1. 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
/japansmall.ps

2. 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 =

3. 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

4. Passive nonlinear
polymer B
saddle point parameter
PDF

5. 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

6. 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

7. 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.

8. 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

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

10. 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