Abstract
Speed of sound waves in gases and liquids are governed by the compressibility of the medium. There exists another type of nondispersive wave where the wave speed depends on stress instead of elasticity of the medium. A wellknown example is the Alfven wave, which propagates through plasma permeated by a magnetic field with the speed determined by magnetic tension. An elastic analogue of Alfven waves has been predicted in a flow of dilute polymer solution where the elastic stress of the stretching polymers determines the elastic wave speed. Here we present quantitative evidence of elastic Alfven waves in elastic turbulence of a viscoelastic creeping flow between two obstacles in channel flow. The key finding in the experimental proof is a nonlinear dependence of the elastic wave speed c_{el} on the Weissenberg number Wi, which deviates from predictions based on a model of linear polymer elasticity.
Introduction
A small addition of longchain, flexible, polymer molecules strongly affects both laminar and turbulent flows of Newtonian fluid. In the former case, elastic instabilities and elastic turbulence (ET)^{1,2,3,4,5} are observed at Reynolds number \({\mathrm{Re}} \ll 1\) and Weissenberg number \({\mathrm{Wi}} \gg 1\), whereas in the latter, turbulent drag reduction (TDR) at \({\mathrm{Re}} \gg 1\) and \({\mathrm{Wi}} \gg 1\) has been found about 70 years ago but its mechanism is still under active investigation^{6}. Here both Re = ρUD/η and Wi = λU/D are defined via the mean fluid speed U and the vessel size D, and ρ, η are the density and the dynamic viscosity of the fluid, respectively, and λ is the longest polymer relaxation time. ET is a chaotic, inertialess flow driven solely by nonlinear elastic stress generated by polymers stretched by the flow, which is strongly modified by a feedback reaction of elastic stresses^{7}. The only theory of ET based on a model of polymers with linear elasticity predicts elastic waves that are strongly attenuated in ET, but elastic waves may play a key role in modifying velocity power spectra in TDR^{7,8}. Using the NavierStokes equation and the equation for the elastic stresses in uniaxial form of the stress tensor approximation, one can write the polymer hydrodynamic equations in the form of the magnetohydrodynamic (MHD) equations^{8}. Then, by analogy with the Alfven waves in MHD^{9,10}, one gets the elastic wave linear dispersion relation as \(\omega = ({\bf{k}} \cdot \hat n)\left[ {tr\left( {\sigma _{ij}} \right){\mathrm{/}}\rho } \right]^{1/2}\) with the elastic wave speed^{7,8} c_{el} = [tr(σ_{ij})/ρ]^{1/2}, where ω and k are frequency and wavevector, respectively, σ_{ij} is the elastic stress tensor, and \(\hat n\) is the major stretching direction, similar to the director in nematics. Such an evident difference between the elastic stress tensor characterized by the director and the magnetic field that is the vector, however, does not alter the similarity between the elastic and Alfven waves, since only uniaxial stretching independent of a certain direction is a necessary condition for the wave propagation determined by the stress value^{7}.
A simple physical explanation of both the Alfven and elastic waves can be drawn from an analogy of the response of either magnetic or elastic tension on transverse perturbations and an elastic string when plucked. As in the case of elastic string, the director is sufficient to define the alignment of the stress. Thus, to excite either Alfven or elastic waves the perturbations should be transverse to the propagation direction, unlike longitudinal sound waves in plasma, gas, and fluid media^{11}. The detection of the elastic waves is of great importance for a further understanding of ET mechanism and TDR, where turbulent velocity power spectra get modified according to ref. ^{7}. Moreover, c_{el} provides unique information about the elastic stresses, whereas the wave amplitude is proportional to the transversal perturbations, both of which are experimentally unavailable otherwise^{8}.
Numerical simulations of a twodimensional Kolmogorov flow of a viscoelastic fluid with periodic boundary conditions reveal filamented patterns in both velocity and stress fields of ET^{12}. These patterns propagate along the mean flow direction in a wavy manner with a speed c_{el} ≃ U/2, nearly independent of Wi. In subsequent studies, extensive threedimensional Lagrangian simulations of a viscoelastic flow in a wallbounded channel with a closely spaced array of obstacles show transition to a timedependent flow, which resembles the elastic waves^{13}. Further, the elastic stress field around the obstacles demonstrates similar traveling filamental structures^{12,13} in ET, interpreted as elastic waves^{7,8}. However, in both studies neither the linear dispersion relation nor the dependence of wave speed c_{el} on elastic stress–primary signatures of the elastic waves–were examined. Moreover, c_{el} was found to be close to the flow velocity, contradicting the theory^{7,8}. Strikingly, an indication of the elastic waves, in numerical studies, originates from observed frequency peaks in the velocity power spectra above the elastic instability^{12,13}. Analogous frequency peaks in the power spectra of velocity and absolute pressure fluctuations above the instability were also reported in experiments of a wallbounded channel flow in a creeping viscoelastic fluid, obstructed by either a periodic array of obstacles^{14} or two widelyspaced cylinders^{15,16}. These observations were in agreement with numerical simulations^{17} and were associated with noisy crossstream oscillations of a pair of vortices engendered due to breaking of timereversal symmetry.
Our early attempts to excite the elastic waves both in a curvilinear flow and in an elongation flow of polymer solutions at \({\mathrm{Re}} \ll 1\) were unsuccessful^{18}. In the ET regime of the curvilinear channel flow, either an excitation amplitude was insufficient and/or an excitation frequency was too high. The reason we chose the elongation flow, realized in a crossslot microfluidic device, is a strong polymer stretching in a welldefined direction along the flow. However, the elongation flow generated in the crossslot geometry has the highest elastic stresses in a central vertical plane parallel to the flow in the outlet channels–analogous to a stretched vertical elastic membrane. The transverse periodic perturbations in the experiment were applied in a crossstream direction from the top wall^{18}, however a more effective method would be to perturb it in a spanwise direction that was difficult to realize in a microchannel. A higher frequency range of perturbations, compared to that found in the current experiment, was used that lead to the wave excitation with wave numbers in the range of high dissipation.
Here we report evidence of elastic waves observed in elastic turbulence of a dilute polymer solution flow in a wake between two widelyspaced obstacles, hindering a channel flow. The central finding in the experimental proof of the elastic wave observation is a powerlaw dependence of c_{el} on Wi, which deviates from the prediction based on a model of linear polymer elasticity^{7}. The distinctive feature of the current flow geometry is a twodimensional nature of the ET flow, in the midplane of the device, in contrast to other flow geometries studied earlier.
Results
Flow structure and elastic turbulence
The schematic of the experimental setup is shown in Fig. 1, where twowidely spaced obstacles hinder the channel flow of a dilute polymer solution (see Methods section for the experimental setup, solution preparation and its characterization). The main feature of the flow geometry used is the occurrence of a pair of quasitwodimensional counterrotating elongated vortices, in the region between the obstacles, as a result of the elastic instability^{15} at \({\mathrm{Re}} \ll 1\) and Wi > 1; \({\mathrm{Re}} = 2R\bar u\rho {\mathrm{/}}\eta\) and \({\mathrm{Wi}} = \lambda \bar u{\mathrm{/}}2R\), where obstacles’ diameter 2R and average flow speed \(\bar u_{}^{}\) are defined in Methods section. The frequency power spectra of crossstream velocity v fluctuations show oscillatory peaks at low frequencies^{15,16} below λ^{−1}. Above the elastic instability, the main peak frequency f_{p} grows linearly with Wi, characteristic to the Hopf bifurcation^{15}. The two vortices form two mixing layers with a nonuniform shear velocity profile and with further increase of Wi their dynamics become chaotic, exhibiting ET properties, with vigorous perturbations that intermittently destroy vortices^{16} and seemingly excite the elastic waves. The ET flow in the region between the obstacles is shown through longexposure particle streaks imaging in Supplementary Movies 1–3 for three different Wi.
Characterization of low frequency oscillations
To investigate the nature of these oscillations we present time series of the streamwise u(t) and crossstream v(t) velocity components and their temporal autocorrelation functions A(u) = 〈u(t)u(t + τ)〉_{t}/〈u(t)^{2}〉_{t} and A(v) = 〈v(t)v(t + τ)〉_{t}/〈v(t)^{2}〉_{t} in Fig. 2a–d. Distinct oscillations in v(t) contrary to weak noisy oscillations in u(t) indicate flow anisotropy. Further, the crossstream velocity power spectra S_{f} (v) as a function of normalized frequency λf for five Wi values in the ET regime are shown in loglin and loglog coordinates in Fig. 3a, b, respectively. The power spectra S_{f} (v) exhibit the oscillation peaks at low frequencies up to λf ≤ 40 with an exponential decay of the peak values (Fig. 3a). These low frequency oscillations look much more pronounced on a linear scale (Supplementary Fig. 1a). Further, these oscillations are also observed in the power spectra of pressure fluctuations S(P) versus λf, though not so regular (Supplementary Fig. 1b). The exponential decay of S_{f} (v) at λf ≤ 40 implies that only a single frequency (or time) scale is identified for each Wi (Fig. 3a). This frequency f_{d}, for each Wi, is obtained by an exponential fit to the data, i.e., S_{f} (v) ~ exp(−f/f_{d}). The variation of f_{d} with Wi is shown in the inset in Fig. 3b; it varies from 0.7 to 2.5 Hz in the range of Wi from 75 to 200, which is comparable to oscillation peak frequency f_{p} (Fig. 4) and larger than λ^{−1}. Strikingly, on normalization of f with f_{d} for each Wi, S_{f} (v) for all Wi collapse on to each other (Fig. 3b). At higher frequencies up to λf ≤ 100, S_{f} (v) decay as the powerlaw with the exponent α_{f} = −3.4 ± 0.1 typical for ET^{5} (Fig. 3c). Contrary to a general case, where the powerlaw decay of S_{f} (v) corresponding to ET^{3,4,5} commences at λf ≈ 1, the low frequency oscillations cause the powerlaw spectra start to decay at higher frequencies 10 < λf < 40, perhaps due to an additional mechanism of energy pumping into ET associated with the low frequency oscillations. In addition, S(P) exhibit the power spectra decay in the high frequency range 10 < λf < 100 with the exponent close to −3 (see the bottom inset in Fig. 2 in ref. ^{16}), characteristic to the ET regime^{19}. It should be emphasized that the power spectra of the streamwise velocity S_{f} (u) do not show the low frequency oscillations and decays with a powerlaw exponent α ≤ 2.
Figure 4 shows the dependence of f_{p} in a wide range of Wi. The first elastic instability, characterized as the Hopf bifurcation, occurs at low Wi, where f_{p} grows linearly with Wi–in accord with our early results^{15}. At higher Wi in the ET regime, f_{p}(Wi) dependence becomes nonlinear at Wi ≥ 60. In the inset in Fig. 4, we present the same data for f_{p} as a function of Wi_{int}. Here, the Weissenberg number of the interobstacle velocity field is defined as \({\mathrm{Wi}}_{{\mathrm{int}}} = \lambda \dot \gamma\) and \(\dot \gamma \left( { = \left\langle {\partial u{\mathrm{/}}\partial y} \right\rangle _t} \right)\) is the timeaveraged shearrate in the crossstream direction in the interobstacle flow region. The parameter Wi_{int} is relevant to the description of elastic waves in ET flow between the obstacles’ region. The inset in Fig. 5b shows a linear dependence of Wi_{int} on Wi.
Dependence of elastic wave speed on Wi_{int}
Figure 5a shows a family of temporal crosscorrelation functions C_{v}(Δx, τ) = 〈v(x, t)v(x + Δx, t + τ)〉_{t}/〈v(x, t)v(x + Δx, t)〉_{t} of v between two spatially separated points, with their distance being Δx, located on a horizontal line at y/R = 0.18 for Wi = 148.4. A gaussian fit to C_{v}(Δx, τ) in the vicinity of τ = 0 yields the peak value τ_{p} at a given Δx. A linear dependence of Δx on τ_{p} (e.g., Fig. 5a inset for Wi = 148.4) provides the perturbation propagation velocity as c_{el} = Δx/τ_{p}. The variation of c_{el} as a function of Wi_{int} is presented in Fig. 5b together with nonlinear fit of the form \(c_{{\mathrm{el}}} = A\left( {{\mathrm{Wi}}_{{\mathrm{int}}}  {\mathrm{Wi}}_{{\mathrm{int}}}^{\mathrm{c}}} \right)^\beta\), where A = 8.9 ± 1.2 mm s^{−1}, β = 0.73 ± 0.12, and onset value \({\mathrm{Wi}}_{{\mathrm{int}}}^{\mathrm{c}} = 1.75 \pm 0.2\). The same data of c_{el} is plotted against Wi (see Supplementary Fig. 3) and fitted as c_{el} ~ (Wi − Wi_{c})^{β} that yields the onset value Wi_{c} = 59.7 ± 1.8.
Discussion
In the light of the predictions^{7}, it is surprising to observe the elastic waves in the ET regime due to their anticipated strong attenuation. An estimate of the wave number k = ω/c_{el} = 2πf_{p}/c_{el} from c_{el} (Fig. 5b) and f_{p} (Fig. 4) provides k in the range between 0.63 and 1.3 mm^{−1} (Supplementary Fig. 2). The corresponding wavelengths (~2π/k) are significantly larger than the interobstacle spacing e − 2R = 0.7 mm. The spatial velocity power spectra S_{k} is limited by a size of the observation window of about 0.7 mm that gives k_{x} ≈ 9 mm^{−1}, much larger than the wave numbers calculated above. Thus, the low k_{x} part of S_{k}(v), where the elastic wave peaks can be anticipated, is not resolved by the spatial velocity spectra (Supplementary Fig. 4b). The powerlaw decay with α_{k} ≈ −3.3 is found at low k_{x} followed by a bottleneck part and a consequent gradual powerlaw decay with an exponent ~−0.5 at higher k_{x} (Supplementary Fig. 4b), unlike S_{f} (v), where the peaks appear at low f and the steep powerlaw decay with the exponent α_{f} = −3.4 at higher f (see Fig. 3b). The spatial streamwise velocity power spectra S_{k}(u), obtained at the same Wi and near the center line y/R = 0.01, are similar to S_{k}(v) at low k_{x} and decays gradually with exponent ~−0.3 at higher k_{x} (Supplementary Fig. 4a).
The observed nonlinear dependence of c_{el} on Wi_{int} differs from the theoretical prediction based on the OldroydB model^{7,8}. The expression for the elastic wave speed in the model^{20} gives c_{el} = [tr(σ_{ij})/ρ]^{1/2} ≈ (N_{1}/ρ)^{1/2}, where \(N_1 = 2{\mathrm{Wi}}_{{\mathrm{int}}}^2\eta {\mathrm{/}}\lambda\) is the first normal stress difference. Then one obtains c_{el} = (2η/ρλ)^{1/2}Wi_{int}. First, c_{el} is proportional to Wi_{int} and second, the coefficient in the expression for the parameters used in the experiment is estimated to be (2η/ρλ)^{1/2} = 4.5 mm s^{−1}. Taking into account that the model^{7,8} and the estimate of elastic stress are based on linear polymer elasticity^{20}, whereas in experiments polymers in ET flow are stretched far beyond the linear limit^{21}, thus it is not surprising to find the quantitative discrepancies between them. Indeed, the value of the coefficient found from the fit (8.9 mm s^{−1}) and estimated theoretical value (4.5 mm s^{−1}) differ almost by a factor of two (see Fig. 5b). Moreover, for the maximal value of c_{el} = 17 mm s^{−1} (at Wi_{int} ≈ 4) obtained in the experiment, an estimate of elastic stress gives \(\left\langle \sigma \right\rangle = c_{{\mathrm{el}}}^2\rho = 0.37\,{\mathrm{Pa}}\) that is lower but comparable with 〈σ〉 ≈ 1 Pa obtained from the experiment on stretching of a single polymer T4DNA molecule at similar concentrations^{21}. Thus, both the c_{el} dependence on Wi_{int} and the coefficient value indicate that the OldroydB model based on linear polymer elasticity cannot quantitatively describe the elastic wave speed and so the elastic stresses. Another aspect of this result is the Mach number \({\mathrm{Ma}} \equiv \bar u{\mathrm{/}}c_{{\mathrm{el}}}\); the maximum value achieved in the experiment is \({\mathrm{Ma}}_{{\mathrm{max}}} = \bar u_{{\mathrm{max}}}{\mathrm{/}}c_{{\mathrm{el}}} \approx 0.3\), contrast to what is claimed in refs.^{22,23} due to a wrong definition based on the elasticity El = Wi/Re instead of elastic stress σ used for the estimation of c_{el} and Ma.
We discuss two possible reasons related to the detection of the elastic waves. As indicated in the introduction, the key feature of the current geometry is a twodimensional nature of the chaotic flow, at least in the midplane of the device (see Fig. 4SM in Supplemental Material of ref. ^{16}), that makes it analogous to a stretched elastic membrane. This flow structure is different from threedimensional elastic turbulence in other studied flow geometries and thus may explain the failure in the earlier attempts to observe the elastic waves. Another qualitative discrepancy with the theory^{7,8} is the predicted strong attenuation of the elastic waves in ET. Below we estimate the range of the wave numbers with low attenuation for the elastic waves and compare with the observed values.
There are two mechanisms of the elastic wave attenuation, namely polymer (or elastic stress) relaxation and viscous dissipation^{7,8}. The former has scaleindependent attenuation λ^{−1}, which at the weak attenuation satisfies the relation ωλ > 1, and the latter provides low attenuation^{24} at ηk^{2}/ρω < 1. The first condition leads to ks > 1, where s = Wi_{int}(2ηλ/ρ)^{1/2} that provides a minimum wave number in the ET regime as k_{min} > s^{−1} = 6.3 × 10^{−3} mm^{−1} for Wi_{int} = 4. The maximum value k_{max} follows from the second condition that gives kΛ < 1 at Λ = (Wi_{int})^{−1}(ηλ/2ρ)^{1/2}. Thus, one obtains in the ET regime k_{max} < Λ^{−1} = 0.2 mm^{−1} for Wi_{int} = 4 and therefore, the range of the wave numbers with the low attenuation is rather broad 6.3 × 10^{−3} < k < 0.2 mm^{−1} and lies far outside of the krange of S_{k}(u) and S_{k}(v) presented in Supplementary Fig. 4, where the range of the wave numbers of the elastic waves is not resolved. However, the range of the observed wave number 0.63 ≤ k ≤ 1.3 mm^{−1} of the elastic waves, shown in Supplementary Fig. 2, is sufficiently close to the estimated upper bound of k.
Methods
Experimental setup
The experiments are conducted in a linear channel of L × w × h = 45 × 2.5 × 1 mm^{3}, shown schematically in Fig. 1. The channel is prepared from transparent acrylic glass (PMMA). The fluid flow is hindered by two cylindrical obstacles of 2R = 0.30 mm made of stainless steel separated by a distance of e = 1 mm and embedded at the center of the channel. Thus the geometrical parameters of the device are 2R/w = 0.12, h/w = 0.4 and e/2R = 3.3 (see Fig. 1). The longitudinal and transverse coordinates of the channel are x and y, respectively, with (x, y) = (0, 0) lies at the center of the upstream cylinder. The fluid is driven by N_{2} gas at a pressure up to ~10 psi and is injected via an inlet into the channel.
Preparation and characterization of polymer solution
As a working fluid, a dilute polymer solution of high molecular weight polyacrylamide (PAAm, M_{w} = 18 MDa; Polysciences) at concentration c = 80 ppm (c/c^{*} ≃ 0.4, where c^{*} = 200 ppm is the overlap concentration for the polymer used^{25}) is prepared using a watersucrose solvent with sucrose weight fraction of 60%. The solvent viscosity, η_{s}, at 20 °C is measured to be 100 mPa · s in a commercial rheometer (AR1000; TA Instruments). An addition of the polymer to the solvent increases the solution viscosity, η, of about 30%. The stressrelaxation method^{25} is employed to obtain longest relaxation time (λ) of the solution and it yields λ = 10 ± 0.5 s.
Flow discharge measurement
The fluid exiting the channel outlet is weighed instantaneously W(t) as a function of time t by a PCinterfaced balance (BA210S, Sartorius) with a sampling rate of 5 Hz and a resolution of 0.1 mg. The timeaveraged fluid discharge rate \(\bar Q\) is estimated as \(\overline {{\mathrm{\Delta }}W{\mathrm{/\Delta }}t}\). Thus, Weissenberg and Reynolds numbers are defined as \({\mathrm{Wi}} = \lambda \bar u{\mathrm{/}}2R\) and \({\mathrm{Re}} = 2R\bar u\rho {\mathrm{/}}\eta\), respectively; here \(\bar u = \bar Q{\mathrm{/}}\rho wh\) and fluid density ρ = 1286 Kg m^{−3}.
Imaging system
For flow visualization, the solution is seeded with fluorescent particles of diameter 1 μm (Fluoresbrite YG, Polysciences). The region between the obstacles is imaged in the midplane via a microscope (Olympus IX70), illuminated uniformly with LED (Luxeon Rebel) at 447.5 nm wavelength, and two CCD cameras attached to the microscope: (i) GX1920 Prosilica with a spatial resolution 1000 × 500 pixel at a rate of 65 fps and (ii) a high resolution CCD camera XIMEA MC124CG with a spatial resolution 4000 × 2200 pixel at a rate of 35 fps, are used to acquire images with high temporal and spatial resolutions, respectively. We perform micro particle image velocimetry^{26} (μPIV) to obtain the spatiallyresolved velocity field U = (u, v) in the region between the cylinders. Interrogation windows of 16 × 16 pixel^{2} (26 × 26 μm^{2}) for high temporal resolution images and 64 × 64 pixel^{2} (10 × 10 μm^{2}) for high spatial resolution images, with 50% overlap are chosen to procure U.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank Guy Han and Yuri Burnishev for technical support. A.V. acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No. 754411. This work was partially supported by the Israel Science Foundation (ISF; grant #882/15) and the Binational USAIsrael Foundation (BSF; grant #2016145).
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A.V. and V.S. designed the experiment. A.V. performed the measurements and together with V.S. analyzed the data. Both authors discussed the results and wrote the manuscript.
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Correspondence to Atul Varshney or Victor Steinberg.
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Further reading

Threedimensional viscoelastic instabilities in microchannels
Journal of Fluid Mechanics (2019)
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