Flexible lament in timeperiodic viscous ow shape chaos and period three Vipin Agrawal1 2and Dhrubaditya Mitra1 1Nordita KTH Royal Institute of Technology and Stockholm University

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Flexible ﬁlament in time–periodic viscous ﬂow : shape chaos and period three

Vipin Agrawal1, 2 and Dhrubaditya Mitra1, ∗

1Nordita, KTH Royal Institute of Technology and Stockholm University,

12 Hannes Alfv´ens v¨ag 10691 Stockholm, Sweden

2Department of Physics, Stockholm university, Stockholm.

(Dated: October 11, 2022)

We study a single, freely–ﬂoating, inextensible, elastic ﬁlament in a linear shear ﬂow: U0(x, y) =

˙γyˆx. In our model: the elastic energy depends only on bending; the rate–of–strain, ˙γ=Ssin(ωt) is

a periodic function of time, t; and the interaction between the ﬁlament and the ﬂow is approximated

by a local isotropic drag force. Based on the shape of the ﬁlament we ﬁnd ﬁve diﬀerent dynamical

phases: straight, buckled, periodic (with period two, period three, period four, etc), chaotic and

one with chaotic transients. In the chaotic phase, we show that the iterative map for the angle,

which the end–to–end vector of the ﬁlament makes with the tangent its one end, has period three

solutions; hence it is chaotic. Furthermore, in the chaotic phase the ﬂow is an eﬃcient mixer.

Keywords: ﬂuid-structure interactions, spatiotemporal chaos, periodic orbit theory

I. INTRODUCTION

The dynamics of ﬂexible ﬁlaments in ﬂows plays a cru-

cial role in many biological and industrial processes [1].

A canonical example is that of cilia and ﬂagella [2, 3]

that takes part in wide variety of biological tasks, e.g.,

swimming of microorganisms, feeding and breathing of

marine invertebrates. In such cases, although the ﬂow

nonlinearities can often be safely ignored, due to its elas-

tic nonlinearities and ﬂow–structure interactions a single

isolated ﬁlament can show surprisingly complex dynam-

ics in ﬂows. Both active and passive ﬁlament, anchored

or freely ﬂoating, in various steady ﬂows have been stud-

ied extensively, see Ref. [4, 5] and references therein. In

steady ﬂows, a single passive ﬁlament has quite complex

transient dynamics [6–14]. For active ﬁlaments, the focus

has been on how a periodic driving can give rise to sym-

metry breaking, e.g., swimming [15] or whirling [16–18].

This year, three papers have focussed on, how periodic

driving, either of the ﬂow or the ﬁlament, can give rise

to secondary instabilities [19] or statistically stationary

state with chaotic/complex dynamics [20, 21]. For the

latter, the shape of the ﬁlament, as described by its cur-

vature as a function of its arc length, is a spatiotempo-

rally chaotic function. Henceforth we call this phenom-

ena shape chaos. Such chaotic solutions are particularly

interesting because they have the potential to be used to

generate eﬃcient mixing in microﬂuidics.

Two eﬀects determine the fate of an elastic ﬁlament

in ﬂow. One is the elastic nonlinearity of the ﬁlament

and the other is the viscous interaction between the ﬁl-

ament and the ﬂow. The latter, in all its glory, gives

rise to non–local and nonlinear interaction between two

diﬀerent parts of the same ﬁlament. Nevertheless, the-

oretical studies [16, 22–24] have often approximated the

viscous eﬀect as a local, linear, isotropic drag. Can this

local approximation to the ﬂow–structure interaction still

∗dhruba.mitra@gmail.com

FIG. 1. Sketch of our numerical experiment. Initially

the ﬁlament is straight and is aligned vertically. The back-

ground shear ﬂow, Equation (1) is shown as arrow: t=T /4

(top panel) and t= 3T/4 (bottom panel).

capture the shape chaos of a freely-ﬂoating ﬁlament ? As

we show in the rest of this paper, the answer is yes; we

prove shape chaos using Sharkovskii and Li and Yorke’s

famous result [25] – existence of period orbits of period

three implies not only the existence of orbits of all periods

but also senstive dependence on initial condition.

In Figure. 1 we show a sketch of our numerical exper-

iment. Initially the ﬁlament is aligned vertically. The

background shear ﬂow is given by

U0(x, y) = ˙γyˆx,and ˙γ=Ssin(ωt).(1)

Here T= 2π/ω is the time period of the periodic shear

arXiv:2210.04781v1 [cond-mat.soft] 10 Oct 2022

and Sis a constant.

II. MODEL

We model the ﬁlament using the bead-spring model [7,

10, 11, 18, 26, 27]: identical spherical beads of diameter

dare connected by over-damped springs of equilibrium

length a. The position of the center of the i-th bead is

Ri, where i= 1 . . . N, the total number of beads. The

equation of motion is:

∂Rα

∂t =−1

3πηd

∂H

∂Rα

+Uα

0(Ri),(2)

where U0is given in (1). Here ηis viscosity of the ﬂuid,

∂(·)/∂(·) denotes partial derivative, U0is the velocity of

the background shear, and His the elastic Hamiltonian

of the ﬁlament. The Greek indices run from 1 to D,

the dimensionality of the space, and the Latin indices

run from 1 to N. The elastic Hamiltonian [18, 28], has

contributions from bending (HB) and stretching (HS):

H=HB+HSwhere (3a)

HB=aB

N−1

i=0

κ2

iand (3b)

HS=H

N−1

i=0

(|Ri+1 −Ri| − a)2; where (3c)

κi=1

a|ˆ

ui×ˆ

ui−1|and (3d)

ui=Ri+1 −Ri

|Ri+1 −Ri|.(3e)

Here Bis the bending modulus of the ﬁlament and His

its stretching modulus. We ignore thermal ﬂuctuations

and torsion. Three dimensionless parameters determine

the dynamics. We call them, the elasto–viscous parame-

ters, the dimensionless frequency and the ratio of stretch-

ing to bending deﬁned respectively as:

µ≡8πηSL4

B,(4a)

σ≡ω

S,and (4b)

K≡Ha2

B.(4c)

In practice, the ﬁlaments are inextensible [29], which

we implement by choosing appropriately high value of

K. We evolve Equation (2) using adaptive Runge-Kutta

[30] method with cash-karp parameters [31]. Our code

is freely available [32] and has been benchmarked against

experimental results [20]. A complete list of the param-

eters of the simulation is given in table I. We study the

problem for a large range of µand σall within experi-

mentally realizable range. Note that, with the local ap-

proximation of viscous forces it is possible for the ﬁla-

ment to cross itself. Such unphysical solutions do appear

in our simulations but for values of µother than that

has been considered in this paper. The computational

complexity of the model, where the viscous interaction

is modelled by the non–local Rotne–Pregor tensor [20],

is O(N2) where Nis the number of beads, whereas the

computational complexity of the model with local vis-

cosity is O(N). This allows us to run our simulations for

much longer times than it was possible in Ref. [20].

III. RESULTS

FIG. 2. (A):Phase diagram in the µ–σplane; We ﬁnd 5 dif-

ferent qualitatively diﬀerent dynamical phases: Straight(•);

periodic (H) with n-period, where n=2(), 3(J), 4(I); com-

plex (F) complex-transients ()(B) Solutions of strobo-

scopic map: The stroboscopic map has many periodic solu-

tions at every point in µ–σplane. We show time period of

the lowest cycle in Sharvoskii ordering.

A rigid ellipsoid in a periodic shear may show chaotic

three–dimensional rotation under certain conditions [33–

36]. Such behavior emerges due to the nonlinearities

present in the Euler’s equations of rigid body rotation.

Here we consider a ﬁlament with no inertia, hence such

chaotic solutions are not present in our system. For a ﬁl-

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Flexiblelamentintime{periodicviscousow:shapechaosandperiodthreeVipinAgrawal1,2andDhrubadityaMitra1,1Nordita,KTHRoyalInstituteofTechnologyandStockholmUniversity,12HannesAlfvensvag10691Stockholm,Sweden2DepartmentofPhysics,Stockholmuniversity,Stockholm.(Dated:October11,2022)Westudyasingle,freely{oa...

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Flexible lament in timeperiodic viscous ow shape chaos and period three Vipin Agrawal1 2and Dhrubaditya Mitra1 1Nordita KTH Royal Institute of Technology and Stockholm University.pdf

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Flexible lament in timeperiodic viscous ow shape chaos and period three Vipin Agrawal1 2and Dhrubaditya Mitra1 1Nordita KTH Royal Institute of Technology and Stockholm University

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