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)