---
title: "Complex fluids: from the conformation tensor to elastic instability, extensional flow and elastic turbulence"
description: "Why a polymer solution flowing too slowly for inertia can still turn unsteady: stress relaxation, coil stretch, extensional flow and elastic turbulence."
canonical_url: "https://sandboxphysics.com/stories/complex-fluids/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/complex-fluids/"
last_modified: "2026-10-29"
---

[Complex fluids](https://sandboxphysics.com/stories/complex-fluids/)

FR26 · An interactive course

# Complex fluids: from the conformation tensor to elastic instability, extensional flow and elastic turbulence

Sandbox Physics · 2026-10-29

A viscous sugar solution flowing slowly through a curved channel stays smooth. Add 80 parts per million of a long polymer and, at the same tiny Reynolds number, the flow can turn irregular, its resistance can rise severalfold, and dye mixes in minutes instead of a day. The cause is elastic stress stored in stretched molecules. This course builds that stress from a spring and a dashpot up to the models used in current research, and ends with what remains unsettled.

[Start with stress relaxation](https://sandboxphysics.com/stories/complex-fluids/stress-relaxation-and-creep/) Open an experiment

Original schematic. In simple shear (left) the velocity grows across the gap; the flow stretches a coil and also turns it, so the coil tumbles. In planar extension (right) fluid arrives along one axis and leaves along the other; a coil held at the stagnation point is stretched without being turned.

## From a spring and a dashpot to elastic turbulence

The first chapter gives a liquid a memory with one spring and one dashpot and defines the Weissenberg number. The second replaces each polymer by two beads on a spring, which turns the memory into a conformation tensor and yields the Oldroyd-B model. The third shows why extensional flow stretches coils that shear only tumbles, and the fourth gives the chains a finite length. The fifth uses that length to read a relaxation time off a thinning filament. The sixth turns normal stresses along curved streamlines into an instability criterion. The seventh follows a computer model of a single stagnation point, where the stress concentrates into a thin strand and the flow loses its symmetry. The last chapter reads the 2000 experiment that named elastic turbulence and the 2001 mixing experiment that followed, and separates what they measured from what later simulations claim.

You need calculus, matrices and first-order differential equations. Tensors appear in the second chapter and are introduced there through their components. [Course plan](https://github.com/yangzichao/sandboxphysics/issues/56)

1. 01 · L1

   ## [Stress relaxation, creep and the Weissenberg number in a Maxwell liquid](https://sandboxphysics.com/stories/complex-fluids/stress-relaxation-and-creep/)

   Use a spring in series with a dashpot to solve stress relaxation, creep and startup shear, then judge with the Weissenberg number when elasticity matters.

2. 02 · L2

   ## [From a bead-spring dumbbell to the conformation tensor and the Oldroyd-B equation](https://sandboxphysics.com/stories/complex-fluids/dumbbells-and-the-conformation-tensor/)

   Treat a polymer as two beads joined by a spring, derive the conformation tensor and Oldroyd-B stress, and solve startup shear with normal stress.

3. 03 · L2

   ## [Why extensional flow unravels polymer coils while shear only tumbles them](https://sandboxphysics.com/stories/complex-fluids/coil-stretch-in-extension/)

   Compare Oldroyd-B coils in shear and planar extension, find the stretch threshold at a Weissenberg number of one half, and check it against DNA data.

4. 04 · L3

   ## [How finite chain length caps stretch and extensional viscosity in the FENE-P model](https://sandboxphysics.com/stories/complex-fluids/finite-extensibility-and-fene-p/)

   Give the dumbbell spring a maximum length and find the FENE-P ceiling on extensional viscosity, shear thinning and the energy balance.

5. 05 · L3

   ## [Reading a relaxation time from the exponential thinning of a polymer filament](https://sandboxphysics.com/stories/complex-fluids/filament-thinning-and-extensional-rheometry/)

   Use the zero-dimensional force balance on the neck to compare Newtonian and polymer threads, derive exponential thinning and read off the relaxation time.

6. 06 · L3

   ## [How normal stress along curved streamlines drives elastic instability](https://sandboxphysics.com/stories/complex-fluids/normal-stresses-and-curved-streamlines/)

   From hoop stress and the cone-and-plate normal force to the elastic instability parameter for curved streamlines, evaluated with experimental data.

7. 07 · L4

   ## [How a four-roll flow loses symmetry at its stagnation point, and how much the numerics change it](https://sandboxphysics.com/stories/complex-fluids/elastic-instability-at-a-stagnation-point/)

   Follow four-roll Stokes–Oldroyd-B simulations through the stress strand, symmetry breaking and oscillation, and ask what stress diffusion changes.

8. 08 · L4

   ## [Reading elastic turbulence: the evidence for chaotic flow without inertia, and what is still open](https://sandboxphysics.com/stories/complex-fluids/reading-elastic-turbulence/)

   How resistance, spectra and mixing in the Groisman–Steinberg experiments show chaotic flow without inertia, and why thresholds differ between geometries.

## Experiments along the way

[01 **Stretch polymer coils in shear and extension** Start the same Weissenberg number in simple shear and in planar extension and watch which flow unravels the coils.](https://sandboxphysics.com/labs/polymer-stretch-startup/) [02 **Thin a polymer filament** Let surface tension squeeze a liquid bridge and read the relaxation time from the exponential stage of the radius.](https://sandboxphysics.com/labs/polymer-filament-thinning/) [03 **Break the symmetry of a four-roll flow** Drive a Stokes flow with four rolls and raise the Weissenberg number until the stress strand at the stagnation point stops staying symmetric.](https://sandboxphysics.com/labs/four-roll-elastic-instability/) [04 **Hold a strain and watch stress relax** Use the viscoelastic hold preset to see stress fall at fixed strain and compare it with an elastic solid.](https://sandboxphysics.com/labs/stress-strain-tester/) [05 **Measure viscosity in a rotational rheometer** Turn torque and rotation speed into a shear rate and a viscosity, the steady-shear measurement every later chapter builds on.](https://sandboxphysics.com/labs/couette-poiseuille/) [06 **Pinch off a Newtonian thread** Watch a viscous thread neck and break to compare with the exponential thinning of a polymer filament.](https://sandboxphysics.com/labs/filament-necking/)

The polymer experiments simulate dumbbell models in homogeneous or idealized flows. Their curves are model output, not measured data; where a chapter quotes an experiment, the numbers come from the cited paper.

## Sources and what we recomputed

1. [A. Morozov and S. E. Spagnolie, Introduction to complex fluids, in Complex Fluids in Biological Systems (Springer, 2015), chapter 1](https://doi.org/10.1007/978-1-4939-2065-5_1)

   Section 3: the Weissenberg number defined as the relaxation time times the shear rate, and Eq. (1.27), the generalized Newtonian liquid. Section 4.1: Eqs. (1.47)–(1.51), the spring, the dashpot, the Maxwell liquid and its memory integral, with a relaxation time equal to viscosity over modulus. Eq. (1.72): the upper-convected derivative; Eq. (1.80): Oldroyd-B; Eqs. (1.86)–(1.87): FENE-P. Section 4.4: the Hookean dumbbell, Eqs. (1.98)–(1.110). Eq. (1.116): the steady-shear normal stress of Oldroyd-B. Eqs. (1.117)–(1.125): hoop stress and the cone-and-plate normal force. Eqs. (1.128)–(1.131): uniaxial extension and the Oldroyd-B divergence at half the inverse relaxation time. The chapter has no startup solution and no planar extension; the course derives both.

2. [E. M. Purcell, Life at low Reynolds number, American Journal of Physics 45, 3–11 (1977)](https://www.math.uwaterloo.ca/~mdunphy/purcell.pdf)

   The Reynolds number written as size times speed times density over viscosity; for swimming microorganisms it is about 1e-4 to 1e-5, and a bacterium that stops pushing coasts about 0.1 ångström. The scallop theorem: at low Reynolds number only the sequence of shapes matters, not how fast it is run. Cited for the low-inertia framing only. Read from a scanned reprint: journal pages 3–6 only.

3. [C. M. Schroeder, Single polymer dynamics for molecular rheology, arXiv:1712.03555v1; Journal of Rheology 62, 371–403 (2018)](https://arxiv.org/abs/1712.03555v1)

   Version 1, 10 December 2017. Section III: single DNA molecules tumble end over end in steady shear (Fig. 4). In extensional flow the coil–stretch transition is extremely sharp for molecules that reach a steady extension, and molecules take different paths from the same start (molecular individualism), reporting Perkins, Smith and Chu, Science 276, 2016 (1997) and Smith and Chu, Science 281, 1335 (1998). Conformation hysteresis: Schroeder et al., Science 301, 1515 (2003), with 1.3 mm DNA that stayed stretched or coiled for more than 12 strain units; the stretched-to-coiled drag ratio is about 1.6 for λ-DNA and about 5 for the 1.3 mm DNA. We cite the three Science papers through this review and did not read them directly.

4. [G. H. McKinley, Visco-elasto-capillary thinning and break-up of complex fluids, MIT Hatsopoulos Microfluids Laboratory report 05-P-04, submitted to Annual Rheology Reviews (2005)](https://dspace.mit.edu/handle/1721.1/18085)

   Eqs. (1)–(2): strain rate and Hencky strain of a thinning thread. Eqs. (4)–(7): Oldroyd-B stress growth in uniaxial extension and the finite-extensibility cap. Eq. (8): elasto-capillary thinning, in which the radius falls exponentially with a time constant of three relaxation times, so the thread runs at a Weissenberg number of 2/3. Eq. (11): linear thinning once the extensional viscosity is constant. Eq. (15): the zero-dimensional force balance on the neck. Eq. (16): the Newtonian similarity solution with prefactor 0.0709; the text after it says the simple zero-dimensional estimate is off by more than 100 %. Section 5.4: the terminal extensional viscosity of a finitely extensible dumbbell. The front factor of Eq. (8) and the one listed in Table 2 differ by the cube root of two; Section 5.4 attributes this to the one-dimensional similarity solution. The course uses only the exponent.

5. [A. Groisman and V. Steinberg, Elastic turbulence in a polymer solution flow, Nature 405, 53–55 (2000); arXiv:nlin/0104052v1](https://arxiv.org/abs/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.

6. [A. Groisman and V. Steinberg, Efficient mixing at low Reynolds numbers using polymer additives, Nature 410, 905–908 (2001); arXiv:nlin/0104050v1](https://arxiv.org/abs/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.

7. [A. Groisman and V. Steinberg, Elastic turbulence in curvilinear flows of polymer solutions, New Journal of Physics 6, 29 (2004)](https://doi.org/10.1088/1367-2630/6/1/029)

   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.

8. [S. S. Datta et al., Perspectives on viscoelastic flow instabilities and elastic turbulence, arXiv:2108.09841v1 (2021); Physical Review Fluids 7, 080701 (2022)](https://arxiv.org/abs/2108.09841v1)

   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.

9. [B. Thomases and M. Shelley, Emergence of singular structures in Oldroyd-B fluids, Physics of Fluids 19, 103103 (2007)](https://doi.org/10.1063/1.2783426)

   Stokes–Oldroyd-B flow driven by the four-roll force without stress diffusion. Near the central stagnation point, the local strain rate times the Weissenberg number acts as an effective Weissenberg number; that strain rate tends to 2/3 as the Weissenberg number goes to zero. Critical effective values 1/3 and 1/2 correspond to Weissenberg numbers of about 0.5 (cusp in the stress) and 0.9 (unbounded stress). Grids up to 4096 points on a side. With a FENE-P penalization the stress stays bounded but cusp-like structures remain.

10. [B. Thomases and M. Shelley, Transition to mixing and oscillations in a Stokesian viscoelastic flow, Physical Review Letters 103, 094501 (2009)](https://doi.org/10.1103/PhysRevLett.103.094501)

    Eqs. (1)–(2): Stokes flow with Oldroyd-B stress in a periodic box, driven by a steady four-roll force; the polymer-to-solvent viscosity ratio fixed at 0.5 to match the cross-slot experiments of Arratia et al.; stress diffusion 1e-3. At a Weissenberg number of 10 the flow breaks symmetry near time 200 in the paper’s units and later develops oscillations with periods of about 317 and 59; below about 5 the flow stays slaved to the forcing.

11. [B. Thomases, M. Shelley and J.-L. Thiffeault, A Stokesian viscoelastic flow: transition to oscillations and mixing, Physica D 240, 1602 (2011)](https://doi.org/10.1016/j.physd.2011.06.011)

    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.

12. [A. Gupta and D. Vincenzi, Effect of polymer-stress diffusion in the numerical simulation of elastic turbulence, arXiv:1809.09648v2 (2019)](https://arxiv.org/abs/1809.09648v2)

    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.

13. [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/)](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr26/four-roll-elastic-instability-model.md)

    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.