Polymeric diffusive instability leading to elastic turbulence in plane Couette flow Miguel Beneitez1Jacob Page2and Rich R. Kerswell1 1DAMTP Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WA UK

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Polymeric diﬀusive instability leading to elastic turbulence in plane Couette ﬂow

Miguel Beneitez,1Jacob Page,2and Rich R. Kerswell1

1DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK

2School of Mathematics, University of Edinburgh, EH9 3FD, UK

(Dated: August 8, 2023)

Elastic turbulence is a chaotic ﬂow state observed in dilute polymer solutions in the absence of

inertia. It was discovered experimentally in circular geometries and has long been thought to require

a ﬁnite amplitude perturbation in parallel ﬂows. Here we demonstrate, within the commonly-used

Oldroyd-B and FENE-P models, that a self-sustaining chaotic state can be initiated via a linear

instability in a simple inertialess shear ﬂow caused by the presence of small but non-zero diﬀusivity of

the polymer stress. Numerical simulations show that the instability leads to a three-dimensional self-

sustaining chaotic state, which we believe is the ﬁrst reported in a wall-bounded, parallel, inertialess

viscoelastic ﬂow.

Keywords: Elastic instability, viscoelastic ﬂows, elastic turbulence

Dilute polymer solutions are ubiquitous in everyday

life (e.g. foods, shampoo, paints, cosmetics) and under-

standing how they behave is important for many indus-

trial processes (e.g. plastics, oil, pharmaceuticals and

chemicals). The stretching and subsequent relaxation of

the polymers introduces new viscoelastic stresses in the

ﬂow which depend on the ﬂow’s deformation history. As

a result, polymer ﬂows can exhibit startlingly diﬀerent

behaviour from that of a Newtonian ﬂuid like water ( e.g.

rod-climbing, die swell and elastic recoil [1] ). Perhaps

most strikingly, a chaotic ﬂow state – so called “elastic

turbulence” (ET) – can occur for vanishing inertia, in

stark contrast to Newtonian ﬂuid mechanics where in-

ertia provides the only nonlinearity. ET has important

applications in small scale ﬂows where, for example, en-

hanced mixing for chemical reactions or heat transfer for

cooling computer chips are highly desirable [2, 3]. Even

though the origin of this behaviour is still not under-

stood, the key ingredients are believed to be ﬂuid elas-

ticity provided by the polymers and streamline curvature

which together give rise to a new elastic linear instabil-

ity [4–7]. Experiments in curved geometries conﬁrm the

linear instability leads to sustained ET [8]. In contrast,

wall-bounded inertialess parallel polymer ﬂow has been

presumed linearly stable, and experimental work there

has focused on triggering a ﬁnite amplitude instability

instead. Obstacles in the ﬂow have been used to provide

the required streamline curvature believed necessary for

this elastic instability and ultimately ET [9–12]. How-

ever, the requirements for both initiating such a tran-

sition and for the existence of a self-sustaining chaotic

state in a planar geometry are unknown.

Recently, by exploring the large relaxation-time limit

of the polymers, a new elastic linear instability has been

identiﬁed for the parallel ﬂow of an Oldroyd-B ﬂuid in

a pipe or channel at ﬁnite inertia [13, 14] and at van-

ishing inertia in a channel only [15]. This instability is

a ‘centre mode’ – concentrated around the centreline of

the channel – and is strongly subcritical, giving rise to

an arrowhead-shaped travelling wave solution [16] over a

large region of the parameter space. This solution has

been seen in simulations at both ﬁnite [17] and vanishing

inertia [18, 19], and may play a role in two-dimensional

elasto-inertial turbulence where inertia is important [20].

While no link has yet been found between this struc-

ture and ET in wall-bounded ﬂows, similar arrowhead-

like ﬂow structures have been observed in doubly-periodic

‘Kolmogorov ﬂow’ [21, 22] where a self-sustaining, two-

dimensional chaotic state is maintained in the absence

of inertia. The chaotic dynamics found here can be di-

rectly connected to a linear instability of the basic state

[23, 24] – though whether this instability is driven by

the same mechanism as the centre mode, or whether the

nonlinear chaotic state is a manifestation of the three-

dimensional ET found in wall bounded ﬂows is an open

problem. In contrast to these conﬁgurations, the simpler

case of constant shear between two diﬀerentially-moving,

parallel plates, known as plane Couette ﬂow, has been

considered linearly stable for all inertia and elasticity pa-

rameters [13].

In this Letter we report that viscoelastic plane Couette

ﬂow is linearly unstable if polymer stress diﬀusion is in-

cluded in the model. This diﬀusion is generally so small

that it is ignored as an important physical eﬀect, but

is reintroduced as a much larger ‘artiﬁcial’ diﬀusion to

stabilise time-stepping schemes if their inherent numer-

ical diﬀusion isn’t suﬃcient. This diﬀusion-induced lin-

ear instability is distinctly diﬀerent from the centre-mode

present in channels, being concentrated instead at the

walls. Signiﬁcantly, there is no smallest diﬀusion thresh-

old below which the instability vanishes: the wavelength

of the instability decreases with the size of the diﬀusion

so the instability could be misunderstood as a numerical

instability. The growth rate of the instability tends to a

non-zero limit as the polymer diﬀusion goes to zero so the

vanishing-diﬀusion limit is singular. The new diﬀusive

instability exists over a very wide area of the parameter

space and is robust to the choice of boundary conditions

on the polymer conformation. Direct numerical simu-

lations (DNS) show that the instability saturates onto

a low-amplitude limit cycle in two-dimensions. Three-

dimensional simulations show a transition to sustained

arXiv:2210.09961v3 [physics.flu-dyn] 7 Aug 2023

spatiotemporal chaos, which we believe to be the ﬁrst re-

ported computation of such a state in a planar geometry.

We consider the inertialess ﬂow of an incompressible,

viscoelastic ﬂuid between inﬁnite plates at y=±hmov-

ing with velocity ±U0ˆ

x. The governing equations are

∇p=β∆u+ (1 −β)∇·T(C),(1a)

∇·u= 0,(1b)

∂tC+ (u·∇)C+T(C) = C·∇u+ (∇u)T·C+ε∆C.

(1c)

where the polymeric stress Tis related to the conforma-

tion tensor Cusing the FENE-P model,

T(C) := 1

W i C

(1 −(tr C−3)/L2

max −I.

This model successfully predicted the phenomenon of

elasto-inertial turbulence (EIT) in 2010 [25] which was

observed a year later in experiments [26, 27].

The equations are non-dimensionalised by the half the

gap width, h, and the plate speed U0, which deﬁnes the

Weissenberg number W i := λU0/h (the ratio of the poly-

mer relaxation time λto a ﬂow timescale). The pa-

rameter β:= µs/µTis the ratio of the solvent-to-total

viscosities while ε:= D/U0his the nondimensionalisa-

tion of the polymer diﬀusivity D[28]. In this conﬁgura-

tion, the laminar basic state is simply U=yˆ

xwith only

Txx = 2W i and Txy = 1 being non-zero stress compo-

nents for an Oldroyd-B ﬂuid (Lmax → ∞). The polymer

equation (1c) changes character from hyperbolic at ε= 0

to parabolic for ε̸= 0 and extra boundary conditions are

then needed. Three boundary conditions are considered:

(i) application of the governing equations with ε= 0 at

the walls [29], (ii) application of the governing equations

with only the term ε∂2

yCij removed [17] and (iii) Neu-

mann, so ∂yCij = 0.

The linear stability of the basic state is examined by

introducing small perturbations of the form ϕ′(x, t) =

ϕ(y) exp (ikx(x−ct)) + c.c., where kx∈Ris the stream-

wise wavenumber and c=cr+icia complex wavespeed,

with instability if ci>0. The linear eigenvalue problem

is solved by expanding each ﬂow variable using the ﬁrst

NChebyshev polynomials (N= 300 is usually suﬃcient

to ensure convergence). An example eigenvalue spec-

trum for an Oldroyd-B ﬂuid with Wi = 100, ε= 10−3,

β= 0.95 and kx= 3 is reported in ﬁgure 1 (the bottom

right inset shows the equivalent spectrum with ε= 0).

The continuous spectra in the absence of polymeric dif-

fusion are regularised with the introduction of ε̸= 0

[30], with a pair of linear instabilities emerging with

wavespeeds cr∼ ±1.

We map out the unstable region for various parame-

ters and boundary conditions in ﬁgure 2(a). In the top

left panel of ﬁgure 2(a) we observe that the instability

persists as ε→0 i.e. this is a singular limit for all three

boundary conditions and occurs at a constant value of

W i for diﬀusivities ε≤10−2, requiring an increasingly

−1.0−0.50.0 0.5 1.0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

−10 1

−0.05

0.00

−1.0−0.50.0 0.5 1.0

−0.0001

0.0000

0.0001

FIG. 1. Spectrum at Re = 0, Wi = 100, β= 0.95, kx= 3

for two diﬀerent resolutions Ny= 300 (orange circles) and

Ny= 400 (purple dots) with ε= 10−3within the domain and

ε= 0 at the boundaries. Top inset: zoom-in to the unstable

eigenvalues ci>0. Bottom right inset: spectrum for the same

parameters but ε= 0 everywhere in the domain and bound-

aries and two diﬀerent resolutions Ny= 300 (red circles) and

Ny= 400 (blue dots). The most unstable eigenvalues with

ﬁnite εhere are c=±1.0052607137 + 4.513293285i×10−5

Note the apparent instabilities in the continuous spectrum

seen in the ε= 0 results is associated with poor resolution of

the (non-smooth) eigenfunctions in our numerics (e.g. [31])

and can be suppressed if the resolution is increased further.

large streamwise wavenumber kx∝ε−1/2/8 ([30] did not

consider these large wavenumbers and so failed to ﬁnd

instability). The imaginary part of the wavespeed scales

with the square root of the diﬀusivity, ci∝ε1/2, so that

the growth rate of the instability, kxci, remains O(1)

as ε→0. The unstable region appears unbounded as

W i → ∞ for boundary conditions (i) and (ii) but stabil-

ity is restored in this limit for (iii). Henceforth boundary

condition (i) is used.

In ﬁgure 2(b)-(c) we also examine the eﬀect of the vis-

cosity ratio and ﬁnite extensibility on the diﬀusive in-

stability. The instability is realised for decreasing values

of W i at ﬁxed εas βis reduced (i.e. increasing poly-

mer concentration), and the marginal stability curves col-

lapse when plotted against W i(1−β)/β in the ultradilute

limit β→1, which is the magnitude of the perturbation

stresses τ′

xx and τ′

xy relative to the diﬀusion terms in

the momentum equation. Furthermore, the instability

survives for realistic values of the polymer extensibility

of Lmax =O(100), in the FENE-P model, though it is

pushed to increasingly low values of βand suppressed

beyond a critical W i. Figure 2(c) shows that this in-

stability is also present in plane Poiseuille ﬂow driven

by a non-dimensional pressure gradient ∂xP=−2, the

neutral curves nearly overlapping when W i is rescaled

by the shear rate at the wall, U′

wall. The quantitative

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PolymericdiffusiveinstabilityleadingtoelasticturbulenceinplaneCouetteflowMiguelBeneitez,1JacobPage,2andRichR.Kerswell11DAMTP,CentreforMathematicalSciences,WilberforceRoad,CambridgeCB30WA,UK2SchoolofMathematics,UniversityofEdinburgh,EH93FD,UK(Dated:August8,2023)Elasticturbulenceisachaoticflowstateobs...

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Polymeric diffusive instability leading to elastic turbulence in plane Couette flow Miguel Beneitez1Jacob Page2and Rich R. Kerswell1 1DAMTP Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WA UK.pdf

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Polymeric diffusive instability leading to elastic turbulence in plane Couette flow Miguel Beneitez1Jacob Page2and Rich R. Kerswell1 1DAMTP Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WA UK

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