|Victor Steinberg (2008), Scholarpedia, 3(8):5476.
|revision #73072 [link to/cite this article]
Long polymer molecules added to a fluid make it elastic and capable of storing stresses that depend on the history of deformation, thereby providing the fluid a memory. Many properties of the polymer solution flows (especially dilute ones) can be understood on the basis single polymer dynamics where the polymer experiences the combined action of the stretching by the flow and the elastic relaxation. The elastic stress created by the polymer stretching in the flow becomes the main source of nonlinearity in the polymer solution flow at low Reynolds numbers, Re. As the result, an elastic instability shows up, when the elastic energy overcomes the dissipation due to polymer relaxation. The ratio of the nonlinear elastic term to the linear relaxation is defined by the Weissenberg number, Wi. In a simple shear flow of a polymer solution the elastic stress is anisotropic and characterized by the difference in the normal stresses along the flow velocity direction and the velocity gradient direction. In a curvilinear shear flow (e.g. Couette flow between rotating cylinders) the normal stress difference gives rise to a volume force acting on the fluid in the direction of the curvature, the "hoop stress". The latter becomes the driving force for a rod climbing effect  as well as an elastic instability. The mechanism of the elastic instability in the Couette flow was first suggested in Ref.  and experimentally verified in Ref. . It was also widely investigated in other flow geometries with curvilinear trajectories, including a curvilinear channel. The streamline curvature is necessary ingredient to reduce the critical Weissenberg number for the instability onset, Wic. On the other hand, a possibility of the elastic instability in a straight channel was suggested and studied theoretically and numerically, though its experimental verification is still lacking.
Above the purely elastic instability, a path to a chaotic flow in a form of irregular flow patterns at Wi> Wic was studied in three flow geometries: Couette flow between cylinders, swirling flow between two disks, and flow in a curvilinear channel. It was reasonable to assume that at sufficiently high Wi and vanishingly small Re a random flow, which exhibits continuous velocity spectra in a wide range of temporal and spatial scales similar to a hydrodynamic turbulent flow, can be excited. Indeed, such random flow was observed first in both Couette and swirling flows of dilute polymer solutions and dubbed "elastic turbulence" . It was identified in the swirling flow between two disks due to three main features: sharp growth in flow resistance, algebraic decay of velocity power spectra over a wide range of scales, and orders of magnitude more effective way of mixing than in an ordered flow. These properties are analogous to those of hydrodynamic turbulence. Elastic turbulence in the swirling flow was observed in dilute polymer solutions of various polymer concentrations down to 7 ppm . Since the elastic energy is proportional to the polymer concentration and should exceed dissipation independent of it, one should expect the minimum threshold value of concentration below which elastic turbulence cannot be generated.
However, the similarities do not imply that the physical mechanism that underlies the two kinds of random motion is the same. Indeed, in contrast with inertial turbulence at high Reynolds numbers Re, which occurs due to large Reynolds stresses, large elastic stress is the main source of non-linearity and the cause of elastic turbulence in low Re flows of polymer solutions. One can suggest that in a random flow driven by elasticity, the elastic stress tensor \(\tau\)p should be the object of primary importance and interest, and that it may be appropriate to view elastic turbulence as turbulence of the \(\tau\)p-field. It would then be more relevant and instructive to explore the spatial structure and the temporal distribution of this field. However, currently no experimental technique allows a direct local measurement of the elastic stress in a turbulent flow. On the other hand, properties of the \(\tau\)p-field in a boundary layer were inferred from measurements of injected power, whereas its local properties were evaluated from measurements of spatial and temporal distributions of velocity gradients.
The key early experimental observation is the power-law decay of the velocity power spectra in all flow geometries with the exponent d>3 (between 3.3 and 3.6). Due to the sharp velocity spectrum decay, the velocity and velocity gradient are both determined by the integral scale, i.e., the vessel size. It means that elastic turbulence is essentially a spatially smooth random in time flow, dominated by strong nonlinear interaction of few large-scale modes. This type of random flow appears in hydrodynamic turbulence below the dissipation scale and called Batchelor flow regime. Further characterization of this random flow was conducted by measuring velocity correlation functions via particle image velocimetry.
These early experimental results initiated theoretical studies. On a molecular level, the onset of elastic turbulence is attributed to the occurrence of a coil-stretch transition in the presence of a fluctuating velocity field at Wi>1. A possibility of the polymer stretching in a random flow was suggested and discussed first by Lumley  in regards to turbulent drag reduction in hydrodynamic turbulence. The more theory predicts that at Wi>1 a dramatic change in the statistics of polymer stretching takes place, and the majority of molecules become strongly stretched up to the full polymer length . This prediction was experimentally verified recently . The coil-stretch transition has remarkable macroscopic consequences on flow: properties of the polymer solution flow become essentially non-Newtonian. Hydrodynamic description of a polymer solution flow and particularly the dynamics of elastic stresses are analogous to that of a small-scale fast dynamo in magneto-hydrodynamics (MHD) and also of turbulent advection of a passive scalar in the Batchelor regime. The stretching of the magnetic lines is similar to polymer stretching, and the difference with MHD lies in the relaxation term that replaces the diffusion term. In all three cases the basic physics is the same and rather general and directly related to the classical Batchelor regime of mixing: stretching and folding of stress field by a random advecting flow. In elastic turbulence a statistically steady state occurs due to the feedback reaction of stretched polymers (or the elastic stress) on the velocity field that leads to a saturation of \(\tau\)p even for a linear polymer relaxation model. The saturation of \(\tau\)p and therefore the rms of fluctuations of velocity gradients in an unbounded flow of a polymer solution are the key theoretical predictions . The same analysis leads to a power-like decaying spectrum for the elastic stresses and for the velocity field fluctuations with the exponent d>3 in a good accord with the experimental results.
Further experimental studies of elastic turbulence confirm theoretical predictions on the saturation of the rms of velocity gradient (or vorticity) fluctuations in unbounded flow (in a bulk flow) though the saturation level found is several times of that predicted theoretically.
But the main message of the experimental study is a surprising similarity in scaling, statistics, and structure of the elastic stresses and of passive scalar mixing in elastic turbulence in a finite size vessel, in spite of the important difference in the dynamo effect. The latter occurs due to the feedback reaction of stretched polymers (or the elastic stress) on the velocity field.
The experiments also reveal a role of elastic stresses in elastic turbulence due to presence of walls in a von Karman swirling flow between two disks. The following features of elastic turbulence are found experimentally .
- The rms of the velocity gradients (and thus the elastic stress) grows linearly with Wi in the boundary layer, near the driving disk. The rms of the velocity gradients in the boundary layer is one to two orders of magnitude larger than in the bulk suggesting that the elastic stresses are accumulated near the wall and are intermittently injected into the bulk.
- The PDFs of the injected power at either constant angular speed or torque show skewness and exponential tails, which both indicate intermittent statistical behavior. Also the PDFs of the normalized accelerations, which can be related to the statistics of the velocity gradients via the Taylor hypothesis, exhibit well-pronounced exponential tails.
- A new length scale, namely the thickness of the boundary layer, as measured from the profile of the rms of the velocity gradient, is found to be relevant for the boundary layer of the elastic stresses. The velocity boundary layer just reflects some of the features of the boundary layer of the elastic stress (rms of the velocity gradients). The measured length scale is much smaller than the vessel size.
- The scaling of the structure functions of the vorticity, velocity gradients, and injected power is found to be the same as that of a passive scalar advected by the velocity field in elastic turbulence.
These properties provide a basis for a model for the dynamics of elastic turbulence, in which elastic stress is introduced into the fluid by the driving boundary, accumulates in the boundary layer and is intermittently injected into the bulk of the flow. The situation is entirely similar to the trapping of a passive scalar in the mixing boundary layer in a finite size vessel, and to its intermittent injection into the bulk.
The non-uniform distribution of the elastic stress, which causes by non-uniform polymer stretching, is indeed verified experimentally by studying the statistics of a single polymer stretching in an elastic turbulent flow generated by the same polymers. These experiments present for the first time direct elastic stress measurements in a flow and confirm the saturation of the elastic stress in a bulk flow and the existence of boundary layer near the wall . Moreover, al larger polymer concentrations the saturation level decreases and approaches asymptotically the theoretically predicted value that is found for linear polymer elasticity. This result elucidates the relation between two theoretical mechanisms that can lead to the stress saturation: (i) feedback reaction of polymer molecules with linear elasticity on the flow, and (ii) nonlinear elasticity of polymer molecules. At higher polymer concentration, the polymer stretching reduces, and the former mechanism shows up. The concentration dependence of flow structure and of statistics of velocity and velocity gradient fields in a wide range of polymer concentrations from 80 till 3000 ppm also reveals a similar tendency: the larger the concentration, the lower the saturation level of the rms of the velocity gradients. On the other hand, fluctuations of the injected power and pressure fluctuations do not show any concentration dependence in the same range of polymer concentrations either in statistics or in scaling properties of algebraic decay of velocity power spectra and also in average and rms of the velocity fluctuations dependencies on Wi .
Analogous studies in a curvilinear channel flow and direct measurements of velocity gradient statistics by homodyne light scattering techniques are on the way. As the next step in this investigation should be direct measurements of statistics and distribution of elastic stress in elastic turbulence in a macroscopic size vessel.
- R. B. Bird, C. F. Curtiss, R. C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, (John Wiley, NY), 1987.
- R. G. Larson, E. S. G. Shaqfeh, S. J. Muller, J. Fluid Mech. 218, 573 (1990).
- A. Groisman and V. Steinberg, Phys. Fluids 10, 2451 (1998).
- A. Groisman and V. Steinberg, Nature 405, 53 (2000).
- J. Lumley, Symp. Math. 9, 315 (1972).
- E. Balkovsky, A. Fouxon, and V. Lebedev, Phys. Rev. Lett. 84, 4765 (2000); M.Chertkov, Phys. Rev. Lett. 84, 4761 (2000).
- S. Gerashchenko, C. Chevallard, V. Steinberg, Europhys. Lett. 71, 221(2005).
- A. Fouxon and V. Lebedev, Phys. Fluids 15, 2060 (2003).
- T. Burghelea, E. Segre, V. Steinberg, Phys. Rev. Lett. 96, 214502 (2006), Phys. Fluids 19, 053104 (2007).
- Y. Liu and V. Steinberg, submitted
- Y. Jun and V. Steinberg, submitted.
- Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
- Søren Bertil F. Dorch (2007) Magnetohydrodynamics. Scholarpedia, 2(4):2295.
- Howard Eichenbaum (2008) Memory. Scholarpedia, 3(3):1747.
- Cesar A. Hidalgo R. and Albert-Laszlo Barabasi (2008) Scale-free networks. Scholarpedia, 3(1):1716.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.