1. 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

2. Experimental motivation: Elastic Turb. (Groisman, Steinberg ’00-’04) Strong permanent shear + weak fluctuations stretching by large scale velocity Elasticity (nonlinear) Thermal noise 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

3. Experimental Setup Regular flow component is shear like: Local shear rate is s=  r/d (Steinberg et.al)

4. 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

5. Angular Statistics Tumbling Time Statistics Hinch & Leal 1972 single-time PDF of the noise + stoch. term deterministic term determ. stoch. constant probability flux depends on veloc. stat.

6. (b) (c) (d) plateau (a) Stretching Statistics

7. 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

8. 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

9. Fluctuation theorem Statistics of the entropy production rate ( independence !!)

10.  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.

11.  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.

12. Tumbling time distribution P(t) t 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)