Stretching and Tumbling of Polymers in a random flow with mean shear

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Stretching and Tumbling of Polymers in a random flow with mean shear

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Stretching and Tumbling of Polymers in a random flow with mean shear

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Sep 3, 2005

Castel Gandolfo

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)

Strong permanent shear + weak fluctuations

Elasticity (nonlinear)

stretching by

large scale velocity

Thermal

noise

Experimental motivation:

Elastic Turb.

(Groisman, Steinberg ’00-’04)

Polymer is much smaller than

any velocity related scale

Questions addressed

theoretically in this study:

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

- 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)

(Steinberg et.al)

Regular flow component is shear like:

Local shear rate is s= r/d

Ballistic vs Stochastic

Angular separation

Time separation:

Fast (ballistic) motion

vs

Slow (stochastic) one

Separation of trajectories

(auxiliary problem in the same velocity field)

Lyapunov exponent

+ stoch. term

deterministic term

constant

probability flux

Angular Statistics

determ.

Hinch & Leal 1972

depends on

veloc. stat.

stoch.

Tumbling Time Statistics

single-time PDF

of the noise

Stretching Statistics

(b)

(a)

(c)

(d)

plateau

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)

In progress:

V.Chernyak,MC,C.Jarzynski,

A.Puliafito,K.Turitsyn

(linear polymer in shear+ thermal fluctuations)

Probability of the given

trajectory (forward path)

(reversed path)

Microscopic entropy produced

along the given path

(degree of the detailed balance violation)

grows with time

Related to work done by shear force over polymer

Statistics of the entropy production rate

( independence !!)

Fluctuation theorem

The distribution of the angle is asymmetric, localized at the positive angles of order twith the asymptotic

P() sin-2 at large angles determined by the regular shear dynamics.

PDF of the angle is also localized at twith 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 the

chaotic velocity component.

P(t)

t

Gerashenko, Steinberg, submitted to PRL

(experiment)

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.