L4 · Chapter 8

Reading elastic turbulence: the evidence for chaotic flow without inertia, and what is still open

Sandbox Physics · 2026-10-29

Add a little long-chain polymer to a thick sugar syrup. In a slow flow with a Reynolds number far below one, the flow resistance can rise to more than ten times the laminar value, the velocity fluctuates irregularly, and dye mixes within minutes. Groisman and Steinberg called this state elastic turbulence. This chapter checks the evidence piece by piece, in the order experiment, spectrum, mixing, criteria and numerical model, and ends with the questions a recent review still lists as open.

No inertia, yet more than ten times the resistance

The 2000 apparatus of Groisman and Steinberg is a cup under a rotating plate: plate radius 38 mm, cup radius 43.6 mm, liquid depth 10 mm. The solvent is a sugar solution of viscosity $0.324\ \mathrm{Pa\,s}$ ; with 80 ppm polyacrylamide added, the relaxation time is $\lambda=3.4\ \mathrm{s}$ . Near a shear rate of $1\ \mathrm{s^{-1}}$ the flow changes abruptly, at $\mathrm{Wi}=3.5$ and a Reynolds number of only about 0.3, and the transition shows hysteresis.

Beyond the transition, the stress needed to keep the plate turning reaches about 12 times that of a laminar flow of the same viscosity, and about 19 times with a 20 mm layer. A Newtonian liquid in the same apparatus needs a Reynolds number of about 20 000 to reach 19 times.

Where does the extra resistance come from? In inertial turbulence it comes from the Reynolds stress. Here the authors estimate from their velocity measurements that the Reynolds stress is less than 0.5% of the resistance, and the averaged viscous stress of the solvent is the same as in laminar flow. The only stress left to carry the increase is the elastic stress of the polymers.

\mathrm{Wi}=\frac{\lambda V}{L},\qquad \mathrm{Re}=\frac{VL}{\nu},\qquad \frac{\mathrm{Wi}}{\mathrm{Re}}=\frac{\lambda\nu}{L^{2}}

The definitions of Groisman and Steinberg (2000), with

\nu

the kinematic viscosity. The ratio does not depend on the flow speed: a more viscous liquid, a slower relaxation or a smaller apparatus makes it easier to reach strong elasticity while inertia stays negligible.

Source: A. Groisman and V. Steinberg

The spectrum says smooth but chaotic

Beyond the transition, the velocity at a point fluctuates irregularly in time. Its power spectrum follows a power law over about one decade, with an exponent of about $-3.5$ in the 2000 swirling flow; in the 2001 curved channel, converting time to space with the Taylor hypothesis, the spatial spectrum has an exponent of about $-3.3$ .

An exponent steeper than $-3$ has a direct consequence: the velocity gradients come mainly from the largest scales. The mean-square velocity gradient is the integral of $k^2E(k)$ , and with $E(k)\propto k^{-\alpha}$ and $\alpha>3$ that integral converges at high wavenumber. The flow is smooth in space and random in time. Groisman and Steinberg (2004) compare it with the part of an inertial turbulent flow below the Kolmogorov dissipation scale.

It also means that elastic turbulence is not a copy of inertial turbulence: there is no cascade of energy from large eddies to small ones, and no $k^{-5/3}$ spectrum at small scales.

\int_{k_0}^{k_{\max}}k^{2}\,k^{-\alpha}\,dk=\frac{k_0^{3-\alpha}-k_{\max}^{3-\alpha}}{\alpha-3}\;\xrightarrow{\;k_{\max}\to\infty\;}\;\frac{k_0^{3-\alpha}}{\alpha-3}\qquad(\alpha>3)

Derived in this course: when the spectrum is steeper than

k^{-3}

, the mean-square velocity gradient is set near the lowest wavenumber

k_0

Source: A. Groisman and V. Steinberg · A. Groisman and V. Steinberg · A. Groisman and V. Steinberg

Mixing three orders of magnitude faster than diffusion

The 2001 channel is 3 mm wide and made of 60 half-rings. The solution relaxes in 1.4 s, and the dye diffuses with a coefficient of about $8.5\times10^{-7}\ \mathrm{cm^2\,s^{-1}}$ . The flow stays steady up to a Weissenberg number of 3.2; at 6.7 the two streams are mixed after about 15 half-rings, about 120 s.

By molecular diffusion alone, mixing across the 3 mm width would take more than a day (see the formula). With a more viscous solvent and slower diffusion at the same Weissenberg number, the mixing is unchanged, so what does the work is the flow the polymers drive, neither inertia nor diffusion.

t_{\mathrm{diff}}=\frac{d^{2}}{D}=\frac{(0.3\ \mathrm{cm})^{2}}{8.5\times10^{-7}\ \mathrm{cm^{2}\,s^{-1}}}\approx1.1\times10^{5}\ \mathrm{s},\qquad \frac{120\ \mathrm{s}}{t_{\mathrm{diff}}}\approx1\times10^{-3}

Computed from the numbers of Groisman and Steinberg (2001); the ratio is their “three orders of magnitude”.

Source: A. Groisman and V. Steinberg

Criteria give a trend, not a universal threshold

The parameter $K$ of Chapter 6 and the Pakdel–McKinley criterion both say that curved streamlines with normal stress drive the instability. The actual transition point depends on the geometry. Groisman and Steinberg (2004) report transitions at Weissenberg numbers of 3.2 and 2 in the flow between two plates for gap-to-radius ratios of 0.26 and 0.52; about 4 in Couette–Taylor flow; and in the curved channel, stationary vortices may appear near 2 while irregular motion starts only at 3.2.

The resistance growth differs too. The paper sums it up as about 20 times between two plates, about 4.2 in Couette–Taylor flow and about 2.8 in the channel. The review of Datta and co-authors gives a local maximum of the criterion of about 4 to 5 in cross-slot simulations, but no critical values for Taylor–Couette or cone-and-plate flow.

So when you read that “the critical Weissenberg number is” some value, first ask: in which geometry, with which definition of the Weissenberg number, and with which solution?

Source: A. Groisman and V. Steinberg · S. S. Datta et al.

How far the four-roll model goes

Experiments cannot see the polymer stress field; a numerical model can. The four-roll model of Chapter 7 gives the symmetry-breaking threshold and the oscillation periods. Thomases, Shelley and Thiffeault also seeded the oscillating flow at a Weissenberg number of 10 with tracer particles, measured an effective diffusivity of about 0.025 and a finite-time Lyapunov exponent of about 0.014, and in one run with a cycling vortex saw the particles from the four quadrants approach a quarter each after t of about 3000.

This course recomputed the model on a coarser discretisation; the numbers below are numerical results of this route, not experimental data. On a $64^2$ grid with $\nu_p=10^{-3}$ the threshold is 4.86; a $128^2$ grid and half the time step also give 4.86, consistent with the papers’ 4.8 to 4.9. At a Weissenberg number of 10 the symmetry breaks near $t=210$ ; between 1000 and 2000 the probe velocity has two periods of about 288 and 56, against the paper’s 330 and 55; the probe curves of the three discretisations differ by less than a few parts in a thousand before $t=2000$ .

Stress diffusion changes the outcome. At $\nu_p=3\times10^{-3}$ the threshold rises to 5.09 and the flow at a Weissenberg number of 10 ends in an asymmetric steady state; at $10^{-2}$ the threshold is 6.64 and the symmetry does not break until $t\approx590$ . In this range, more diffusion makes breaking harder. That points the opposite way to the conclusion of Gupta and Vincenzi that diffusion creates the breaking, but they compared zero diffusion with a different force, and this course did not test the limit $\nu_p\to0$ .

One check did not pass. On $64^2$ with $\nu_p=10^{-3}$ , the smallest eigenvalue of the conformation tensor falls to about −4.4 during start-up, a sign that this grid does not resolve the thinnest stress strand; on $128^2$ it stays positive. The mixing is also only partial: after the tracers are released the mixing index settles near 0.66, and particles from the diagonally opposite quadrant make up only about 6% of quadrant one.

Source: B. Thomases · A. Gupta and D. Vincenzi · Sandbox Physics FR26

What is still open

The 2022 review by Datta and co-authors summarises a workshop discussion. Three of its points bear directly on this course.

First, constitutive models. Elastic instabilities usually appear at high Weissenberg number, where Oldroyd-B is almost never quantitative in shear or in extension, and the same holds for extensions such as FENE, Giesekus and PTT. Understanding elastic instability has to go hand in hand with better rheological models.

Second, the eigenvalue spectrum of two-dimensional elastic Taylor–Couette flow has hardly been studied, although the onset of instability has been seen there near a Weissenberg number of 10 at a large gap ratio.

Third, a Lagrangian view. Every purely elastic instability mechanism worked out so far relies on shear, in viscometric flows. Yet a single polymer in extension shows a coil–stretch transition that is sharp, hysteretic and dependent on its history. How the history along a fluid element’s path enters the instability remains to be studied. The stress-diffusion debate of Chapter 7 is another question the numerics have not yet settled.

Source: S. S. Datta et al. · A. Gupta and D. Vincenzi

Worked example: is the channel really free of inertia?

Groisman and Steinberg (2001) use $\mathrm{Wi}=\lambda\,(4Q/d^{3})$ and $\mathrm{Re}=2Q\rho/(d\eta)$ . Dividing one by the other removes the flow rate: $\mathrm{Wi}/\mathrm{Re}=2\lambda\eta/(\rho d^{2})$ .

With $\lambda=1.4\ \mathrm{s}$ , $\eta=0.198\ \mathrm{Pa\,s}$ , $\rho=1320\ \mathrm{kg\,m^{-3}}$ and $d=3\times10^{-3}\ \mathrm{m}$ , $\mathrm{Wi}/\mathrm{Re}\approx46.7$ . At $\mathrm{Wi}=3.2$ that gives $\mathrm{Re}\approx0.069$ , against the 0.06 the paper reports. The difference comes from shear thinning: the viscosity was measured at a shear rate of 4 per second.

Conclusion: at the transition, inertial forces are a few per cent of viscous forces, so the disorder can only come from elasticity.

Source: A. Groisman and V. Steinberg

Try it in the Lab

Watch the oscillating flow mix the four quadrants

The experiment opens on the oscillating flow at a Weissenberg number of 10. Release the tracers again, watch the four colours cross the quadrant boundaries, and note the mixing index after 1000 time units.
Switch to the Symmetric stress strand preset (Weissenberg number 4) and release the tracers. Does the mixing index leave zero? Explain why a symmetric flow cannot carry fluid from one quadrant into another.
Choose the Too much stress diffusion preset, which starts from rest at a Weissenberg number of 10 with ten times the published stress diffusion. See how much later the symmetry breaks and how weak the oscillation becomes, and compare with the time of about 590 given in this chapter.

Back to a single polymer

In planar extension, move the Weissenberg number from 0.4 to 0.6 and watch the stretch go from bounded to unbounded. The stress in elastic turbulence comes from fluid elements passing through such stretching regions again and again.

Exercises with explained answers

The swirling-flow apparatus of Groisman and Steinberg (2000) has a Reynolds number of about 0.3 at $\mathrm{Wi}=3.5$ . If every dimension is reduced four times, with the same solution and the same shear rate, what are the Weissenberg and Reynolds numbers?
Show the answer
The shear rate and relaxation time are unchanged, so the Weissenberg number stays 3.5. The velocity falls four times with the size and so does the length, so the Reynolds number falls sixteen times, to about 0.02. The paper reports the same resistance curve in the smaller apparatus, with velocities and Reynolds numbers lower by factors of 4 and 16.
One velocity spectrum falls as $k^{-5/3}$ at high wavenumber, another as $k^{-3.5}$ . In which one do the velocity gradients come mainly from the small scales?
Show the answer
The first. The integral of $k^2k^{-5/3}=k^{1/3}$ grows with the highest wavenumber, so the smallest scales set the gradients; the integral of $k^{-1.5}$ converges, so the largest scales do.
Why can the critical Weissenberg number of 3.2 measured in the curved channel not be compared with the four-roll threshold of 4.86?
Show the answer
The two Weissenberg numbers are defined differently: the channel uses a shear rate from the flow rate and width, the four-roll model the extension rate at the stagnation point. The geometries differ too: the four-roll cell has no walls, no through-flow, no channel width and no radius of curvature. The elastic-instability parameter also depends on the polymer share of the viscosity and the gap-to-radius ratio, and there is no universal critical value.
In the four-roll model, how would you tell whether stress diffusion creates the symmetry breaking or only delays it? Which half can this course’s data answer?
Show the answer
You need the trend of the threshold and the breaking time as the diffusion coefficient goes to zero, ideally checked with a scheme that needs no artificial diffusion. This course covers only 1e-3 to 1e-2: more diffusion raises the threshold and delays the breaking, so in that range diffusion suppresses it. The zero-diffusion limit was not tested, so the data cannot say whether diffusion changes the conclusion at smaller values.

Sources and what we recomputed

A. Groisman and V. Steinberg, Elastic turbulence in a polymer solution flow, Nature 405, 53–55 (2000); arXiv:nlin/0104052v1
arXiv version 1, read in full. Fig. 1: swirling flow between a rotating plate of radius 38 mm and a cup of radius 43.6 mm, liquid depth 10 mm, 12 °C. Methods: 65 % saccharose and 1 % NaCl solvent of viscosity 0.324 Pa s with 80 ppm polyacrylamide of molecular weight 1.8e7; solution viscosity 0.424 Pa s at a shear rate of 1 per second; relaxation time 3.4 s. Fig. 2: transition near a shear rate of 1 per second (Weissenberg number 3.5, Reynolds number about 0.3), with hysteresis; resistance up to about 12 times laminar for a 10 mm gap and 19 for 20 mm. Fig. 3: snapshots at Weissenberg number 13. Fig. 4: brightness spectra. Fig. 5: velocity spectra, a power law with exponent −3.5 at 4 per second over about one decade. Text: Reynolds stress below 0.5 % of the resistance.
A. Groisman and V. Steinberg, Efficient mixing at low Reynolds numbers using polymer additives, Nature 410, 905–908 (2001); arXiv:nlin/0104050v1
arXiv version 1, read in full. Fig. 1: a channel of width 3 mm made of 60 half-rings with radii 3 and 6 mm. Solvent viscosity 0.153 Pa s, density 1320 kg per cubic metre; 80 ppm polyacrylamide; solution viscosity 0.198 Pa s at a shear rate of 4 per second; relaxation time 1.4 s; dye diffusion coefficient about 8.5e-7 square centimetres per second. The Weissenberg and Reynolds numbers are defined from the flow rate and the channel width. The flow stays steady up to a Weissenberg number of 3.2 (Reynolds number 0.06); mixing is studied at 6.7. Fig. 2: velocity spectra as a power law with exponent −3.3 in wavenumber. Fig. 4: mixing over about 15 units, roughly 120 s, three orders of magnitude faster than the diffusion time.
A. Groisman and V. Steinberg, Elastic turbulence in curvilinear flows of polymer solutions, New Journal of Physics 6, 29 (2004)
Section 2: the elastic instability in curvilinear flows is controlled by K, built from the polymer share of the viscosity, the gap-to-radius ratio and the square of the Weissenberg number, the elastic analogue of the Taylor number; the paper gives no universal critical value of K. Section 4.2: the Couette–Taylor transition occurs near a Weissenberg number of 4, against 3.2 and 2 between two plates for gap-to-radius ratios 0.26 and 0.52; at the highest Weissenberg numbers explored the flow resistance grew by a factor of 4.2 in Couette–Taylor flow against 11 and 19 between two plates. Section 5: in the curved channel stationary vortices may appear near a Weissenberg number of 2, and irregular motion starts at 3.2 (Reynolds number 0.06); in a 100 µm microchannel the resistance grows from an estimated Weissenberg number of 1.3 and reaches a factor of 2.8 near 9. Section 6 summarises the resistance growth as factors of 20, 4.2 and 2.8 for the flow between two plates, Couette–Taylor flow and the channel.
S. S. Datta et al., Perspectives on viscoelastic flow instabilities and elastic turbulence, arXiv:2108.09841v1 (2021); Physical Review Fluids 7, 080701 (2022)
arXiv version 1, 22 August 2021; the journal version was not checked. Section II, Eq. (1): the Pakdel–McKinley criterion, which combines streamline curvature with the ratio of normal stress to shear stress. No critical value is given for Taylor–Couette or cone-and-plate flow. Section III: simulations of cross-slot flows lose symmetry or become time dependent when the largest local value of that criterion exceeds about 4 to 5; Eq. (2) and Table II write Oldroyd-B, FENE-P and other models with a conformation tensor, which allows a positive-definiteness check and log-conformation schemes; a two-dimensional Taylor–Couette simulation became chaotic near a Weissenberg number of 10. The workshop summary states that Oldroyd-B is almost never quantitative at high Weissenberg number.
B. Thomases, M. Shelley and J.-L. Thiffeault, A Stokesian viscoelastic flow: transition to oscillations and mixing, Physica D 240, 1602 (2011)
Eqs. (1)–(2) and (6): the same model with stress diffusion 1e-3 for Weissenberg numbers up to 10. Eqs. (4)–(5): the polymer strain energy balance. Section 3.1, Fig. 2: a small perturbation of the symmetric steady state decays at 4.8 and grows at 4.9; 256 by 256 grid, and doubling the resolution does not change the dynamics. Fig. 6(d): at 10 the velocity spectrum has periods of about 330 and 55. Section 5: between about 4.9 and 9 the asymmetric flow settles down again; above about 9 a vortex unpins and persistent oscillations mix the fluid; periodic at 12 and 15, aperiodic above about 20. Appendix B: stress diffusion is calibrated against FENE-P; its effect on the threshold is not reported.
A. Gupta and D. Vincenzi, Effect of polymer-stress diffusion in the numerical simulation of elastic turbulence, arXiv:1809.09648v2 (2019)
Version 2, 7 March 2019; the journal version was not checked. Two-dimensional Oldroyd-B flow driven by a cellular force at Deborah number 10, solved with a Cholesky decomposition and the Kurganov–Tadmor scheme. With the stress diffusion used by Thomases and co-workers (Schmidt number 1e3) the flow stays almost frozen for a long time and then breaks the cellular symmetry (Figs. 2–4); without diffusion it becomes chaotic quickly and keeps the symmetry of the forcing (Fig. 5). Section 3: in their setup the symmetry breaking is a spurious effect of the diffusion. Section 4: the same holds for FENE-P; for forcings that mix strain and vorticity the effect may be weaker.
Sandbox Physics FR26, four-roll elastic instability: numerical model (docs/frontier/fr26/four-roll-elastic-instability-model.md, with long-run records in docs/frontier/fr26/records/)
Model version fr26-four-roll-elastic-instability-v1. The model note states that the doubly periodic four-roll cell replaces the curved channel of the 2001 experiment: it has curved streamlines and a stretching stagnation point but no walls, no through-flow, no channel width and no radius, so channel thresholds and spectra must not be compared with it. It records the equations, the pseudo-spectral discretisation and the diagnostics shown in the Lab. Each run is stored with its grid size, time step, stress diffusion and perturbation seed. These are numerical results of this route, not experimental data.