20, 054105 (2008).

[27]C.-Y. Chen and L.-W. Lo‚ “Influences of Field Conditions to the Rupturing Instability of a Circular Magnetic Thin Film‚ Magnetohydrodynamicd”‚42‚1‚31‚(2006).

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