L2 · Chapter 3

Why extensional flow unravels polymer coils while shear only tumbles them

Sandbox Physics · 2026-10-29

At the same Weissenberg number, shear stretches a coil to less than twice its size while extensional flow can stretch it tens of times. The difference is rotation. Shear stretches and rotates at once, so a coil that has just been stretched is soon turned away from the stretching direction. Extensional flow has no rotation, and once the stretch rate exceeds half the relaxation rate the coil keeps being pulled. Single-molecule DNA experiments see this transition, and also a hysteresis that the linear-spring model cannot explain.

Shear is extension plus rotation

The velocity gradient splits into a symmetric rate of strain $\mathbf D$ and an antisymmetric rotation $\mathbf W$ . In simple shear, $\mathbf D$ stretches along 45° at the rate $\dot\gamma/2$ , while $\mathbf W$ turns everything at the same rate. As soon as a coil is stretched it is turned toward the flow direction, where the stretch rate is zero.

Follow a material line moving with the fluid. It starts along $y$ with length one, and after a shear strain $\gamma$ its length is $\sqrt{1+\gamma^2}$ : linear growth only. Planar extension $u=\dot\varepsilon(x,-y)$ has no rotation, and a material line along $x$ grows as $e^{\dot\varepsilon t}$ . A polymer that does not relax follows the material lines.

The review by Schroeder records single DNA molecules in steady shear: a molecule repeatedly stretches, flips and recoils, tumbling end over end (Fig. 4). There is no such tumbling in extensional flow.

\mathbf L=\begin{pmatrix}0&\dot\gamma\\0&0\end{pmatrix}=\underbrace{\begin{pmatrix}0&\dot\gamma/2\\\dot\gamma/2&0\end{pmatrix}}_{\mathbf D}+\underbrace{\begin{pmatrix}0&\dot\gamma/2\\-\dot\gamma/2&0\end{pmatrix}}_{\mathbf W}

The velocity gradient of simple shear. The eigenvectors of

\mathbf D

lie at 45° and 135° with eigenvalues

\pm\dot\gamma/2

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.

Source: C. M. Schroeder

The steady coil in planar extension

Insert $\mathbf L=\mathrm{diag}(\dot\varepsilon,-\dot\varepsilon,0)$ into the conformation equation and each diagonal component obeys its own equation. Along the stretching direction the flow amplifies $A_{xx}$ at the rate $2\dot\varepsilon$ and the spring pulls it back at the rate $1/\lambda$ . A steady state needs the spring to win: $\mathrm{Wi}=\lambda\dot\varepsilon<1/2$ .

The stress difference along the stretch divided by the stretch rate is the planar extensional viscosity $\eta_P$ . At low Weissenberg number it equals $4\eta_0$ , the Newtonian value; for uniaxial extension the Newtonian value is $3\eta_0$ . As $\mathrm{Wi}$ approaches one half, $\eta_P$ diverges, while the shear viscosity at the same parameters stays $\eta_0$ .

Morozov and Spagnolie find the same divergence in uniaxial extension and note that it is sometimes miscalled the coil–stretch transition. A linear spring can stretch without limit, so the divergence only says that above this rate the spring cannot keep up with the flow; it does not say what a real molecule does.

A_{xx}=\frac{1}{1-2\,\mathrm{Wi}},\qquad A_{yy}=\frac{1}{1+2\,\mathrm{Wi}},\qquad \eta_P=4\eta_s+\frac{4\eta_p}{1-4\,\mathrm{Wi}^2},\qquad \mathrm{Wi}=\lambda\dot\varepsilon

In planar extension

A_{zz}=1

. For the uniaxial counterpart see Eqs. (1.128)–(1.131) of Morozov and Spagnolie, where the extensional viscosity also diverges at

\lambda\dot\varepsilon=1/2

Source: A. Morozov and S. E. Spagnolie

Above the threshold: exponential stretch

Start planar extension from rest. Then $A_{xx}$ obeys $\lambda\dot A_{xx}=(2\mathrm{Wi}-1)A_{xx}+1$ . Below the threshold it approaches the steady value of the previous section; above it, it grows exponentially at the rate $(2\mathrm{Wi}-1)/\lambda$ .

Eqs. (4)–(6) of McKinley give the same structure for uniaxial extension: above a stretch rate of $1/(2\lambda)$ the Oldroyd-B tensile stress grows without bound in time. What stops it is the finite length of the chain, the subject of the next chapter.

A_{xx}(t)=\frac{2\,\mathrm{Wi}\,e^{(2\mathrm{Wi}-1)t/\lambda}-1}{2\,\mathrm{Wi}-1}

Initial value

A_{xx}(0)=1

. For

\mathrm{Wi}<1/2

the exponential decays and the result approaches

1/(1-2\mathrm{Wi})

Source: G. H. McKinley · A. Morozov and S. E. Spagnolie

What single-molecule experiments see

Section III of the Schroeder review collects experiments on fluorescently labelled DNA in extensional flow. Counting only molecules that reach a steady extension, the coil–stretch transition is extremely sharp. Starting from the same coil, different molecules take different paths, some through a dumbbell shape, some folded into a hairpin, some kinked; the review calls this molecular individualism.

Longer molecules also show conformation hysteresis. The review describes a 2003 experiment by Schroeder and co-workers with 1.3 mm DNA: at the same stretch rate, molecules that started stretched stayed stretched for more than 12 strain units, while molecules that started coiled stayed coiled. The review ties the hysteresis to drag: a stretched chain feels more drag than a coiled one, by a factor of about 1.6 for λ-DNA and about 5 for the 1.3 mm DNA.

In the dumbbell model the drag coefficient is a constant, so a stretched and a coiled molecule feel the same drag. The model has no two stable states and therefore no hysteresis. The Oldroyd-B threshold describes when the average coil starts to be stretched without limit; the single-molecule experiments describe how each molecule actually gets there.

Source: C. M. Schroeder

Worked example: two flows at the same Weissenberg number

Take $\beta=0.5$ and $\mathrm{Wi}=0.4$ , below the threshold. Steady planar extension gives $A_{xx}=1/(1-0.8)=5$ , $A_{yy}=1/1.8=0.556$ , $A_{zz}=1$ , and a stretch of $\sqrt{6.556/3}=1.48$ . The planar extensional viscosity is $\eta_P=4\times0.5+4\times0.5/(1-0.64)=7.56\,\eta_0$ , almost twice the Newtonian value.

The same $\mathrm{Wi}$ in shear gives $A_{xx}=1+2\times0.16=1.32$ , a stretch of $\sqrt{3.32/3}=1.05$ , and a shear viscosity still equal to $\eta_0$ . The extensional flow has visibly stretched the coil; the shear has barely touched it.

Raise the number to $\mathrm{Wi}=1$ and let the linear spring run for $8\lambda$ . In planar extension $A_{xx}=2e^{8}-1=5961$ and $A_{yy}$ tends to $1/3$ , so the stretch is about 44.6 and still growing exponentially. In shear $A_{xx}=1+2\,(1-9e^{-8})=2.99$ and the stretch is 1.29. These are the two numbers the Lab’s Unbounded linear spring preset should show when it finishes.

Source: A. Morozov and S. E. Spagnolie

Try it in the Lab

Push the coil over the threshold

Choose the Too slow to stretch preset (FENE-P, Weissenberg number 0.3) and then the Coils unravel preset (Weissenberg number 0.8). Compare how the ratio of the two stretches changes.
Choose the Unbounded linear spring preset (Oldroyd-B, Weissenberg number 1) and confirm that after $8\lambda$ the stretch is about 44.6 on the extension side and about 1.29 on the shear side.
Keep Oldroyd-B and set the Weissenberg number to 0.45, then 0.55. Predict which one settles at a finite stretch, and use $A_{xx}=1/(1-2\mathrm{Wi})$ to estimate the steady value at 0.45.

Exercises with explained answers

An Oldroyd-B liquid with $\beta=0.5$ is in planar extension at $\mathrm{Wi}=0.25$ . Find the steady $A_{xx}$ and the planar extensional viscosity.
Show the answer
$A_{xx}=1/(1-0.5)=2$ and $\eta_P=4\times0.5+4\times0.5/(1-0.25)=4.67\,\eta_0$ , about 17% above the Newtonian value $4\eta_0$ .
A material line of length one starts perpendicular to a shear flow. How long is it at a shear strain of 10? How long is a line along the stretching direction of planar extension at $\dot\varepsilon t=10$ ?
Show the answer
In shear $\sqrt{1+10^2}=10.05$ ; in planar extension $e^{10}=2.20\times10^{4}$ . Exponential beats linear by a factor of more than two thousand.
Use Eq. (1.131) of Morozov and Spagnolie to compute the uniaxial extensional viscosity at $\beta=0.5$ and $\lambda\dot\varepsilon=0.25$ , and check it independently with the uniaxial conformation components.
Show the answer
Eq. (1.131) gives $3\,(1-0.125-0.0625)/(0.5\times1.25)=3.9\,\eta_0$ . In uniaxial extension $A_{zz}=1/(1-2\times0.25)=2$ and $A_{rr}=1/(1+0.25)=0.8$ , so $3\eta_s+\eta_p(A_{zz}-A_{rr})/\mathrm{Wi}=1.5+0.5\times1.2/0.25=3.9$ . The two agree.
Why does 1.3 mm DNA show clear hysteresis while the shorter λ-DNA does not?
Show the answer
The review gives a stretched-to-coiled drag ratio of about 1.6 for λ-DNA and about 5 for the 1.3 mm DNA. The more the drag grows with extension, the more easily the flow holds a stretched molecule stretched and the harder it is to open a coiled one, so the two states can coexist at the same stretch rate.

Sources and what we recomputed

A. Morozov and S. E. Spagnolie, Introduction to complex fluids, in Complex Fluids in Biological Systems (Springer, 2015), chapter 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.
C. M. Schroeder, Single polymer dynamics for molecular rheology, arXiv:1712.03555v1; Journal of Rheology 62, 371–403 (2018)
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.
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)
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.