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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). Jou r nal of Fluid Mechanics , 531 , pp. 251-260 (2005).

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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)

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Strong permanent shear + weak fluctuations

Elasticity (nonlinear)

stretching by

large scale velocity



Experimental motivation:

Elastic Turb.

(Groisman, Steinberg ’00-’04)

Polymer is much smaller than

any velocity related scale

Questions addressed

theoretically in this study:


  • 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)

Experimental setup l.jpg


Experimental Setup

Regular flow component is shear like:

Local shear rate is s= r/d

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Ballistic vs Stochastic

Angular separation

Time separation:

Fast (ballistic) motion


Slow (stochastic) one

Separation of trajectories

(auxiliary problem in the same velocity field)

Lyapunov exponent

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+ stoch. term

deterministic term


probability flux

Angular Statistics


Hinch & Leal 1972

depends on

veloc. stat.


Tumbling Time Statistics

single-time PDF

of the noise

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Stretching Statistics






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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)



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Tumbling – view from another corner

(non-equilibrium stat. mech)

In progress:



(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

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Statistics of the entropy production rate

( independence !!)

Fluctuation theorem

Angle distribution l.jpg
angle distribution

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.

Angle distribution11 l.jpg
angle distribution

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.

Tumbling time distribution l.jpg



Tumbling time distribution

Gerashenko, Steinberg, submitted to PRL


The characteristic tumbling times are of order t=1/(st ), 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.