# Presentation: 696764

## 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, 2005
- Castel Gandolfo

## Strong permanent shear + weak fluctuations

- Experimental motivation:
- Elastic Turb.
- (Groisman, Steinberg ’00-’04)

_***Strong ***__***permanent shear ***__***+ weak ***__***fluctuations***_

_**Questions addressed **_
_**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:***_

- 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 than
- any velocity related scale

## Experimental Setup

- Regular flow component is shear like:

- Local shear rate is *s=**** r/d*

- (Steinberg et.al)

## Time separation:   

_*Time separation:*_
- Fast (ballistic) motion
- vs
- Slow (stochastic) one

_*Angular separation*_

_*Separation of trajectories*_
- (auxiliary problem in the same velocity field)

- Lyapunov exponent

_***Ballistic vs Stochastic***_

## Angular Statistics

_***Tumbling Time Statistics***_

- Hinch & Leal 1972

- single-time PDF
- of the noise

- determ.

- stoch.

- depends on
- veloc. stat.

## Stretching Statistics

_**Stretching Statistics**_

## 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)*

- 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

_*In progress:*_* *
*V.Chernyak,MC,C.Jarzynski,*
*A.Puliafito,K.Turitsyn*

## Statistics of the entropy production rate

- 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 asymptotic
*P(****) **** 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 the
- chaotic velocity component.

## Tumbling time distribution

- 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.

- Gerashenko, Steinberg,  submitted to PRL
*(experiment)*