Stretching of Polymers r Superimposed on

Prasad Perlekar

Description

Stretching of Polymers r 2 Superimposed on ? (16384 2 ) Where : ? ? (? 2 ? ? 2 )/8; ? > 0 ? vortical region, ? < 0 ? strain-dominated region. ? 2 ? ∑ i ,j ?ij?ij and ?ij = ∂iuj + ∂jui . ! Polymers stretch predominantly in strain-dominated regions. Two dimensional turbulence with polymer additives

• kolmogorov-like scaling


• cc liquid


• clearly ob- served


• yag laser slaved


• coherent structures


• soap film


• linear polymer


• polymers stretch

Sujets

• Soap bubble

Informations

• Publié par: pefav
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

2 2Stretching of Polymers r Superimposed on ! (16384 )
Two dimensional turbulence
with polymer additives
2 2Where : !! (! "" )/8;
!> 0# vortical region, !< 0# strain-dominated region.!
2" ! " " and " =# u +# u .ij ij ij i j j ii,j
! Polymers stretch predominantly in strain-dominated regions.Acknowledgement
1.  Dhrubaditya Mitra

2.  Dario Vincenzi

3.  Roberto Benzi 2d turbulence
2∂ ω+u ·∇ω= ν∇ ω−µω+Ft ω
−5/3E(k)∼k
How polymer additives affect
forward and inverse cascade?
µ
−3+δ(µ)E(k)∼k
Perlekar et al., PRL (2011); Ray et al., PRL (2011);
Wiki: Linear polymer molecule Boffetta et al., ARFM (2012). 2d turbulence: Topological structures
2 2Λ=(ω −σ )/4
How polymer additives affect
the topological properties?
Expts: Daniel and Rutgers, PRL (2002);
Simulations: Perlekar and Pandit, NJP (2009). Soap-film experiment-1/4
week endingPHYSICAL REVIEW LETTERS
PRL 96, 024502 (2006) 20 JANUARY 2006
2/3
100
l
inj
1.810
4
2
<# >
2
3 <" >
1
0 5 10 15 20 25
! (ppm)
0.1 1
*FIG. 1. Experimental setup. A voltage difference V!V ) l (cm)
)V is applied to the film generating a uniform current densityJ.
Beneath the film is a set of bar magnets with alternating poles. FIG. 2 (color online). The second-order structure function
S #l$. In increasing order of turbulence intensity, the curves2
correspond to "! 24, 12, 9, 3, and 0 ppm. The inset is theKolmogorov forcing generates turbulence in soap-films. goes a sequence of instabilities and becomes turbulent
2 2plot of h' i (circles) and h! i (squares) vs".
whenV>20 V. The soap solution was made of a mixture
of four components (5 cc liquid detergent, 80 g ammonium
Jun et al., PRL, 96, 024502 (2006) chloride, 40 cc glycerol, and 400 cc distilled water). Linear becomes flat forl>l ,indicatingthetruncationofenergyinj
6polymers (polyethylene-oxide, M ! 8"10 , R ’ transfer to these large scales. The blockage of energy tow g
0:4 !m) of varying concentrations (0<"<25 ppm) large scales can also be seen by the decreasing total kinetic
2were used. Within this concentration range, there is no energy v =2, which is the asymptotic value of S #l$ forrms 2
2overlap between polymer coils as evidenced by a small l' l . Figure 2 shows thatv =2 drops sharply for"(inj rms
" dependence of the kinematic viscosity of the soap solu- 10 ppm.
2tion, which we determined to be# ’ 0:02 cm =s. To mea- The abrupt change of turbulent behavior when " in-
sure the velocity field v~#x~$, the film was seeded with creases is suggestive, indicating that there may be a critical
3hollow glass spheres (diam! 10 !m, $! 1:05 g=cm ). polymer concentration " #’ 10 ppm$ for quenching tur-C
A 12 mJ double-pulsed Nd:YAG laser slaved to a CCD bulence. The observation prompted us to examine other
camera (Redlake,1016"1008 pixels) was used to illumi- signatures that may be used to quantify the effect. One of
2nate the soap film. Images (4:5"4:5 cm ) were acquired the prominent features of 2D turbulence is the coherent
at the center of the soap film at 30 fps, yielding typically structures, such as vortices and saddle points in the flow. In
410 vectors per velocity field. a previous study [10], we investigated the distribution of
1 2 2In the following discussion, five different polymer con- centers and saddles via the quantity !! #' )! $,2
2 2centrations "! 0;3;9;12;24 ppm were used and the en- which is related to the pressure by)r p!!. Here! !
2 3 1 2 1ergy injection rate " ! 201:4 cm =s was kept fixed byinj #@v )@ v$ and ' ! #@v *@ v$ characterize thei j j i i j j i2 2
maintaining a constantV across the film. Figure 2 shows a center and the saddle structures in the flow with the re-
2set of second-order structure functionsS #l$!h%v i mea-2 peated indices implying summations. Statistical distribu-l
2 2sured using different ", where %v is the longitudinall tions of! and' were measured for different" and their
velocity difference on scale l. We found that in all cases probability density functions (pdf) are displayed in Fig. 3.
2 2there is a well-developed enstrophy range (l<l ) where It is shown that for "<" , P#! $ and P#' $ are unaf-inj C
1:8%0:2S #l$ /l . This scaling relation agrees reasonably fected by ", but they become significantly narrower for2
2well with the theoretical prediction S #l$ /l and persists ">" , indicating that strong centers and saddles are2 C
down to the smallest scale (&300 !m) resolvable by the suppressed. The effect is represented by the inset of
2 2PIV. Aside from small changes in the amplitude, the poly- Fig. 2 where h' i and h! i vs " are plotted. We noted
mer appears to have no effect on this scaling behavior. For that in all cases of different",the‘‘topologicalcharges,’’
large scales (l>l ), two classes of behaviors can be averaged over space, are not strictly conserved. The dif-inj
2 2ferences between h' i and h! i result from the film beingidentified: (a) For 0<"<10 ppm, S #l$ increases with2
slightly compressible. Since polymers are mostly de-l and is reminiscent of an inverse energy cascade. Despite a
large Taylor-microscale Reynolds number Re ’ 153, the formed by saddles, it is instructive to compare the&
2=3 distribution of the strain rate ' with the Zimm relaxationKolmogorov-like scaling S #l$ /l was not clearly ob-2
3time(!)R =#k T$ of the polymer. For our system withserved due to the limited inertial range. In spite of this g B
) ’ 0:02 cP and R ’ 0:4 !m, we found ( ’ 16 ms orshortcoming, the magnitude of S #l$ was observed to de- g2
2 3 )2crease as " was increased. (b) For ">10 ppm, S #l$ 1=( ’ 3:91"10 s , which is delineated as the vertical2
024502-2
2 2
S (l) (cm /s )
2
2 2 3 -2
<" >,<# > ($10 s )Soap-film experiment-2/4
week endingPHYSICAL REVIEW LETTERSPRL 96, 024502 (2006) 20 JANUARY 2006
polymer is present, the situations are somewhat different;200
v increases initially, levels off, and then increases again.rms
It forms a plateau for a small range of V between 50 and(a)
150 " 55 volts. This measurement suggests that there exist twop
thresholdsV andV marked by two arrows in the figure.C1 C2
100 We believed that the lower threshold V corresponds toC1
the onset of the turbulent suppression and the higher"
#50 threshold V corresponds to the saturation of the elasticC2
" field. However, for the entire range of V, the measured$
3energy transfer rate (/h$v i) remains positive, indicating0 l0 10 205 15 25 an inverse energy cascade but with a reduced transfer rate
! (ppm) when the polymer is present.
To summarize, the polymer effects on forced 2D turbu-16
lence in freely suspended soap films were investigated
quantitatively using two independent control parameters
14 ! and" . The measurement shows that when" is fixed,inj inj
turbulent suppression has a sharp threshold! !’ 10 ppm".C
12 However, when! is fixed, two thresholds can be identified,
but the transitions in this case are much weaker. We found(b)10 that turbulent suppression occurs concurrently with theV V
C1 C2 elimination of strong saddles. Inspection of Fig. 3(a) re-
veals that those saddles that are eliminated have strength45 50 55 60 65
2 2determined precisely by the relation% * 1=& , indicatingV (volts)
that the time criterion is strictly obeyed in the experiment.
Since polymer-turbulence interactions are primarily viaFIG. 5 (color online). (a) The energy budget vs!. (b) v vsrms
saddles and the weakening of saddles by polymer stretch-V for !$ 0 (circles) and for!$ 15 ppm (squares).
ing has the drastic effect of quenching turbulence, it sug-
gests that this hydrodynamic structure may play a role in
of turbulence and is whereS !l" levels off (see Fig. 2) [12].2 transferring energy from scale to scale. It remains an
An appreciable amount of energy (#30%) is also trans- intriguing possibility that the same mechanism operates
ferred to small scales and consumed by molecular viscos- in 3D as well as in 2D turbulence.
ity. In the absence of polymer, this partition of energy on We acknowledge helpful discussion with Dr. B.
small and large scales is consistent with a previous study Eckhardt. This work is supported by the NSF under
[9]. It is interesting to note that when ! is increased, the Grant No. DMR-0242284.
fraction of energy consumed by the fluid viscosity remains
almost constant [see the heights of the dark-hatched area in
Fig. 5(a)] until! crosses! , where " suffers a jump ofC "
#28:7%. The effect is more dramatic for the energy trans-
[1] B. A. Toms, in Proceedings of the International Congressfer to large scales as indicated by the heights of the light-
on Rheology (North-Holland, Amsterdam, 1949).hatched area in the same figure. Here one observes that"#
[2] J. L. Lumley, J. Polym. Sci. 7, 263 (1973).keeps decreasing with ! and drops precipitously at ! .C
[3] T. Tabor and P.-G. de Gennes, Europhys. Lett. 2, 519Such a strong ! dependence is due to the significant
(1986).change in v when the polymer was introduced into therms [4] R. Benzi, E. De Angelis, R. Govindarajan, and I.
flow as seen in Fig. 2. Since" is constant, it follows thatinj Pr