La Jolla 070700 Polymer Stretching by Turbulence Elastic

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

MC, PRL 05/00 Polymer Stretching by Turbulence Balance of forces Models of Elasticity A B linear (Hook) dumb-bell Scale Separation The question: to describe statistics of passive polymer ? Passive statistics = of is given smallest scale of the flow C nonlinear chain nonlinear dumb-bell >> stretched polymer length >> equilibrium polymer length Advection >> Diffusion Statistics of passive scalar advected by the large scale “Batchelor” velocity is understood 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

Passive linear polymer A 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. scales Elastic part of the stress tensor in the kinetic theory approximation >> Inter-polymer Stretched Equilibrium >> polymer length distance n- is the polymer solution concentration N>>1 - is the dimensionless polymer length

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

Pure Elastic Turbulence (experiment) Groisman, Steinberg ‘ 96 -’ 99 d=20 mm Swirling flow between two parallel disks transition to turbulence d=10 mm pure solvent 80 ppm polyacrylamide+ 65% sugar+1% Na. Cl in water Power spectra of velocity fluctuations Wi=13 Re=0. 7

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

Increase of n - polymer density Inertia-elastic Turbulence (instead of conclusions) Energy containing 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) 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 Viscous (Kolmogorov) scale