Polymer Stretching by Turbulence + Elastic Turbulence Theory La Jolla, 07/07/00 Polymer Stretching by Turbulence + Elastic Turbulence Theory Misha Chertkov Los Alamos Nat. Lab. Polymer Stretching by Turbulence (Statistics of a Passive Polymer) Pure (Re<<1, Wi>>1) Elastic Turbulence of dilute polymer solution Inertia-Elastic Turbulence (Re>>1,Wi>>1). Drag reduction. Thanks A. Groisman, V. Steinberg, E. Balkovsky, L. Burakovsky, G. Falkovich, G. Doolen, D. Preston, S. Tretiak, B. Shraiman http:/cnls.lanl.gov/~chertkov/polyprl.ps /japansmall.ps

Polymer Stretching by Turbulence MC, PRL 05/00 Balance of forces A B C Models of Elasticity linear (Hook) dumb-bell nonlinear dumb-bell nonlinear chain smallest scale of the flow stretched polymer length equilibrium polymer length Scale Separation >> >> The question: to describe statistics of passive polymer ? Advection >> Diffusion Batchelor ‘59 Kraichnan ‘68 Shraiman, Siggia ‘94,’95 MC,Falkovich,Kolokolov,Lebedev ‘95 MC,Gamba,Kolokolov ‘94 Balkovsky,MC,Kolokolov,Lebedev ‘95 Bernard,Gawedzki,Kupianen’98 MC, Falkovich, Kolokolov ‘98 Balkovsky,Fouxon ‘99 Statistics of passive scalar advected by the large scale “Batchelor” velocity is understood statistics of is given Passive =

Passive linear polymer A Nonlinearity beats the stretching !! CLT for the Lyapunov exponent statistics at (saddle point parameter) First order transition: the polymer stretches indefinitly if advection exceeds diffusion PDF Lumley ‘72 Balkovsky,Fouxon,Lebedev’99 PDF Nonlinearity beats the stretching !! diss. scale

Passive nonlinear polymer B saddle point parameter PDF

Passive nonlinear chain C linear conformations are dominant N (number of segments) >>1 is an additional saddle parameter Notice the nonlinear dependance coming from the “equilibration” of the stretching by the nonlinearity

Non-Newtonian hydrodynamics of a dilute polymer solution Navier-Stokes equation Scale separation Hydrod. Inter-polymer Stretched Equilibrium scales distance polymer length polymer length >> >> >> Elastic part of the stress tensor in the kinetic theory approximation n- is the polymer solution concentration N>>1- is the dimensionless polymer length

Rate of strain --- Stress Tensor Relation weak elasticity (linear stretching) =>OldroydB model constitutive equation extremely strong elasticity (nonlinear stretching) => local relation between and : the maximal tension Weissenberg number the largest eigenvalue of the direction of the eigenvector nondeg. deg.

Pure Elastic Turbulence (experiment) Groisman, Steinberg ‘96-’99 d=20mm Swirling flow between two parallel disks transition to turbulence d=10mm pure solvent 80ppm polyacrylamide+ 65% sugar+1% NaCl in water Power spectra of velocity fluctuations Wi=13 Re=0.7

Pure Elastic Turbulence (theory) Elastic dissipation >> Viscous dissipation, Advection Nonlinear diffusion + constitutive equation poor-man scaling K- is the pumping amplitude of

Inertia-elastic Turbulence (instead of conclusions) Energy containing scale Viscous (Kolmogorov) scale Dissipation due to elasticity at the Kolmogorov scale is less then the viscous dissipation The drag reduction (dissipation dominated by the elasticity onset) The energy is dissipated at the elastic scale Polymers start to overlap each other (the kinetic approximation fails) Increase of n - polymer density According to Lumley’69 the increase in bulk dissipation (viscosity) is accompanied by a swelling of a boundary layer, that leads to the drag reduction