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Stretching and Tumbling of Polymers in a random flow with mean shear
M. Chertkov (Los Alamos NL)
I. Kolokolov, V. Lebedev, K. Turitsyn (Landau Institute, Moscow)
Journal of Fluid Mechanics, 531, pp. 251-260 (2005)
Sep 3, 2005Castel Gandolfo
Experimental motivation: Elastic Turb.
(Groisman, Steinberg ’00-’04)
Strong permanent shear + weak fluctuations
iiijji RRvRtR )(
stretching bylarge scale velocity
Elasticity (nonlinear)
Thermalnoise)(ˆˆ)(ˆ tst
Questions addressed Questions addressed theoretically in this study:theoretically in this study:
For velocity fluctuations of a general kind to find
•Statistics of tumbling time•Angular statistics•Statistics of stretching
(all this -- above and below of the coil-stretch transition)
Pre-history:Pre-history:
•Shear+Thermal fluctuations
theory - (Hinch & Leal ’72; Hinch’77) experiment - (Smith,Babcock & Chu ’99) numerics+theory - (Hur,Shaqfeh & Larson ’00)
•Coil-Stretch transition (velocity fluct. driven – no shear) theory - Lumley ’69,’73 Balkovsky, Fouxon & Lebedev ’00 Chertkov ‘00
Polymer is much smaller thanany velocity related scale
ijij v
Experimental Setup
Regular flow component is shear like:
Local shear rate is s= r/d
(Steinberg et.al)
Rt
t
t
sRR
s
s
sincoscos)(ln
cossincossin
sin
2
2
Time separation:Time separation:
Fast (ballistic) motionvs
Slow (stochastic) one
1~ s
1~ ts
Angular separationAngular separation
1~
t
Separation of trajectoriesSeparation of trajectories(auxiliary problem in the same velocity field)
1
)0()( ]exp[~
t
tt
ts ~
Lyapunov exponent
Ballistic vs Stochastic
Angular Statistics
Tumbling Time Statistics
Hinch & Leal 1972
single-time PDF of the noise
+ stoch. term
23 cossin
)sintan(),(
UP
deterministic term
aP
2
|)(determ.
stoch.
1t
2sin)( tCPt
tP )( constantprobability flux
depends onveloc. stat.
a
sP
sP
2
2
3
22)(
ts 1
tP ~)(lnt
)(P
t
01;1 Wi
1;1 Wi 1||;1 Wi
(b)
(c) (d)
*
1~ RR
tR **R
*R
)(' R
plateau
0;1 Wi(a)
TRR 1~
*1)/(~ RR t
Stretching Statistics
)0(
Wi
Conclusions:
We studied polymer subjected to Strong permanent shear + Weak fluctuations in large scale velocity
We established asymptotic theory for
• statistics of tumbling time -- (exponential tail)
• angular (orientational) statistics -- (algebraic tail)
• Statistics of polymer elongation (stretching) – (many regimes)
K.S. Turitsyn, submitted to Phys. Rev. E (more to the theory)A. Celani, A. Puliafito, K. Turitsyn, EurophysLett, 70 (4), pp. 439-445 (2005) A. Puliafito, K.S. Turitsyn, to appear in Physica D (numerics+theory)Gerashenko, Steinberg, submitted to PRL (experiment)
Shear+Therm.noise
Tumbling – view from another corner (non-equilibrium stat. mech)
(linear polymer in shear+ thermal fluctuations)
)(ˆ2
1exp~)(
2
stPdttP Probability of the given
trajectory (forward path)
)(ˆ2
1exp~)()(
2
stPdttPtP (reversed path)
)()(log)()(
2ˆˆ
2
1ˆˆ
2
1
)()(
log)(
22
st
stP
Pdtdt
tPtP
tS
Microscopic entropy producedalong the given path
(degree of the detailed balance violation)
grows with time
Related to work done by shear force over polymer
ˆ
In progress: V.Chernyak,MC,C.Jarzynski,A.Puliafito,K.Turitsyn
2
ˆˆ
w
21
2
))(4()(
/)(exp~)|(
22 w
s
swswQ
wTQTwP
wwQwQ )()(
Fluctuation theorem
Statistics of the entropy production rate
( independence !!)
angle distribution
The distribution of the angle is asymmetric, localized at the positive angles of order t with the asymptoticP() sin-2 at large angles determined by the regular shear dynamics.
angle distribution
PDF of the angle is also localized at t with algebraic tails in intermediate region t<<||<<1. These tails come from the two regions: the regular one gives the asymptotic P() -2, and from the stochastic region, which gives a non-universal asymptotic P() -x, where x is some constant which depends on thestatistical properties of thechaotic velocity component.
Tumbling time distribution
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
P(t)
t
The characteristic tumbling times are of order t=1/(st ) , however due to stochastic nature of the tumbling process there are tails corresponding to the anomalous small or large tumbling times. The right tail always behaves like P(t) exp(-c t/t), while the left tail is non-universal depending on the statistics of random velocity field.
Gerashenko, Steinberg, submitted to PRL (experiment)