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

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

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  1. 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)

  2. 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)

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

  4. 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

  5. + 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

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

  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) 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

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

  10.  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.

  11.  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.

  12. P(t) t Tumbling time distribution Gerashenko, Steinberg, submitted to PRL (experiment) 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.

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