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 d F friction u t ; Felastic Ftherm dt A B linear (Hook) dumb-bell Felastic Scale Separation The question: to describe statistics of passive polymer ? Passive statistic =s (t ) of nonlinear chain nonlinear dumb-bell

U '() smallest scale of the flow C >> stretched polymer length u (t; ) (t ) U ' ( i;i 1 ) U ' ( i;i 1 ) >> equilibrium polymer length Advection >> Diffusion Statistics of passive scalar advected by the large scale Batchelor velocity is understood 0 Batchelor 59 Kraichnan 68 Shraiman, Siggia 94,95 MC,Falkovich,Kolokolov,Lebedev 95 MC,Gamba,Kolokolov 94 Balkovsky,MC,Kolokolov,Lebedev 95 Bernard,Gawedzki,Kupianen98 MC, Falkovich, Kolokolov 98 Balkovsky,Fouxon 99 t (t ) (t ) Passive linear polymer A t t t ' 1 (t ) e W (t ) (0) e W (t ' ) (t ' )dt ' 0

n (t ) First order transition: the polymer stretches indefinitly if advection exceeds diffusion n/2 1 max t W (t ) T exp (t ' )dt ' 0 n/2 exp t nn S (n , n/2 n W ln W 2t exp tS d SG 2 2 CLT for the Lyapunov exponent statistics at t 1 / (saddle point parameter) , 2 2

2 2 2 Lumley 72 Balkovsky,Fouxon,Lebedev99 PDF PDF S ' (n ) n (0) (t ) 2 (t ) a 1 2 ~ a Nonlinearity beats the stretching !! diss. scale Passive nonlinear polymer B t (t ) U ( ) (t ) dr P exp S ' r r U ' (r ) S (r ) r r S ' (r ) PDF 2 Psp exp saddle point parameter Rstr Req dr r U ' (r ) r 2 Req Rstr

Passive nonlinear chain C t i (t ) i U ( i;i 1 ) U ( i;i 1 ) linear conformations are dominant N (number of segments) >>1 is an additional saddle parameter 2 2 2 exp PGauss;U ( x )qx 2 R (2 ) Rstr eq 1/ 2 q Rstr 1/ 1 Notice the nonlinear dependance coming from the equilibration of the stretching by the nonlinearity 4 3 N 4 2 1 2 2 1 2 /

Non-Newtonian hydrodynamics of a dilute polymer solution 0 t u u u p f Navier-Stokes equation u n 3 N Scale separation N ds s F el 0 Elastic part of the stress tensor in the kinetic theory approximation Hydrod. scales >> Inter-polymer Stretched Equilibrium >> >> distance polymer length polymer length n- is the polymer solution concentration N>>1- is the dimensionless polymer length Rate of strain --- Stress Tensor

Relation t U ' ' (0) p weak elasticity (linear stretching) =>OldroydB model Wi 1 constitutive equation extremely strong elasticity (nonlinear stretching) => local relation between : and z* N / 2 0 z* dzU ' ( z )U ' ' ( z ) 0 z* yU ' ' ( y)dy z 0 z xdxU ' ' ( x) z* z* yU ' ' ( y)dy

nondeg. k k , k 2 k 2 / 2, deg. z* k the maximal tension the largest eigenvalue of Wi 1 yU ' ' ( y)dy x dzU ' ' ( z ) z n 3 N the direction of the eigenvector || || Wi U ' ' ( 0) Weissenberg number Pure Elastic Turbulence (experiment) Swirling flow between two parallel disks Groisman, Steinberg 96-99 / lam transition to turbulence d=20mm d=10mm pure solvent R2 43.6mm R 38mm

80ppm polyacrylamide+ 65% sugar+1% NaCl in water Power spectra of velocity fluctuations Wi=13 Re=0.7 Pure Elastic Turbulence (theory) t Elastic dissipation >> Viscous dissipation, Advection p f + constitutive U ( x) x 2 qx 4 2 N 3n q P dt exp it u 2 t Nonlinear diffusion equation poor-man scaling r ~ r ~ Kt qK ~ 3 r2 N n 1/ 3 2r r 2 N 3 P 3 / 2 7 / 2 ~ ~ f q

K- is the pumping amplitude of Increase of n - polymer density Inertia-elastic Turbulence (instead of conclusions) Energy containing scale ( L) r ( ) r ~ 1/ 3 / r 2 / 3 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) Viscous (Kolmogorov) scale 1/ 2 nN 3 ~ 1 3/ 2 q 1/ 2 n* N 3 ~1 3/ 2 q 3 L ~ nN / q n** ~ N / R3 R ~ N 2 / q According to Lumley69 the increase in bulk dissipation (viscosity) is accompanied by a swelling of a boundary layer, that leads to the drag reduction 1/ 8