Turbulent mixing and beyond ICTP, Trieste, GrigNANO Mare 2007

Rayleigh-Taylor Instability in Micro and Nano Hydrodynamics

Eduard Son

©Moscow Institute of Physics and Technology

Physical Mechanics

Department

Talk outline

• Dispersive relations for finite Kn numbers• Size effects in transport properties (viscosity, diffusion,

heat conductivity) • Creating shock waves in a gas-piston system in MNFM• Gas-Liquid interface and Surface tension in micro- and

nanofield modification of dispersive equations• Features of rarefied flows and connection to turbulent

flows• Boltzman Equation – NS equations• Vortexes dynamics in 2D and 3D• Thermal Induced vortexes (experiment and theory) • Air dynamic flow control by creating vortexes• Plasma (DBD) induced vortexes• Micro- and Nano Rheology properties of liquids and

Viscous-Plastic Fluid Turbulence

Micro- and Nano Hydrodynamics

• One phase flows– Internal Hydrodynamics – gas flows in micro- and

nanochannelsand pipes (including CNT – carbon nanotubes)

– External Hydrodynamics – flows around micro- and nanoparticles

• Two phase flows– Flows of solid micro- nano particles

• Gas flow around particles• Gas flow around particles

– Micro- and nano bubble flows• Multi Phase hydro- and electrodynamics

Applications of micro- and nano hydrodynamics

(NT boundary = 100 nm = 0.1 µm)– HDD (Hard Disc Drives d=50 nm)– Micro- and nano electronic devices (MEMS)– Micro- and nano separators– Microchannels, micropumps, microvalves– Micronozzles, microgiroscopes, micro- and nanosatellites– Medicine (blood flow in the human body, etc.)– Flow Sensors – Auto- car Industry (ABS, fuel, etc.)– Electro hydrodynamics super sensitive sensors for micro- and

nano fields (MIPT)– Micro- Nano hydrodynamics of porous media– Oil – hydraulic fracture– Rheology (micro- nano structure properties of liquid, viscous-

plastic and elastic non Newtonian media

Features micro- and nanoflows (mesodynamics)

• Channel size compared with mean free path, interparticle distance and even with particle size

• Density fluctuations are large• Transport properties (viscosity, diffusion, heat

conductivity) could have size effects (like in turbulence)• Nano flow - wall interaction may be main factor• There are no exact boundary conditions• Continuum flow approximation is not valid• Some phenomena does not exist in microdynamics: flow

memory after penetrating the holes• Some quantum effects are essential in nanoflows• Laplace law for surface tension pressure is modified

GAS MICROFLOWS: What happens at

microscale?

• Compressibility (density variations)

• Rarefaction (low pressure)

• Viscous heating (work done by viscous forces)

• Thermal creep (gas molecules go from cold to hot)

• Non-continuum effects (Kn>0.1-0.5)

LIQUID NANO FLOWS: What happens at micro/nanoscale?

• Hydrophobic vs. hydrophilic surface (wetting)

• Adsorption of species on wall-induced roughness

• Electrokinetic effects• Intrinsic surface

roughness• Non Neutonian media• Different rheology

Couette free molecular flow

yq yq

Particle distribution functions

Stress

Heat flux

Fourier Law (q ~ - grad T)Heat flux (q ~ ┴ grad T)Does not exist in hydrodynamics,But could exist in turbulent flow !!!

yq

xq

1 /S U h h

Boltzmann equation for instabilities

)1( 20 bKnaKnff

2D simulation, Infinite order schemeEnergy spectrum vs. time Re=infty, Re=40000

What are main features of kinetic approach to HD instabilities

Why tau-approximation reasonable for HD instabilities• Use opportunity compare with DNSC (SMILE)• Nonlinear Euler equations are taken into account• Exact viscosity is not essential• Maxwellization is underestimated

1. Nonlinear transport and finite correlation size are taken into account2. Boundaries in rarefied gas become smoothed for MFP length – 3. No singularities in vorticity and other parameters4. Effective viscosity appears 5. Analogy to turbulence (long correlations, size effects)

BE solution at arbitrary Kn number

Onuphriev A, Sapharov R., Son K. Son E.

Semiempirical models of turbulence,

Theory and Experiment, Russia 2003,

TBT: Begell House, 2007

Feature rarefied gas flows

• Stress and heat flux are not depend on the distance between plates and velocity and temperature profiles, but depend on differences of velocities and tyemperature

• Near walls creating Knudsen layers with jumps of temperature and velocities• Directions of heat fluxes and temperature gradients not coincide• Highly transport anysotropy: under action of transversal temperature difference

arise heat flux in the direction of homogeneous (along x) Possible arising «negative thermal conductivity» (heat flux directed along the temperature gradient), i.e. Fourie Law is invalid.

• At decreasing mean free path (in the limit of Kn<<1) Stokes law and Fourier Law follows, along x heat flux limits to zero, the nonzero x- q-flux appears in next order of expansion on fields gradients

Micro Gas Dynamics2

4

, ,

10 1 0

p

S

a

cUL pPe a

alPe dS

55

2

1 / *1010 1

1 /a w

Ul см с смPe T const T

см c

0

( )

: , w

t

pt

УРС p RT T const T

v

vv v

0 0

0

( )

: ,k

p

t

pt

cpАдиабата k

p c

v

v

vv v

Classical gas dynamics

Micro-nano gas dynamics

Compression shock waves – Follows fron 2nd Thermodynamics LawNo rarefying shock waves

Riemann problemIn Nano Gas Dynamics2nd Thermodynamics Law invalid

Moments equations in rarefied gas – expansion on Kn number

• Zero approximation in Kn – thermodynamic equilibrium• First approximation on Kn – dense gas – Navier –

Stokes equations, Fourie, etc – linear relations between thermodynamic forces and fluxes

• Second approximation on Kn – Barnett approximation, nonlinear relations between forces and fluxes

• Higher approximations on Kn – complex problem• Limit high Kn numbers – free molecular regime (FMR) • Numerical simulation Boltzmann Equation (SMILE),

there are some exact solutions• Lattice Boltzmann• Molecular Dynamics for nonequilibrium

nonhomogeneous system (Supercomputing)

Vortex creation in Nuclear explosion in atmosphere

p E

1( )

1( )

pt

pt

vv v g

Ωv Ω

Apply nabla to momentum eqn:pressure illuminatesgrad p acts as external forcelike suppose volume force independent on density

Fundamentals of Thermal Actuator

1 1 1 v 1

v

1( )

p p

pT T T

T pt T

Ω

v Ω

Plasma DBD Actuators Flow Control

2

0 0

( )

1

( )

1 1, ,

c

cc

cc

ext ind ind ind ext extc c

pt

D

Dt

t

vv v E

ΩE E

Ωv Ω E E

E E E E E

Important:Vorticity creates only in sheets of space charge gradient,interacts with external and inducedElectric fields

Vorticity transportby flow

Vorticity sourcefrom plasma

DBD Plasma actuator equations

* ( ) ( )*

( )

( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

c

p e i extp p p p e i e e i e e p r e e n r e

i enn n n n e a e p n r n p d

t

pt

nn n n n nk T n n k T n n k T n n k T S

tn

n n n n nk T n n k T n n k Tt

n

v v

vv v σ E

v v U

v v U

*** * * * *

0

( ) ( ) ( ) ( )

( )

exte e e i

c p n e

n n n nk T n n k T St

e n n n

v v

E

compressibility

Flow transport

* ( )*

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) (ln )

ee e e

e exte i e e i e n d e p r e e a e e

e e e e e e e e

nn n

t

n nk T n n k T n nk T n n k T n nk T S

n n D n D n n

v U

U E

Electrontransport

Internal, Vibrational and Electron Temperature equations

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

3 3( )

2 2

mTR TR TR V V V e TR V e

VT

mm V V m V V V V V V e V e e

VT

e e e e e e e e u e e

d nnU nU p U T U T n T kT

dt

d nn U T n U T U T U T n T kT

dt

dn kT n kT p n T kT

dt

v q

v q

v j E q

' ' ' '' '

0

( , ) *( ', ') 2arctan

( / ) 2 ' 'i v v st v v

ph ph e exieff A R

k x y h dx dyS x y n

k E N

External source Ionization could be photoionization

Problem: in strong E-fields equation for electron temperature not valid, It is extra equation. Electron energy is controlled by external or

selfsustained E-field

Plasma-wall vortex formationNumerical simulation (GDT – parallel code)

Plasma flow control streamer size less 1 micron

Multiscale problems

• Turbulence: L/l=Re^3/4, T/t=Re^1/2, for Re=10^4 L/l=1000, N=(L/l)^3*(T/t)=10^11

max Re=20000 for DNS (Livermore, Blue Jin)

predictions (Spalmaret, Boeing) 2040 – LES,

2080 - DNS• In DBD L/l=10^5, T/t=10^6, Multiscale is

nesessary

Numerical simulation Boltzmann EquationВ мире существует три основных кода ПСМ:

• DS2V (скалярный код G. Bird)• DAC (NASA, только для NASA)• SMILE (ИТПМ СО РАН).

Результаты для трех кодов:

• давление• напряжение трения • тепловой поток

Метод прямогостатстического моделирования

Surface tension depends on curvature

V.Byakov, S. Stepanov (ITEP)

Rayleigh-Taylor Instability in MNHD

RTI in a magnetic field

2T Tv v

3 (1 )3 2( )w Kn

Hydrodynamics MEMS Technology (MIPT)Molecular-electronic transducer

Instability in electronic~new device1. Ceramic or glass

channel;2. Clamping rings;3. Electrolyte;4. Ceramic spacers;5. Anodes;6. Cathodes7. As does a pentode,

an MET provides extremely high power gain (>108)

• The relationship between the pressure drop across the cell and external acceleration is the following:

where - is electrolyte density, l – is the length of the cell, - is the external acceleration. So the input signal for MET is acceleration.

extalp

Oil microhydrodynamics

• Oil moves in pores 1 µm size. For hydraulic fracturing nesessary to have media with oil density and viscosity – microbubbles media with 1 µm bubble diameter

• We developed experiments and theory for micro- nano bubble media, methods of generation and applications for hydraulic fracturing

2

2 2

v=0

Re 1: =0

, , 1 1

p

pd

Kp K d D m

v

v

Darci Law

Stability and mixing in bubble flows(AFM) Atomic – Force microscopy

Size spectra distribution

Bubble Diameter, µm

10 12 144 6 80

100

200

300

400

500

600

700

Число пузырьков в заданном интервале диаметров

Statistics 1560 bubbles.

1 2 34 5 67 8 9 1011 12 131415 16 17 18 19 20 2122 2425 2627 28 29 3031 32 33 34 353637 38 3940 41 4243 4445 46 4748 49 50 51 5253 5455 5657 5859 6061 62

63 6465 66 67 6869 70 71 7273 74 75 76777880 8183 84 8586 87 88 8990 919293 94 95 9697 98 99100 101 102 103104 105 106107 108 109110111 112 113 114115

116117118119 120 121 122123 124125 126 127 128129130 131 132133 134135 136 137

138 139 140 141142143 144145 146 147148 149150 151152153 154 155156 157 158

159160 161162 163164 166 167168 169 170171 172175 176177 178179180 181 182183 184 185 186187 188189 190191192193 194 195 196 197198 199200 201 202203 204 205 206207

208 209 210211212 213214215 216217218 219 220 221222 223 224225 226 227 228 229230 231232 233 234 235236 237238 239 240241 242243 244 245246 247 248 249250251 252 253 254255 256257 258 259 260 261262 263 264265

266267 268269 270 271 272 273274 275 276277 278 279 280281 282 283284285 286 287 288 289290 291 292 293294295 296297 298 299 300 301302 303304 305 306307 308 309310 311 312313 314 315 316317318 319320321322 323324 325 327328329 330 331332 333 334 335336 337 338 339340 341342 343344 345346 347348 349 350351352 353354 355 356

357 358359 360 361 362363 364 365 367368 369 370 371372

373 374 375376 377 378379 380381 382383

384385 386 387 388 389390391 392 393394 395 396 397 398399400401 402403 404 405 406 407408

409410411 412 413 414415 416 417418 419 420421 422423 424 425426427 428 429 430 431432 433434435436 437 438439440 441442443 444 445 446447 448 449 450451 452 453 454455 456 457458459 460 461 462 463 464466 467468 469470

471 472473 474475 476 477478 479 480 481 482483 484 485 486487 488 489490 491 492 493 494495 496497 498499 500 501 502503 504 505 506

507508

509 510511 513 514515 516 517518 519 520521

522 523524 525527 528529 530531 532533 534 535 536 538539 540541 542543 544 545546 547548 549 550 551552 553554 555 556557 558 559560 561 562 563 564 565566567 568569 570 571 572 573574 575 576 577

578 579 580581 582 583 584585 586587 588 589590 591592 593 594 595 596597 598 599600 601602 603 604 605606 607608 609 610 611613 614615 616617618 619620 621 622 623624 625 626 627 628629 630

631632 633634635 636 637638 639 640 641642

643644 646 647

648 649650 651 652 653 654655656 657 658 659660 661662 663 664 665 666667668 669 670 671672 673 674675 676677 678679 680681682 683 685 687 688689 690 691692 693 694 695 696697698 699 700 701702 703 704 705706 707 708 709710 711712713 714 715 716717 718 719 720 721 722723

724725726 727 728729 730 731 732 733734 735 736 737738739 740

741 742 743744 745 746747 748 749750 751 752 753754 755 756757758 760761 762 763 764 765766 767 768 770 771772 773774776 777 778 779780 781 782783 784 785 786787 788789 790 791792793 794795 796 797 798799 800801 802 803 804805806 807 808 809

810 811812 813 814 815 816817818 820 821822 823 824 825826 827 828829830 831833 834 835 836 837 838839 840841 842843 844845 846 847848 849850 851852 853 854855 856857 858859 860861862 863864 865 866 867868 869 870 871872 873 874 875876 877 878879 880 881 882883 884 885 886887888 889 890892 893 894895 896897 898 899900 901

902 903904 905 906 907 908 909910 911912 913914 915 916917 918919920 921922923 924

925926 927 928929 930 931 932933 934 935936 937 938 939940 941 942 943945 946947 948 949950 951 952953 954 955 956957 958 959960 961 962963 964

965 966967 968 969 970971 972 973974 975 976 977978 979 980981 982 983 984985 986 987988 989990 991 992 993994995 996 997 998999 1000 1001100210031004 1005 1006 10071008 100910101011 1012 1013

1014 1015 1016 10171018 10191020 10211022 1023 1024 1025 10261027 1028 1029 103010311032 103310341035 1036 1037 10381039 1040 10411042 1043 10441045 1046 1047 1048 1049 10501051 1052 1053 10541055 10561057 1058 1059 1060 10611062 1063 10641065 1066 106710681069 1070 10711072 1073 107410751076 1077 10781079 1080 10811082 10831084 1085 108610871088 1089 1090

1091 109210931094 1095 109610971098 1099 11001101 1102 1103 11041105 1106 1107 1108 11091110 1111 1112 11131114 1115 1116 1117 11181119 1120 1121 11221123 11241125 1126 1127 11281129 1130 1131 1132 11331134 1135 1136 1137 11381139 1140

11411142 1143 114411451146 11471148 1149 1150 11511152 1153 1154 115511561157 1158 1159 11601161 1162 1163 11641165 1166 11671168 11691170 1171 11721173 1174 117511761177 1178 11791180 1181 11821183 11841185 11861187 118811891190 1191 1192

1194 1195 11961197 1198 11991200 1201 12021203

1204 12051206 1207 12081209

12101211 12121213 12141216 1217 12181219 1220 1221

1222 12231224 1225 1226 1227 122812291230 1231 1232 1233 12341235 1236 12371238 1239 12401241 1242 1243 1244

12451246 1247 12491250 1251 12521253 125412551256 1257 1259 12601261 1262 12631264 126512661267126812691270

12711272 1273 1274 1275 1276 127712781279 1280128112821283 1284

12851286 1287 1288 12891290 129112921293 129412951296 12971299 13001301 1302 13031304 1305130613071308 1309131013111312 13131314 1315 1316 1317 13181319 1320

1321 1322 1323 1324 1325 132613271328 132913301331133213331334 1335 1336 13371338 1339 134013411342 134313441345 1346

13471348 134913501351 1352 1353 1354 13551356 13571358 136013611362 13631364 136513661367 13681369 1370

1371 1372 1373 1374 137513761377

13781379 13801381 1382 13831384 1385 1386 13871388 13891390 1391

1392 13931394 13961397 13981399 1400 1401 14021403

140414051406 14071408 1409 141014111412 14131414 1415 1416 141714181419

1420 14211422 1423 14241425 1426

14271428 14291430 1431 143214331434 1435 14361437 14381439 14401441 1442144314441445 14461447 1448 1449 14501451 14521453 1454 14551456 145714581459 1460 14611462 1463 146414651466 1467 146814691470 1471 147214731474 147514761477 1478 1479 14801481 14821483 14841485 1486 14871488 1489 14901491 149214931494 1495 1496 14971498 14991500 1501 1502

1503 1504 1505 15061507 150815091510 1511 151315141515 15161517 1518 1519 15201521 1522

15231524 1525 15261527 1528 15291530 1531 1532 15331534 1535 15361537 1538 1539 1540

154115421543 15441545 15461547 1548 1549 15501551

155215531554 155515561557 15581559 156050 мкм

Spectra and image microdisperse water gas mixture (MDWGM)

NANORHEOLOGYViscous – Plastic Media

0

vt

gτvvv

pt

)(

l

l

i

k

k

i

x

v

3

2

x

v

x

v

2

1

2

ikikS

Sτ

0

00

Epp

ikn

ik SSS 10

102 0

0ikS 0

SS :2

12 S

ττ :2

12

,0

pppLxx

0

SauBoundary conditions:

Bingham and Hershel-Bulkley flows (3D pipe)

0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

1.2E-01

1.4E-01

-6.0E-02 -4.0E-02 -2.0E-02 0.0E+00 2.0E-02 4.0E-02 6.0E-02

R (m)

V (m/sec)

Theory

11*11 cells

21*21 cells

41*41 cells

Micro- nano- structure VPF

Micro-Nano rheology analysis (Stress-Deformation rate)

2

Thermal gradients render the magnetic force nonuniform through the temperature-dependent magnetic susceptibility

Fluid control in microgravity

Edwards B. F. et al. Magnetothermal convection in nonconducting diamagnetic and paramagnetic fluids. Proceedings of the 3-d International Microgravity Fluid Physics

Conference, Cleveland, USA, 1996.

Odenbach S. Microgravity experiments on thermomagnetic convection in magnetic fluids. J. Magnetism and Magnetic Materials, 149 (1995)

Diamagnetic controlDeLucas L.J. et al. Diamagnetic control of convection during crystal growth. (Protein experiments on DC-9 Reduced Gravity Aircraft) Science ,1989, 246. Yin D. C. et al. Formation of protein crystals in quasi-microgravity environment obtained by superconducting magnet. J. Crystal Growth, 2004, 270. Ramachandran N., Leslie F. W. Using magnetic fields to control convection during protein crystallization – analysis and validation studies. J. Crystal Growth, 2005,  274.

Control of oxygen content in the air… on board ISS

Ageikin D.I. Definition of heat emission by magnetothermal convection. Proceedings Academy of Sciences the USSR, 1950, 74.

Magnetic field effects on the morphology and orientation of lysozyme crystals crystallized with NiCl2. Pictures were taken 2 days after supersaturating the solution with NiCl2: under 0 T and under10 T

Fig. 2. The superconducting magnet which can provide effective gravity ranging from mG to 1.8G simultaneously.

The crystals obtained inside the magnet (15 T) show better quality than obtained outside.This result is in agreement with the previous results of orthorhombic lysozyme crystals. The improvement of crystal quality was not clear at the center of the magnet (10 T).

Formation of protein crystals (orthorhombic lysozyme) in quasi-microgravity environment obtained by superconducting magnet.D.C. Yina, N.I. Wakayamaa, K. Haratab, M. Fujiwarac, T. Kiyoshia,

H. Wadaa, N. Niimurad, S. Araid, W.D. Huange, Y. Tanimotoc

Tsukuba Magnet Laboratory, National Institute for Materials Science, Tsukuba, Japan; Biological Information Research Center, National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan; Institute for Molecular Science, Okazaki, Japan; Neutron in Biology, Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Japan; Stake Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Shaanxi, China

Journal of Crystal Growth 270 (2004)3

differential magnetic susceptibilities

protein media

paramagnetic melts

magnetic fluid

-10-5

10-3

1

Lysozime crystals immersed in paramagnetic solutions MnCl2, CoCl2 etc.:

is several orders greater

Single domain ferromagnetic particles (~10 nm) are coated with long chained molecules to prevent particle agglomeration and suspended in an appropriate carrier fluid;kerosene-based magnetic fluid: magnetic saturation MS = 48 kA/m,  = 5.7, concentration 10 %, Pr =100

Ferromagnetic nanoparticles

)1(

)( 2

a

hRa mo

m3Th

a

gRag

Finlayson B. A. Convective instability of ferromagnetic fluids. J. Fluid Mech. 40 (1970) 

Т - temperature difference across fluid h - layer thickness g - acceleration due to gravity - thermal expansion coefficient а – thermal diffusivity - kinematic viscosity m - relative pyromagnetic coefficient М – magnetization - differential susceptibility - density of magnetic fluid o – permeability of free space

h ~ 1mm

CONTROL PARAMETERS

6

2Mh

~Ra

Ra

g

m N.B.!

(v v = Р v (GrТ2 еz – Grm Т2H2);div v = 0;

0 =T1; Pr vT2 = T2;rot Hi = 0; divBi = 0; B1 = H1; B2 = (H2 + M);

The conditions at infinity and in the center of ball are:r: Н1 =Н0 еz; Т1 = еz;r = 0: all function are limited r = 1: v = 0; Т1 =Т2; Нn1 = {– Grm(T2/H2)}Нn2; H1 = H2

The convection instability in spherical cavity heated from below (above) in the presence of magnetic fields

i=1,2: the parameters and functions relating to body (fluid) are marked inferior index 1 (2).

11

PLASMA CRYSTAL IN -GRAVITY

PKE - NEFEDOV

INTERNATIONAL MICROGRAVITY SPACE EXPERIMENT PKE - NEFEDOV

ISS-1 PKE - NEFEDOV Yu. Gidzenko, S. Krikalev

Dusty plasma space experiments

Conclusion

Nanofluidistics is very fast developments area which need basic research, both theory and experiments and will be one of the main fields of

Nano National Initiatives with different applications

• Science

• Technology

• Micromechanics

• Optics

• Space equipment

• Clothes

• Toys

•Art (ballet, painting, …)

•Household

• Others

•Oil Industry

•Plasma Technology