Title nonlinear shear viscosity

Author(s) , ; ,

Citation (1967), 7(6): 452-478

Issue Date 1967-03-20

URL http://hdl.handle.net/2433/85994

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

IvriOllin8arShearviseosity

()

()

-

(2 16)

Abstrac

t,Zj Shearviscosityj

A- 1 ShearvSC0Siy d

17, 0= (,Y,

effechve LSbearvise,0.8ity q(d) d

( Sbea-rviscosi.t.y)tKawasaki n g

-qr,+qld2+q2d4+--

i E-9 . n 1.aj

OTnStein-Zerrlike ra_nge

--. 7. l

( (

O BrJtclFixman

-O 72 ; 0 772

kk. k3O

2 71 Ogk

. 2 bulkviscosity k

0-3 ghost

(a) d-0nOn-analytic

-452-

L

nonil7_ear ShearVisGOity

1 Inrou/3ion

cTiica poinh aCrO f-ilCuation non -

near

W.Boch M.Fixman binaryixtuTe bTiticalp.

h Shearviscosity nonlinear velociygradien

( (ol) Fixman 2) n-pnlnear

Fixmarl binaryixure r_ViscosH y cor1-

cenration fluctuaioriUjCritical p

/lc(.2) staionary Btate enropy prodllC - .

ion rae veociy gra-dient 2 viscoy

o L Viscc)sity Navier-S+uokeB.

awasaki(3 any,id4k vscoLSiy fl

.- anOrnalc)uB concenrai.Dn flu.-

Cuaion_ Fixi-ilrl Ju

flux bFixman

COrrelao1funconapproacb

ressensor no1,1ocal

t3)

'vl Sationary Stress terLSOr nOrl-

1irlear cOrr.elation fullCion aPPrC}-

ach linear nOninea_I

stress tensor

velocity gradieri, correlaion

funcion o

Themal disturbunce correatJioA functi~orl

(i)loca equilibrium 5)(iij

Fokker-Planck(6) (jindirecKubomehodr7)(8)(9)(ivjreg

453-

resBion hypothesis(") (i)MPrifo,muation no1inear a .52

Kawasaki(3),( Bcch-Fixznan 3 , O

4

2 Sress ensorriOinear

matCrO

tpransierl

ma

O,Jqlequilibrium di ribuHon

local eq:uilibriulndisrilburr,iop.E B5)

r)- eJ 1H/fRf " 1

dxN-dxl-dXN,Xi(i- 1!2 tN fI03ialni1-boniaI ujlparticle2.pC)enial

-A- + jfLujl

N

3-1 2

U "cannonical - OPerator .

Li)G

brium average . (1)

fz"-U-1-GPT/Ie-p-qdxNT(equilibrium distri-bution Cion) (311

dxN Cannonical

fTT locallvelociy O

< )>l= O=poy(r)

(I) locai velocity

l- O-p. ,,-oi poifT)2I

j(),p(rhH(r) operaor(bSI

i(r)-p rj-r')(mass~flow denLqi'Lty)

p(r)= mj(rj~r)(massdensity)

Ir1-

(4)

-

H(rJl (rj-t,+i ?,Elujl,n(rj-r(ene- nsity)1

U

upj-Pj+V{rjjUrj.-rj(*)i

(5)

(6)

() exac canno cal _T.'

0

-455-

Jii

u-1=H+A

R- j(r)Y(r)dr+ifp(r),(r)2dr

LiouvileoperaT,oT

iL- - .( - )

imi

G

ei G - )

fN(U)-e-itL7-e~pIR(-i)

fN() = tjfN

(7)

(S)

(9)

nteracton

fl" -Jdsn S2)h(-sn (10,0/

n- 1 erm b

stres.stensor Nairer-Stokes

Local Lqjress+ueSOr J()O.

J(r pjPj6(rj-r rjl 8( r)

-pl+j()V(r)+V()j(rr)V(r))

+I;(I)-456-

(11

nonnearsheaViscosy

dt

>

1ulni dyadic ,

P=.(equilibrium pressurej

-Jv*(r)-ipj-nV'rjj]lPj- rj) -r)

rjl 8(r rj-pl t (r)-J(r p,

UIJ(r)-Jv*(I)+pl

(12)

(13j

( reSS T,enS.T J (tl

h

Sress ensor VisccjSity P*(r (iOj

.

p*(r)-fJv*(r)F()dxN

-

--Jarlja(r va(rDvp(r) O

a P(r,) j(f,)

j(r,)J(r,)

Ova(r)

aI/9drJvTtr):V(r) (16)

(17)

FixInant2) (shear fW

vx-vz-0,vy-()-d-,( Ld veociy gTadientj (18)

q6)

JvXy(r)dra

:JvXy*-Jy (11 (14)

*vl_J(-dPjn ,,_i,_ _. ._1__p*xy -

(19)

fdtlfdrlIf dtnJrn< JvXy( , o ) JvT -y ( r 1,-I-~rF l n ! O A Jo uJ - V . V

JvX'y(rri 1- nj>l (30)

Cut off JXy r

ilriPicit PrO.jecion. (Ap-

pendixA)

Stress ensoT

nonHnearviscosi+uy d) O

(20)(2 (13) O(,JXy-UJvjCY

t-jt rtT-(-podI)n/._l ,TJI-dtl/'drl-fdtnfdrnJXy(,)luJvXy(rl,1)n = l rH "a o

-458-

noninearshearvisGOSity

xluJvXy(rn,n>.C) D)fdtlfdrl-fdtn/ldrnJXlr(r, y(r1,-1)LJTXy(rn, /(-djn

n = l n ! O

(2i)

2 y(ri,I-ti)JXy(ri,)Liouville

operatoriL i(i+dL') o

L'

iL.-" xi - C21 0 iLJ AppenixB .

(21)(,

1pxy*(r)-

F l n ! v Jo drid

_...

1 (djn/V rI-& l n ! Jo~L JoV

IXy-fdrJXy(r)

xljdrlfdtLiJdrnOl rl (23)

Vh maCTOSCOPic SyStem

volue

5'3 Non-linearViscosity

Kawasaki bina.ry mixture Shearvisoosiy

long rangeconcentration fuctuaion CriicalpT_

flux corlee.ntration

fuctuation , O(Rfere1Ce

3, Ref.I)

-459-

Ref. fux

JXy()-f y(2i)

C;concentration,f;Ref. cons'

(23) IX y

xl -ffdr - fkxkyckC-k- CkC-k,] (25,

k-space concenration Ck( Appendixa

RefI .

)'Jk

dkx~ ek= -rk2(k2+K2)ok(i,5'.

JC cTiJa.ipt. 0 C,Orrelaion engh

(26) (25) Ckelk(k;definition)

- r Ak 2rk2(k2+K2jAk( >0)(2'7)

di,ssipaion

(27j

Ak dky IAkt,--2 2+K2)k2Ak) (>0 27,,

(23) (

7

EiiZE Liouvile c,p.i(L+a.L A =

A(i) 0

-460-

nonlinearshearviBCObliy

Cx

fdtAk')-( d +d2 SS)Ak(0.)0

-;a(k o) (28)

-A-2r(kZ+K2)k2

.

S

( OPra A ,B Tvl

1 1- --i B+Bi Bi--A+ A A A

(23) oX

pxy* ( (dDd- - :Ll_=

n(dt)-djr .1n!

if3

(29)

(=x3 CQ ij'dtriIXy y y(-tri>.(3Qj00

e9) dh h n

IXy(-i,i) d O

Ref.I RPA n

j >O, C(m rleClbedav'erage > OC h f c, cOrlneCed.aJVerage

I(3 y (B.P.A. )

0 (Cunuanaverage)

AppendixC3 O

R.P.A.cOncenraticn f-1ucuaion 2

OTBein-ZeTrJikeype. (Ref.I)

-Vfl

-461-

I(3i)

AppendixD.( n(a)

- --1vP- ky0-pd ky 2,2i

n2

n(a)-2 lpn(-dln'-1

(32)

frl+lkxkyCSdikxky

x

:-- 25kBTd23212zcr369

(353 (34)

125 d2

1 2 -- 10

d2

03255 X- 7 10

t>

r10nli_T3earShearviscosiy

5)

(36)

(36) BochFixITnarl gH*)

2,3 nonli-nea.T Vj-sculSjty d

d2 ,J 200

- --IL----:-;:--- :-1

kBTd3 /1-_kxikv2 f 1fclk

1

24H3r3 ' W-k i 2+C2)5Ik22+ nT2)

BTd2i5212T3E9

k)Td2 li-I-r - ;-:---I-:-1-:-

kBTd2

15213Hr3E9

,

(37j

(38)

(37,,(3Bj 12 ( 03%)

10nlineaVise)osiy d2 OTder

(*)Botch-Fixman k

disribuion ( k

pa .

O 4

- 4 6 3 -

d)-kBT

160G

d2

(1-O.3268- --+-,)r268 (39)

nOninear d2K-9 Criical p.

nOni.near critical p.

d b t l dimen-

LSiona analysis d

d dieBnSic)nle,ss paramer

(a/r4) . d2 - E~8o

Tc COrlCelTation flucuaon

) d4 b

5'4 higher ordeT

7((i)d d2 Order 3O C)rd,er

d-L rder k

o

(32) 71d4 OrdeT divergeI,-2qt MD kT3Td4

14 lVIx24r510@2+2)2k2 kirk28kyk2kv k2 ky k2

kBITd4 . 16OH,(PC d

._ j:..._I?____.:7-__._. . i: _:__i_:Il:. j.

- T X (- xf

(40)

24r5(2H3:17 1287Vo (i-ill+12

k- (In-) k~3 ,

2 .

, d6 1 7o rea

d mOSdvergen

-464-

noninearshearviscosiy

d order

)d-0 analytic

2 binaTynixre shearvicosity d2

ordeT .

12:=~~ 2kBTd2(2JT)2(2r)3JCO

- -_; :I- -

2kBTd2

(27r(2310

m2(m2+1~AIm4tm2+14

4(2nP+1)(4m2+1)m8(2+16

pn m G

nyi)

(A1)

S>

m o .

n.onlineaT b

(26) diffusionrype

o 2 buk viscosityl.bulk visco-

siW , referenCe 4 ido Order

in r og O

i, flux 1 me (oTreatiorl_

h life'ime

_ P.A. lifeime

3 Hnear negligible

conTibutio O

- con.uinllt3,a ra.Ciorl(12) 2 bulkTJiB-

cosity (13 3

corlnued fracion,

, COnirlued fraciorl

.b

birlarymixture. Iinear shearviscosity

cotinued.fracon oriicaanomaly

blhearviscoBi-cy 1

ar10maly o

(40),(4

1 ( hQ

AppendixA

.62 streSSensor (2-3_)

Mc' PrOJecionop(14) I,I

CO ent

[- 0Lh ocalequilibrilJlneSemble(

)

f"(-0-f-eT R(o) > 0 .

/

- e H+

(A-i)

(A-2)

Mori (14) nOnlinear

problemmacroscopicparameter(velocitygradient

eC.j consiBen o

L macroBCOPicparameer D

qllaSi-Btaionarycondition - foN flNMori 0

-466-

nonlinearshearvisoosity

4

()-fL+R'(-)ii: i

R-R(), (-)-(1-LPl)fi()L

LPl (A-7)prOjecionbop

tL-O

fL-R,R'(0)-Oo

Mori

RJ(-)ifdse-i(t- )L(1-LPi)LLJ-DIR S0

LPLGA*JW*>;lAC->ii:

fo

:J1-51)

-6j

'-]'..-7)

rl-LR(0 quasi ationary flN

AmaCr,JVaables crJlun

atrixA' cOnjuga.emat.rixo

fN(

-fl" fte-i(t (1-PL( Pli'J (-S)ds

0

R

- V()dr-+/ V2(r)dr

vir)dr-aEPl3'at'r)-

vp(r)vd(r)

rpdr

(LLt.-L,

(A-9)

(A-10)

Projecon operatorLZ)l A

467-

OA,A*= U~ lA ,

U-1A+, (U 22 -cannonica.1 OP.

-1LPlCT-U-1-4*< (U~lAjCu--1A*)>1l

-uT-1A*AA*>:1AUG>.-Trl,uG (A-1-

32(2) tZ'O ensemble ptroje ionop

quaSi-sf,aionaryBtaeI/1

iiJZ'lR(-S)-fL L

t

1

r

l

1

nonlineaTIShearviSeOSity

,I vy =dq ?

jy-XyeSPTudsew i(t-Bj(1-LPJ(i+dL'jiIXyd>O 7)0

(IXy JJXydr)

ill-" xi - )01

.JTV-IXyPt-?/dB is(1-LPJ(ll+dLJjlIXydl>O ~-I I V ~' o

0 ..

r S -is(1-Lan)(i+dLJjiIXyd>. 0

- .)a

ti

11 -

IXy1- -dfds(eBp-is(1-LPJ(i+dIJJjIXylij>.0

'll:i11neLIl:~I n= l V n !

-18)

t._

0 XyrjdB is(1-JDo)(i+IIXy>o 19)0

/

LPo O

y translaional invariance O

y(ti- ri+a)-IXy(ri) 20)

O

(iL)a(i (iL)i(ill)PLIXy -2)

translanalinvariance = o

poGA*.AoAJlAoG>

469-

Ao, A q maCrO Ab,A q70 Iii

3oiLa-AAA05 1Aod>--A;AA.> -1-*)

macro Aq A0-0 _r

. ( 18) (1-LJZ)o) i iF

( LPo)L'LJLno o (A-18) dL

F(:tl)(1-LPoiL(",(>o t20,1>0 (A-22)

1---

i::: i=;F('rLi)(1-LPo)iL'LIT(-h2)>O-F(t1,1>.(1-(Z'O)iLJG(2)>. 23)

(1-Lj1.) o

J

F(1JD.iL'G(2)>6-F(tlA=,oAoA>-1AoiLTG(TJ2)>o

-.AAo*>wol.

.F(tl_)Ao't>O- .tA;(-tl)>O-F(0)A;>o

(A-24)

(A-24) tl 1-CX3 .

F(tl)iL'G(ib.> tl- F(I,ljiL'G-(2)>.

LA-23) O cance1 O 1- -

5iZ i:!

(1-LPo)iL LPo

Appendix Gi)Liovil.le opiL

-470-

ETI

noninpJar ShearviscoBiy

( .( diffusion

eq. )

Appe1dix B

I

(21) 2

UiLG(riPi;I)-1pirq_V(ri)

-Tni

I

+Fi.-

XG-(ri,pi+miV(ri)i

pimiV(ri)

(.let:'.-i)Fjj-1) 8/ariG V OPerae ho

operae

pi+V(ri), _/ ___,__. ,pT,.

e mi(-iTiVCrij) +Fi- -i)T J'

mi

cTT(ri,Pi+miVri))-i(L+dL'jU-

Fi i .-Particle

Vy(ri)= jrld

ill-u={j)JEl

densiy func on G(r,)

G-(r) gi8(ti-r1

-2.

-3)

gi i ri,Pi (mass,

Charge,eC.)o

LOvi1e op.ri,pi (Liov11e

op.i(L+dL) ) ri ,pi

-471-

L,L' G i(L+dL')

JIJ(r,T_)

i(i+dLJ)(,J(r,)=i(Lt+dLG(r,)

-iLtG(,- dx G(r,

O-(,)i_1__ / --J,tn,adxT G(,iL ,t

1, ~~- y l'J

-4) 1 2 o

iLJtG-(r,)-iL it-r i

- gia(ti+ulr

-- giS(r )

= ,)

4)

-4)

Q3-5)

-4) idL' ,

dri-ft term C,

ATJPendix C

Connected avJeTage(Cumulan average)

10cal equ1ibriu dstrbution

fdxNfti-fdxNfLN

-472-

(lil-1)

1

nonlinearshearv.iscoSiy

(10)

ISdBl/dsn 811)(-sit)0(C3-2) (23)

S'()nlq ,q_JTY/ _ .F l n ! -0 -Jo ~ro (23) connecedaverage

(-Pdjn

n= l n !

==

+

q

E3

CX3 C_-o

CoOC

+ -+nci.CO

+ - i

)

- (-TPdjfdslrXyTxy(-sJ >oc(1+ (-,qEi)Tl~1

2 (a-i)ifds2dB1-y(-82)-I

-'rm v(>o)

+.+(-GtOlfdsl-dsi.i (-3djn-i

n=li+i(n-i)i!

(-Pa)1!

oi+

L=(-.Jc'i_lil

fd + =

OC=i

-473-

n! fdsl- n

+-+(-?d)1i.F (-d)a

Il=1 n!

fds.dBi< =XyfXy sljy(-si)>oc(1+

fdsldsno)+

(C -3) ((3-4)

ci3(-djnn-1rlIfdsl.dsnoc

(C!-4)

IXy y(-sl) sn-)>oc si) cumu.ant aTJer;3Lge .

APPF?ndi.xD

bL-fn. ITje)

f()(;Xfl.-f 8(-)

-iI(xoh(-xJjf=- 8(-(32)

(3Oj 125) (28j

1(a,- f)2k,k kxkyOTpd(k,oc

RP.A.

< (ckC-k-o)(Ck.C-k,-Ck,a-k>o)>oc

- < Lckt2>2i8(+k+(k-k)

-1)

-474-

p-ll)

Ili17)

nonlinearshearviscosity

i:l-:(27E)6 f2 xkyO;a(k)(k-k7j

x< ick12> 2 (I)-3:)

facT,Or2 P-2fiP.A. kk'

k-k'2f) h

bh-fn, P-3)

1(d'-i- f) ,k (k-kJ,

k sk. 8(k-k

-k . sk. LMk-,' (,- b'

-k skS:(k-k (.:- b)

skk 8{k-k- sk kyS(k-k ( b)

-

k Oh,dCkJ)(-k

-og(k) ky(-k

18h(D L-6)

kfk, kyk)I(k kyOgr(k ky8(k-kJ)

-kxkyOT,a(k k b-g(k)kxkyi

LD-7)

p

p-9

p-9_)

,.) hh

(33) -5j (25)(2.5)n(a)

Pn(-djn~1

n! - ~J kl,k2,-+11 :fin+1._._.:klXklyk12k2y-kn.1,Xkn+y

Xo-nd(kl)0-pn(kn+I)< (CklC-k CklC-kl>O)(Ckn+i-kn+1

>)>'< ckn+lC kn+1 0 9C

-476-

-

nd)n~1n!

&

nonlinear shearViBCOSiy

fln" snkl,.kn y-kn-knIy0-p?(kl)

-0-pd(kn+Ds*1- 2-k3)8 +icki2>n-1 p-10)

Szl PA. 2

coneCedaverage 2

icki2 1

~ O

SnE. (a+ 1)

ri! b L

A.f/2 (a+D 1k

2 paiT 1

2n b( + or -

)

n!

sn= 2n=2n~1n!

p-9)- P-1) LD-1OjEJ

L.Jij:=2n-1nd)n~1

fin+1kxky0-ikxky tCk2,i?'1

kxkyin~1

fiefereriCe

1) WBotch & M.Fixman,JChemPy-Si6 3100(62J)

2j MIFixman,JChemPhys. 310(62

3) Kawasaki,PhysRev291(661)

4) JDeuch andfi.Zwanzig,(obepublished)

5j H.Mor,PhysRe,Vii2182958-477-

-ii)

p-12)

6j JClKirkood,JChemPhys. ii 180(46I)MSGreen,JOhelnPhysB 1281(52J);ibd 39854 .

7) J.MLu inger,PhysRev13il5=1505(64

Bj J.LJackson&Pzur,PhysicaQ_2295(64'):9j a,.W.MolltrOll,Re,ndicontiDella Scuoa Inernaional

DiFiLSicaBlnricoFerlTliX 217(Sociei:a aliaTia )iFisi-

ca,Bologna,1960)

1(),b KuboYokotla&Nakajll-ila,I.PhysSocJapa:rl1212)3

1IKKawaLSaki& M.Tanak_,J.Phys.Soc.London(topub-

lish

12 brProg.Theo-JPbys 399(65r)

1 (

1.4) fl.Mori,PrgTheor.'Phys.33423(1965)

-478-

I