Pinching instability of convective rolls in an anisotropic fluid : first step to chaos

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Pinching instability of convective rolls in an anisotropic fluid : first step to chaos

R. Ribotta, A. Joets

To cite this version:

R. Ribotta, A. Joets. Pinching instability of convective rolls in an anisotropic fluid : first step to chaos.

Journal de Physique, 1986, 47 (5), pp.739-743. �10.1051/jphys:01986004705073900�. �jpa-00210256�

PINCHING INSTABILITY OF CONVECTIVE ROLLS ^{IN AN} ANISOTROPIC ^{FLUID :} FIRST STEP TO CHAOS

R. RIBOTTA and A. JOETS

Laboratoire ^de Physique ^des Solides, Bât. 510, Université de Paris-Sud, ^F-91405 Orsay ^{Cedex, France}

(Reçu ^{le 5} juillet 1985, révisé le 6 mars 1986, accepté ^{le 10 mars 1986)}

Résumé

Nous décrivons une nouvelle instabilité de la structure convective de rouleaux dans un nématique

sous champ électrique. ^{La structure} alors obtenue montre un pincement périodique ^le long des rouleaux et apparait semblable à la structure dite varicose

oblique ^{observée en} Rayleigh-Bénard ^{dans un fluide} isotrope. Le résultat nouveau déduit de l’étude des écoulements montre que ^cette instabilité qui ^induit

un nouveau mode de rotation est la première étape importante ^{dans la} désorganisation ^{de la convection}

avant le chaos.

Abstract

A new instability ^{in the} convective roll struc- ture of a nematic subjected ^{to an AC} electric field is presented. ^It brings ^{a new structure characte-}

rized by ^a periodic pinching of the rolls along their axis. This structure has features similar ^to those of the so-called skewed varicose in the Rayleigh-

Bénard convection of isotropic ^{fluids. The} study ^of

the streamlines shows that the pinching ^{creates a}

new rotation mode and that this is the first impor-

tant step ^{in the} disorganization ^{of a} convective flow before the onset of chaos.

Classification

Physics ^Abstracts

47.20 - 61.30 G

We have recently ^shown (1,2) ^{that a convective} anisotropic ^fluid subjected ^{to an} increasing ^cons-

traint undergoes ^a well-defined succession of insta- bilities before the onset of the chaotic state. In this sequence, the lowest-threshold structure is a set of parallel convective rolls (the "Normal Rolls")

and the last ordered ^structure before the onset of chaos is a set of rectangular closed cells. The Normal Rolls correspond ^{to a rotation} "mode" around a well- defined direction and the rectangular ^{cells corres-} pond ^to two orthogonal ^{rotation modes} decoupled ⁱⁿ

space (the "bimodal"). In between these two structures we have found two new instabilities : the first one

destabilizes the Normal Rolls into a set of Oblique

Rolls after an undulation deformation (1,3), ^{and the}

second one which occurs between the Oblique ^Rolls

and the rectangular structure, will be described in this letter. This instability ^{leads to a} stationary

structure which has never been observed previously.

We shall describe its main features and show by ^the study of the flow streamlines that the velocity ^field

looses its translational invariance along ^{the roll} axis. The rolls become periodically pinched,and ^we

shall show that this pinching instability ^{is the} key mechanism in the evolution towards the bimodal rec-

tangular ^{structure which} precedes ^{the chaotic state.}

We shall try ^to relate this instability ^{to the so-}

called "skewed-varicose" observed in the thermal convection of isotropic fluids with a moderate Prandtl number (4).

A layer of nematic liquid crystal ^of negative ^die-

lectric anisotropy ^{(either MBBA or an} eutectic mixture of azoxy by Merck : Phase V) is sandwiched between two glass plates ^{that are coated with} semi-transparent

electrodes. The typical sample thickness is d = 50 um.

The nematic molecules are oriented parallel ^{to the} plates along ^the a direction by rubbing ^the plates

coated with a polyimid (Kerimid). ^The structures are

observed under a polarizing microscope ^{and the measu-}

rements are made either on photographic plates ^or by microdensitometry. ^{The flow} streamlines are visualized by tracking glass spheres , ^{3-5 Um in} diameter, immersed in the nematic. The frequency ^{of the AC} electric field is set at some fixed value f - 120 Hz in the so-called conduction regime (5). ^The voltage ^V will be increased smoothly ^{from zero} up ^to 0,1 ^volt below threshold then by small steps of 25 mV every

two minutes.In the rest state (V-0) the sample ^is transparent. Then after some well defined threshold (V-7 volts) the first convective instability appears and the structure is a set of parallel ^rolls along

y. ^{These are} the so-called Williams Domains of the conduction regime (6). ^Further increasing ^{the vol-}

tage destabilizes ^{the roll structure at some new}

higher ^threshold (Vz - 7,5 volts).One ^{then obtains}

undulated rolls which transform with increasing voltage ^{into the} Oblique ^{Roll structure} (or zig- zag) composed ^{of domains of} parallel rolls tilted

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004705073900

740

symmetrically by ^{t e}

on y

it has been shown that the streamlines of the flow

nearly ^{lie in} planes ^{normal to the roll} axis, and, ⁱⁿ addition,a ^small axial component is present.

Now, ^{as the} voltage ^is again increased, ^{a new} pertur- bation of the roll structure appears inside a tilted domain at some threshold (V * ⁸ volts).It is a static modulation of small period x ^{along the roll axis}

(fig. 1). ^This period Àv ^{is slightly} larger ^{than a} convection period ^{a(= 2d) of the} unperturbed ^struc-

ture (the zig-zag), Àv: ^{1.2 X. The} amplitude ^{of the}

deformation measured normally ^{to the} roll axis grows

continuously ^{from zero as} the voltage ^increases (fig. 2), ^{as for a direct} bifurcation, ^while the pe- riod Xv remains almost constant. Even close to threshold, it is observed that the modulation is not purely sinusoidal but is rather asymmetric (sawtooth-like).

Moreover from ^one roll to the next one,this ^modula-

tion is of ^same amplitude ^but slightly ^{out of} phase (the zero-phase plane being ^{normal to the roll} axis).

Hence the rolls ^are periodically pinched ^{along their}

axis with the periodicity Àv. ^{The dephasing between}

the rolls results in a second periodicity a ^along

an oblique ^direction (fig. lb).

The whole structure can be defined entirely by

two wave

vectors k S,v . 27r/X S,V oriented

with respect ^to the former axes set Ox, Oy. ^{We shall}

define another set of axes

Oxl, 3Yl

lative to the former one. Such a deformation _appears

to be quite ^{similar to the} "skewed varicose" tran-

sient structure first observed and described by ^Busse

and Clever (4).

This structure can be approximately ^{described as} resulting ^{from a} perturbation of the rolls (of ^wave

vector q - 27r/X) ^{in the} zig-zag ^{which is the sum of}

the two elementary ^modes (2) :

a) ^a longitudinal ^{mode of wave vector}

qE

to the roll wave vector q (direction

xl).

ponds ^{to a} compression-dilation of the roll diameter.

b) ^a transverse mode qu corresponding ^{to an undula-}

tion of the rolls (with ^{a constant} diameter)along ^their

axis (y 1) .

Figure 1

a) Periodical pinching ^{of a set of} parallel convective rolls in a structure tilted by ^{6 over} y.

The phase plane ^{of the} deformation is oblique.

Figure ¹

b) The two wavelengths ^{of the} pinched structure :

a ^{is along the oblique} direction and Xv ^{is the modu-} lation period along ^{the roll axis.}

Figure ²

A typical plot ^{for the} amplitude A of the modula- tion measured normally ^{to the roll} axis, ^{as a func-} tion of the relative

voltage

threshold Vv. ev . (VZ _

V2v)/v2v.

close to ev . 0. appears ^to be due to the long-range influence of the grain boundaries still present in the zig-zag ^structure.

The flow, visualised by the tracks of the glass

spheres, ^{now becomes} complex along ^{the roll axis} yl.

Along 11

deformation amplitude increases, ^this portion ^of

roll is slightly rotated around its center,and yl ^gets

closer to the y direction. Inside this portion, ^the

flow is mainly ^{a rotation around the local}

axis yl

as in a usual roll. Inside the pinching ^{the stream-}

lines are now almost normal to the local axis which is parallel ^{to the skew direction} (fig. 3). ^{In the} elbows the streamlines distribution is divergent

around the inflexion points ^{of the} distorted lines for the up and down flows. Thus, ^the velocity ^field

has lost its continuous translational invariance along the roll axis. In addition,singularities ^{appear in}

the convective motion ^{and are} located in the pinched

zones (see fig. 3c). ^{As the} pinching amplitude ^increa-

ses further, ^{the former} portions ^{of rolls rotate such}

as to

align 11

Figure ³

a)b)c)d) Evolution of the streamlines in the pinched ^{zone as 6 and the} pinching ^increase.

In step c) ^{a new local} vorticity

Z6x

former wy

two stagnations points ^{S. The final}

step

bimodal{k, W-Y}.

formed into a staircase-like succession of rectangu- lar cells connected by ^{one corner. The} diagonal ^of

the cells is precisely ^the

initial yl

Oblique ^{Roll. This is the bimodal} rectangular ^struc-

ture. During ^the process,it appears clearly ^from the glass spheres ^motion that, ^{inside a} small volume within the pinched zone, there exists now a rotation

Wxl

axis

XI,

rotation Wyl

.the initial unaltered portion ^{of roll} (fig. 3). ^Thus the effect of the complex perturbation

lqE,qu} on

Oblique ^{Roll is to} crete periodically ⁱⁿ space ^{a new} rotation component

WXI

from the initial rotation "mode". As a result, ^at these places,the initial convection mode

myl

pressed. This implies ^{the creation of two} stagnation points ^S (fig. 3). ^We show elsewhere (lb,7) ^{that this}

process ^can be described as the destabilization of

singular ^{lines in a vector field.}

In order to understand the coupling between the

director n ^{and the} velocity gradients, ^the knowledge

of the streamlines must be complemented by ^{that of} the molecular alignment. ^The director orientation

can be evidenced by birefringence effects. For ins- tance the tilt angle (P ^{over the} xOy horizontal plane

can be measured using interferometry ^{with a} parallel monochromatic light. ^In fig.4 ^{the isoclines} ((p- (p 0

appear inside each roll and they ^{show a} small gra- dient

a(D

close to zero. This small value is compatible ^-di7th

a small velocity (or vorticity

(S

since w ^and

’w Yi are

of the relative height ^{of the} focalisation planes ^for the rays transmitted through ^the pinched ^{zone. In} effect,the focusing height ^{h is a function of the} spatial gradient ^of (P:h -- IV(pi-1. ^On the other hand,

742

Figure 4

Fringe pattern in ^{a He Ne} parallel light. ^The fringes correspond ^{to lines} (isoclines), ^where

the molecular angle (p ^{in the vertical xOz} plane ^is

constant.

it is difficult to measure by optical ^means any azimuthal angle 6 of the director out of the initial orientation plane ^{xOz. This is due to the} Mauguin

condition (8) which states that the polarisation ^vec-

tor of the light ^{wave follows} adiabatically any ^rota- tion 6 of the director around the vertical axis t,

when the gradient

ae

fringence ^{and X is the} light wavelength. Up ^to now, we can

only

wy

periodical ^decrease in (p may be associated to a elas- tic mode of twist for the director n, ^{which is now}

added to the bend mode associated to the initial

deformation responsible ^{for the} convective rolls. The wavevector

kT

fact, ^{the skew direction} (where

iz y

bution of additional elastic modes of less importance

has to be considered. For instance, ^{in our} experiements

the real skew direction might ^be explained by ^{an addi-} tional twist around i of order 30° with respect ^to x.

Once this periodical pinching ^{structure is esta-}

blished it is absolutely stationary ^{and reversible}

without hysteresis. ^{As the} voltage is further increa- sed, the pinching becomes total and focal lines for the up and down motion appear connected together,

as previously explained. ^{We obtain the} rectangular

structures which has often been observed in liquid crystals (9) and which may be compared ^{to the one}

in Rayleigh ^{-Bénard of} isotropic ^fluids (10,11).

We name it bimodal structure because, now, the ^two rotations mx ^and

wy

x in order to form a distinct mode (fig. 3d). ^{It must}

be noticed that the optical ^{caustics are no} longer lines but rather have changed ^{at the} transition into

an array of focal points (lb,7). Measuring ^the perio- dicity ^{in the bimodal structure one finds that the} period ^{of the} rectangular one, measured along x, ^is

the same as that of the Normal Rolls structure within 2 X, whereas the rolls had a smaller period ^{in the} oblique structure, i.e. before the pinching ^instabi- lity.

Defects may exist inside the.pinched ^{roll struc-}

ture. They ^{come from two different} origins. ^{The first}

type ^{comes from real} edge-dislocations ^{that were trap-}

ped ⁱⁿ the former structure at lower voltage (the Oblique Rolls). The second and new type is ^{a defect} which comes from a fault in the period ^{of the} pinching

modulation along ^{the roll axis} (fig. 5). ^In fact, ^it is localized transition between two regions ^with period differing by ^one unit, ^and separated by ^{an area where}

the amplitude of the modulation vanishes. By continui- ty, this jump ^{in the} modulation period ^will produce, in the rectangular structure, ^{a defect} in the form of an extra row meeting ^{an extra column} (1) ^{at the} core, since ^{one extra} modulation period ^{means an}

extra

wx

extra wy

Therefore, it is found here that the new pinching instability ^of convective rolls produces ^a stable structure and builds a new rotation mode of the flow.

The symmetry ^{of the new} flow is then lowered and this instability appears ^as the first important step in the evolution of the flow field before reaching

the bimodal and next the chaotic state.

Figure 5

Defect in the "varicose" structure. It is a

localized jump by ^one period ^{in the modulation} along the rolls.

This pinching instability ^{occurs here with a well-}

defined wave-vector, in order to homogeneously (in space) destabilize the Oblique ^Roll structure. We have also found (12) that the pinching ^can appear well-localized in space inside a Normal Rolls struc- ture. In that case and when the pinching ^is total, it leads to the nucleation of a dislocatious pair. ^We

also show that topology ^{of a} dislocation core is indeed quite ^{similar to that of the} pinched zone, and this result is important ^to understand the inter- action of the defects with the structures to the rec-

tangular ^structure (bimodal) ^{and next to the chaos} (12).

A structure having ^{similar features to the} perio-

dical pinching ^had already been observed in the thermal convection of isotropic ^fluids (10),was ^named "skewed- varicose", ^and contrary ^{to our} case, ^was usually obtained as a transient state which destabilizes

an initially imposed ^roll structure. In addition, ^the precise local evolution of the flow lines was not studied in such experiments. The fact that in our sys- tem the structure is stationary might ^{be due to the}

elastic restoring ^torques specific ^{of the} liquid crys- tal. However the particular periodically ^localized change ^induced by ^the pinching ^{in the flow structure} might ^{not be} specific ^{of the} anisotropy ^{of the fluid.}

In conclusion, ^{we have} found in the convection of an anisotropic fluid, ^{a new} instability ^of pin- ching,of the rolls that leads to a stationary ^struc-

ture. The pinching ^{of a} convective roll breakes the translational symmetry ^{of the} flow and induces a new

local rotation mode which becomes decoupled ^{from the} initial mode as the amplitude ^{of the} instability

increases. Therefore this pinching instability ^makes

the transition from a bidimensional to a three-dimen- sional flow which in our system will be the last ordered structure before the chaos. It is the main step in ^the disorganisation ^{of a} convective flow and the question is raised of its possible counterpart in isotropic ^fluids.

This work was supported by the Direction des Recherches et Etudes Techniques.

References

(1) a) ^A. Joets, ^R. Ribotta in "Cellular Structures in Instabilities". Lecture Notes in Physics,

210 ed. by ^J.E. Wesfreid, ^S. Zaleski, Springer (1984), p. 294.

b) ^A. Joets, Thèse 3ème cycle, ^{Paris VII} (1984).

(2) A. Joets, R. Ribotta, ^to appear in J. Physique ⁴⁷ (1986) 595.

(3) R. Ribotta, ^{A. Joets, Lin} Lei, Phys. Rev. Lett.

56 (1986) 1595.

(4) ^F.H. Busse, ^R.M. Clever, ^J. Fluid Mech., 91, 319 (1979).

(5) Orsay Liquid Crystal Group, Phys. Rev. Lett. 25

1642 (1910).

(6) R. Williams, ^{J. Chem.} Phys. 39, ^{384 (1963).}

(7) ^R. Ribotta, ^{A. Joets, to be} published ^(1986).

(8) R. Cano, Bull. Soc. Fr. Minér. Cristall., 90,

333 (1967).

(9) ^S. Kai, ^K. Yamaguchi, ^K. Hirakawa, Jap. ^J. Appl.

Phys. 14, ^{1385 (1975).}

P.H. Bolomey, ^C. Dimitropoulos, ^Mol. Cryst. Liq.

Cryst. 36, ^{75 (1976).}

(10) ^F.H. Busse, Rep. Prog. Phys., 41, ¹⁹³⁰ (1978).

and in "Hydrodynamic Instabilities ^{and the transi-}

tion to Turbulence", ^ed. by ^H.L. Swinney,

J.P. Gollub, Springer (1981).

(11) R. Krishnamurti, ^{J. Fluid.} Mech., 42, ^{295 (1970).}

(12) R. Ribotta, ^A. Joets, ^X.D. Yang, ^{to be} published.