Transition from circular Couette flow to Taylor vortex flow in dilute and semi-concentrated suspensions of stiff fibers

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flow in dilute and semi-concentrated suspensions of stiff fibers

Blaise Nsom

To cite this version:

Blaise Nsom. Transition from circular Couette flow to Taylor vortex flow in dilute and semi- concentrated suspensions of stiff fibers. Journal de Physique II, EDP Sciences, 1994, 4 (1), pp.9-22.

�10.1051/jp2:1994112�. �jpa-00247953�

Classification

Physics

Abstracts

47.20-47.50-47.55K

Transition &om circular Couette flow to Taylor vortex flow in dilute and semi-concentrated suspensions of stiff fibers

Blaise Nsom

Laboratoire des Ecoulements

G40physiques

_et

Industriels,

Institut de

M4canique

de

Grenoble(*),

B-P-

53X,

38041 Grenoble

Cedex,

France

(Received

11 May 1993, revised 28

July1993,

accepted 21

September 1993)

R4sumd. Nous _{proposons une} 4tude th40rique de la stabilit6 de l'6coulement de Couette

de suspensions dilu6es _ou semi-concentr6es de fibres

rigides.

En utilisant

l'6quation rh4010gique

d'4tat du fluide

anisotropique

d'Ericksen dans le _cas de l'entrefer

large,

_nous calculons num4ri- quement ^le nombre de

Taylor

et la

longueur

d'onde critiques, _en fonction du rapport

d'aspect

des fibres et de la

largeur

de l'entrefer _pour

une

suspension dilu4e,

et _en fonction de la _concen- tration

volumique

des fibres, ^de leur densit4 et de la

largeur

de l'entrefer _{pour une} suspension

semi-concentrde,

Abstract. We report the results of _a theoretical

study

of the

stability

of _a Couette flow of dilute and concentrated

suspensions

of stiff fibers. The Ericksen

anisotropic

fluid

rheological

equation of state in the

wide-gap

situation is used to compute ^the critical

Taylor

number and wavenumber of the

suspension

_{as a} function of the aspect ^ratio of the fibers and the

gap-width

in the dilute case; and _{as a} function of the volumic concentration of the fibers, their

density

and the

gap-width

in the semi-concentrated _case.

1

Background.

Since

Taylor's original

work

[I]

on flow between concentric

cylinders,

the theoreticians have

generally

concentrated their researches in three main directions.

I) Pointing

out the different

instabilities,

after the onset of

Taylor

vortices.

The first attempts were made in the _{same way as} for the onset of the

Taylor

vortex flow: I-e-

superimposing

a small

perturbation (with

_a form which is

suggested by

the

experiments)

_on the trivial Couette state

[2]. But,

the mathematical

developments rapidly

become _so

complicated

that the

possibilities

of

extending

this

procedure

_appear

quite

limited.

(*)

CNRS-UJF-INPG ENSHMG.

Then,

_{a more}

poweriul procedure

_was undertaken in

[3, 4], through

^the biiurcation

theory,

which does not

postulate

_a

priori

^the iorm oi the

perturbation

and

thus,

_can

point

out all the instabilities obtained irom the solution oi the

equation

oi motion and

then,

a

posteriorj,

manages ^to observe them

experimentally.

ii) Characterizing

the process oi _appearance oi turbulence. In Landau's

theory,

turbulence is the consequence oi the

amplification

oi _a small

perturbation

in the

flow, through

an

exchange

oi energy between the

perturbation

and the main

flow,

and

thus,

turbulence may occur aiter

an infinite number oi instabilities

[5, 6].

In

1979,

Fenstermacher et al. [7]

proved

that this _{was not} the _correct scheme. In

iact,

turbulence _occurs aiter _a finite number oi instabilities.

Also,

Barcilion et al. _[8] showed

that,

at very

high Taylor numbers, (about 400.T~ ).

^G6rtler

vortices formed close to the

cylinder wall, coexisting

with the

Taylor

vortex flow.

iii) Taylor

vortex flow in

complex

fluids.

Most of the papers are devoted to

polymeric solutions,

which exhibit viscoelastic behaviour

[9-12].

^In these

fluids,

the two

components (solid

and

liquid)

do not exist

individually. They

form a

unique liquid,

whose constitutive

equation

_can ^{be built} from _a

microscopic theory

_or

from _a network

theory, according

to the concentration of the

polymer.

The _case oi the

suspensions

oi

macroscopic

fibers in _a Newtonian

liquid,

which is the

subject

oi the present _{paper, may} ^be

applied

to many industrial processes

(paper manuiacture,

water

treatment...).

It differs irom that oi the latter fluids in

that,

because oi the effective

physical

presence oi the

dispersed phase,

_say, the

fibers,

the width oi _a confined container and the

dimensions oi the solid

particles

must be taken into account.

Our attention iocussed _on the

hydrodynamic ieatures,

in order _to

point

out the iundamental aspects oi these flows. Thus _we

neglect physico-chemical

iactors such _as chemical and _van der Waals

interactions, deiormability

and

polydispersity

oi the fibers. In

addition,

the medium is

submitted _to three

types

oi interactions:

"fiber-fluid",

"fiber-fiber" and "fiber-boundaries".

The "fiber-fluid" interaction is used to build the constitutive

equation

^oi the

suspension

in the dilute range oi

concentration,

I,e, the fluid behaviour about each solid

particle

is unaffected

by

the _presence oi the others. Ii this condition is _no

longer

met

(undilute suspension),

the

"fiber-fiber" interactions have to be taken into account ior the elaboration oi the constitutive law oi the

suspension.

These

approximations

will _now be discussed.

1 I THE FIBER-FLUID INTERACTIONS.

Jeflery

₍₁₃₎ studied the motion oi _a

single ellipsoid

in _a Newtonian

fluid,

_at low

Reynolds

numbers. He iound the

iollowing equation

to govern the material derivative oi the orientation vector oi the

ellipsoid,

d

(unit-vector along

the direction oi the

length)

d=Q.d+I(@-d:ddb)d (I,I)

where fl is the

vorticity,

the rate oi strain

tensor,

b the unit tensor, ^{and I is} a

parameter.

Batchelor

[14, 15] proposed

the

iollowing

modified Newtonian constitutive

equation

ior _a dilute

suspension

a =

-pb

+

2qo£

_{+ co}

Lap (1.2)

where _{no is} the

viscosity

oi the

suspending liquid,

_co

i

the volumic concentration oi _one

particle,

p the

isotropic pressure-field,

_ap the

"particle-stress",

which is the contribution to the stress in the medium _a oi _a

single

fiber and the summation is

brought

_on all the

particles.

Ericksen

[16],

proposes a

rheological equation

^ior

anisotropic

fluids

a =

-pb

+ 2qo@ ⁺

(nidd +1l2@dd)

dd ₊ 21l3(@ dd + dd _@)

(1.3)

where the

il(s (I

=

1, 2, 3)

_are

rheological

coefficients. Note that _an alternative iorm oi this

equation

^is

given by

Tucker

[17].

^This constitutive

equation (1.3) requires

not

only

the orien-

tation oi _a

single fiber,

but the orientation oi all the fibers in the medium. This orientation state oi the fibers is

given by

the orientation distribution iunction ~l

(d~, t)

It is the

probability

oi

finding,

at time t, a fiber such that its orientation-vector is contained in the solid

angle

oi axis

d',

and

hali-angle ~ (d~,t),

in the

configuration

_space

(d-space).

It

obeys

the

iollowing

conservation

equation

dfi(d~ t)j

^d

(~l(d, t) 'd,j

dt

ad, ~~'~~

Substituting equation (I,I)

ior the motion oi _a

single

fiber into

(1.4),

leads to _a Fokker- Planck

equation

ior

~ (d,, t).

It is not necessary ^to solve this

equation explicitly. Following

Advani and Tucker

[18],

Grmela and Carreau

[19],

we introduce the orientation distribution function to

perform

the

summation of

(1.3)

_on all the fibers:

Lap

= nco

/ ap~(d, t)dd (1.5)

where _n is the number of fibers _per unit volume.

Then, denoting by

_a2 and _a4 the orientation tensors defined

by

a2 "

/ _{dd~(d)dd (1.6)}

a4 "

/ _{dddd~(d)dd (1.7)}

introducing

them in

(1.3), gives

Lap

₌

lap

₌

-pb

_{+ 2ilo@} + ilia2

+1l2@

a4 + 21l3_(@ a2 +

a2£) (1.8)

Hand

[20],

^{Hinch and Leal}

[21]

^{introduce the}

equation

of motion of _a

single

fiber

(I,I),

and the

equation

of

continuity

^for

~(d, t) (1.4),

in the definition of _a2, to

get

the

equation

of evolution of that _tensor. The

equation

which is obtained contains the fourth-rank tensor a4.

Similarly,

the evolution

equation

oi _a4 contains _a sixth-rank tensor, and so on. In order to

generate

a closed set oi

equations,

it is necessary to make _a

closure-hypothesis. Generally,

this is done

by approximating

a

higher-rank

tensor in terms of _a lower

[22],

^{for instance}

a4 * a2 a2

(1.9)

The limitation oi this

procedure

is that the solution is

dependent

on the closure

approxima-

tion which has been used.

A second scheme is

proposed by

Dinh and

Armstrong

_{[23] ior}

homogeneous

flows.

Using

different

hypotheses, they

iorm _a constitutive

equation

ior the stress in terms oi _an

integral

_over

a iunction oi the

Cauchy

strain tensor, ior the orientation.

So, they

obtain

integral expressions

ior the transient

viscosity

and normal stress coefficients in

start-up

^oi

steady

shear flows and for the transient

elongational viscosity.

The limitation oi this

procedure

is that it cannot be used ior all flow situations.

A third

scheme,

which is _more

natural,

is to _assume that all the fibers lie

along

the _stream- lines

[24]. Then,

the orientation _vector is

locally given by

the

velocity-field.

This scheme is valid _as

long

_as the fibers

align

with

respect

to the streamlines.

Indeed,

this is the intrinsic

limitation oi this

procedure,

which

requires experimental

verification that the fibers _are indeed

aligned along

the streamlines. These "fluid-fiber" interactions _are the

only

_ones which _are to be considered

ij

a dilute

suspension, I,e.,

when the total concentration oi the solid

particles

is

less than ~

,

[23].

L

To close this

sub-section,

_note that when the

particles

_are

cyclindrical

rods oi

large aspect-

ratio

(

=

~,

_which

is the _case considered in the present paper, the

rheological

coefficients in D

the above relations _are such that [25]

~li, Q3 ^" o

(I.lo)

q~~L3~~

~~

61n(2()

^~~~~

(l,ll)

where

e =

(In(2())~~ (l.12)

ior finite

_aspect

ratio

_fibers and

f(e) = _I

(~Qo (1,14)

~~

ln(r)

1.2 THE "FIBER-FIBER" AND "FIBER-BOUNDARIES" INTERACTIONS When the total _con-

centration oi the solid

particles,

_c, lies in the range

ID

~ D

i~ ~

_~

~

L

(1,15)

the

suspension

is said _to be

semi-concentrated,

and the

dynamics

oi the flow _are affected

by

the "fiber-fiber" and "fiber-boundaries" interactions.

Shaqieh

et al. [26] ^{have calculated} these interactions.

They

established the two

iollowing

main results:

the

"fiber-boundary"

interactions _are

negligible

with

respect

to the "fiber-fiber" interac-

tions;

introducing

the "fiber-fiber" interactions in the

dynamics

oi the

medium,

its constitutive

equation keeps

^{the iorm}

(1.8),

but the

rheological

coefficient _{1l2 now} has the

iollowing expression

~rnL~~o

~~ 3

('n (I/C)

+ In

('n I/C))

₊

A) ^~'~~~

with A

= -0.66 for _a

suspension

^{in which all} orientations _are

equiprobable,

and A

= 0,16 when

all of the

particles

_are

aligned

in _{a common} direction.

In

fact,

^from Stover's

experimental study [25],

even at the

highest concentrations,

the fibers

are well

aligned

with the streamlines. This

study

_was

performed

for _a

large

_range of _concen- trations and aspect ^{ratios in} a

cylindrical

Couette device.

This

experimental

evidence is the

justification

for _our choice of the third closure _approx- imation.

Thus,

in the

equation

oi

state,

^the

velocity-field

will

simultaneously

describe the

kinematics and the "orientation-state" in the flow.

Also,

it is clear that the dilute and the semi-concentrated

suspensions

_can be treated in _a similar _way since

they

_are described

by rheological equations

oi the _same iorm. The details oi

the two _cases will _appear in the numerical

results,

in the

respective

_ranges oi the concentration.

2. Circular Couette flow.

We consider the flow between _two coaxial

cylinders,

the outer, with radius

R2 being

^fixed and the

inner,

with radius

RI

in rotation, ^and

angular speed Q;

_q denotes the ratio

~~.

R2

In _a

cylindrical system

oi coordinates with _z

along

the _common axis oi the

cylinders,

the circular Couette

regime

exists when the

Reynolds

number is low. It is described

by

_a

tangential velocity-field,

I-e- oi the iorm

V =

V(r)eo (2,1)

and the orientation-state is

given by

d

= eo

(2.2)

while the

components

of the rate of strain tensor

(which

is

symmetric)

_are

and _zero otherwise.

Then, using

the constitutive

equation (1.8)

the

components

of the stress-tensor

(also

_sym-

metric)

_are

given by

all " a22 "'33 "

~P(~)

_~~'~~

a13 " a23 " 0

(2.6)

The

equation

^oi

continuity

is

identically satisfied,

while the

equations

^of motion reduce to

~))~

⁺

2')~

^{= °}

^(2.8)

Substituting (2.4-2.6)

ior the

a,j's,

a

straightiorward

calculation leads to

V = AT ₊ ^~

(2.9)

~2

~2

P = p

(A~-

_~ ^{+ 2A B}

^n(r))

⁺

^{Po (2.10)}

2 2r

A and

B,

the constants oi

integration,

are determined

by

the

no-slip

condition of the fluid _on the

cylinders

walls

V = 0 ior _r

=

RI

and

R2 (2,ll)

So,

a~2 QR)

(2,12)

A =

~i

B "

1 _~/2

while

Po

is _an

arbitrary

constant.

Indeed,

this solution also holds ior the

general equation

oi state

(1.3)

and without _any closure

approximation [27, 28].

^{In this} way, Leslie [28] studied the

stability

oi _an

anisotropic

fluid oi Ericksen

type

with the

small-gap approximation.

His results _are in

good agreement

with the

experimental study

_on dilute

polymer

solution oi Rubin and Elata

[29].

As stated in the first

section,

^the

small-gap approximation

^is not

appropriate

ior _{our case} oi

macroscopic

solid

particles.

Thus a

wide-gap study

is

required

where consideration oi the

aspect

ratio oi the fibers and their volumic concentration is taken. In

iact,

it is not

surprising

that this

study

had not been made earlier

because, firstly,

the calculation oi the

rheological parameters

^ior dilute

suspensions

oi stiff

cylinders

was

only

made in 1986

by Lipscomb _[30],

and

secondly,

the summation oi the "fiber-fiber" interactions in _a semi-concentrated

suspension

was carried out ion 1990

by Shaqieh

et al.

[26].

3.

Equations governing

the transition.

Ii the

angular speed

oi the internal

cylinder

is

progressively increased,

the flow

develops

_ax-

isymmetric counterrotating

toroidal

cells, periodically

distributed

along

the _common axis oi the

cylinders.

This is

Taylor

vortex flow and is described

by

the

superposition

oi _a small

perturbation

oi the iorm

ur(r, z)

=

u(r, z)

=

fi(r) cos(kz) (3.I)

uo(r, z)

=

u(r, z)

₌

@(r) cos(kz) (3.2)

uz(r, z)

=

w(r, z)

₌

I(r) sin(kz) (3.3)

P(r, z)

=

fl(r) CDS(kz) (3.4)

on circular Couette flow.

The introduction oi these

perturbations

into the

equation

^oi

continuity

leads to

tb(r)

=

~"

(3.5)

In the rest oi the paper, we shall

drop

the bars _over the iunctions

u(r), u(r), w(r)

and

p(r),

to

simpliiy

the notation.

Ii s~ ^{denotes the}

components

^{oi the} stress-tensor oi the

perturbation,

let _us introduce the

same

superposition

in the

equations

of motion of the

flow,

_we find the

following equations

Using again the roperty oi

lignment of

it-vector

to ^V

di

=

) (3.9)

d3

=

d2

= 1 ^{- d(} - d( (3,ll)

iii

₌

~"~~~cos(kz) (3,12)

dr

@22 =

"~~~cos(kz) (3.13)

r

@33 "

kw(r)cos(kz) (3,14)

@23 "

~Ku(r)sin(kz) (3.15)

2

The stress-tensor and the rate of strain tensor of the

perturbation

_are related

by

the rheo-

logical equation

oi state oi the fluid

(1.8).

Then the s,j ^{have the}

following expressions:

sir "

-P(r)

₊

2qo (~~~

cos(kz) (3.18)

r

822 "

~P(T)

+

~)~

^{(2Q0 +}

^{2))CDS(kZ) (3.19)}

s33 =

(-P(r)

₊

2iJokW(r))CDS(kz) (3.20)

s12 "

llorj ~~~)cos(kz)

(3.21)

r r

s13 " llo

(-k ^~1(r)

⁺

^(~~~ _lsin(kz) ^(3.22)

r

s23 "

-llok u(r)sin(kz) (3.23)

Introducing

the

expressions

of the orientation vector

(3.9-3.ll)

and those oi the _{stress tensor}

(3,18-3.23)

in the

equations

oi motion

(3.6-3.8),

and

taking

into account the

hydrodynamic

field oi the circular Couette flow obtained in the second

section,

we

get

~~~

~ ~°

~ ~~~~

^~

^~~"~~~

^~

_~~~~~

^{~ ~}

_~(~~ ^~~~

~~

~~~ ^~~~~~~

(3.24)

~Jo

ITS _(~l~~)

+

31 ^{~l~~ ~U(r))}

=

2PA~I(r) (3.25)

From

(3.26),

_{we can}

_get

the

expression

oi the pressure field

p(r)

and introduce it in

(3.24, 3.25), using

then the

equation

^oi

continuity (3.5),

_we obtain

(DD*)~

2k~ (DD*

+

k~

₊ ^~~

~

~1=

2p~

A _{+ u}

(3.27)

llo

~

1~

r

(DD* k~)

u = 2A ^~ _~1

(3.28)

llo where

D =

~

and D*

=

~

+

(3.29)

To non-dimensionalize this

system,

we use the reduced _wave number _a, defined

by

a =

kR2 (3.30)

and _{we measure} the radial coordinate _r in units oi the radius oi the external

cylinder R2. So,

it il denotes the ratio oi the radii oi the

cylinders Ri/R2

(

_"

ii

11 I

(

I 1

(3.31)

Then,

_we make the substitution

~~"~~~

~

- u

(3.32)

llo

and _we introduce the

Taylor number,

with its usual definition

4AB

2~2 (3.33)

~~

il(

^{~ ~}

Then the

system (3.27, 3.28)

^beconles

liDD* ^~~)~

⁺

ii ill

₁₁

-

-Ta~ II _ii

U

(3.34)

(DD* _a~)

_{u = ~1}

(3.35)

These _are the

equations

^of motion of the

Taylor

vortices. For _a

given

Couette apparatus and _a

given aspect-ratio

oi the

fibers, they depend

_on two

parameters:

T

(Taylor number),

and _a

(wave number),

which determine the appearance of the

instability.

They

_are

completed by

the

no-slip

condition _on the

cylinders

walls

u(()

=

u(()

=

D11(()

₌ 0 at

(

= il and 1

(3.36)

Putting1l2

" 0 in that

system (3.34, 3.35), (no

fiber in the

medium),

_we obtain

again

the

equations joveming

^{the transition in a Newtonian}

^liquid.

So,

^~~

~j

~1 can be _{seen as a}

perturbation

_term,

characterizing

the presence oi the fibers in 110

f

the medium. This

perturbation

term

depends

_on the size and the concentration of the

particles, through

the

rheological

coefficient _q2.

4. Solution.

Indeed, substituting

_u in

(3.34)

in the form

given by (3.35),

the

system

established in section 3 reduces to the

iollowing single equation

liDD* ^a~)~

⁺

II

iD~ ~)I

U -

-Ta~ II _ii

U

(4.1)

We solve this

equation

oi motion

by

the method oi Chandrasekhar [31] ^which

expands

the

expression ((DD* a~) u)

ⁱⁿ terms of the

following

set oi

orthogonal

functions

Cl (ajr)

m

(DD* a~)

u =

£ _{( Cl (aj() (4.2)}

j=1

determined

by

the characteristic value

problem

described

by

the

iollowing equation

(DD*)~

-

IS

+

II II

^~

=

°~Y (4.3)

and the

boundary

conditions

hese haracteristic functions are

xpressed

as

inear ombinations

of

the Bessel iunctions

Cl (aj()

=

Aj ^Ji (aj()

₊

_B( Yi (oj()

+

C(Ii (oj()

+

D(Ki (aj() (4.5)

where aj ^is a root oi _a transcendental

equation

which is

given

in

[31].

Thus,

the

iollowing

secular

equation

has been obtained ior the

stability

Nj

°~

~~

T

~~

~

bjk

+

2a)A~)~

+ ~~

Mjk

T

(pj _(1)~~(a) -1)~~~(a))

⁺

a

a~

^{a J} @

+ qj

(K)~~(a) K)~J(a)) ^~°~

_~

(Aj)~ Aj[~~)

~~

~

Mjkl"

⁰

^(4.6)

a~

a

o~

^a

where

Njbjk

=

/ _(nj

+

vj) ink

+

vk)

rdr

(4.7)

~

i

A))~

=

/ _(nj

vj

ink

+ vk rdr

(4.8)

~

A)[~l

=

/~ (nj vj) ink

+

vk) ~dr _(4.9)

r

Mjk

₌

/~ ^(nj

+

vj) ink

+

vk)

dr

(4.10)

~ r

1)~~(a)

=

/ _{Ii(ar) (nk}

+

vk)rdr (4.ll)

~

I)~~~(a)

₌

/ _{Ii(ar) ink}

+

vk) ~dr _(4.12)

~

r

K)~~(a)

₌

/~ Ki(ar) ink

+

vk)

rdr

(4.13)

K)~~~(a)

₌

/ _{Ki(ar) ink}

+

vk) ~dr _(4.14)

~

r

and the Bessel functions _are calculated

numerically according

to [32]

figures

I and 2

give

the variation of the critical

Taylor

number and _wave number with _q

=

~~

in the absence of the

R2

fibers

(Newtonian fluid). They

show that the critical parameters increase with ratio q, in _a

good agreement

^{with the values obtained}

by

Roberts

[33].

Figures

3 to 6

give

the relative variation of the critical

Taylor

number with the various

parameters

of the

problem.

From these

graphs,

the effect of the _presence of the fibers in the medium _can

readily

been

pointed

out. In all the situations which have been considered

(dimensions

of the

particles

and ratio

q),

in the dilute _range of

concentrations,

the presence of the fibers does not alter the critical

wavenumber;

the critical

Taylor

number increases with ratio q;

the critical

Taylor

number increases with the

aspect

^ratio of the fibers.

while,

in the semi-concentrated _range,

again,

the _presence of the fibers does not alter the critical

wavenumber;

the critical

Taylor

^{number increases with the}

^particle

^volumic

concentration,

for _a

given

ratio q and

density;

120000 8

iooooo

~

80000 T~ 6

~~~o~ ac

5 40000

20000 ⁴

0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0,6 0.7

n n

Fig.

I

Fig.

2

Fig.

I. Critical

Taylor

number

against

_q in the absence of the fibers.

Fig.

2. Critical wavenumber against _q in the absence of the fibers.

Tc/Tco-1 ~_~~ ^Tc/T~-1

0.3 O.25

~~~~

O.25

O.3 _o,2

i~ O.3

O.4

o_5 ^o-1

~'~

O.5

~'~

O.6

0 , , ,

0 20 40 60 80 0.00 0.01 0.02 0.03 0.04 0,05 0.06

(

_c

Fig.

3

Fig.

⁴

Fig.

3. Relative variation of the critical

Taylor

number with the aspect ^{ratio of} the fibers in _a dilute suspension, for diverse values of _q.

Fig.

4. Relative variation of the critical

Taylor

number with the volurnic concentration of the fibers in _a semi-concentrated suspension, for diverse values of _q and for _an

assigned

value of the

density (nL~

=

45).

the critical

Taylor

number increases with the

particle density,

for _a

given

ratio _q and

concentration;

the critical

Taylor

number increases with

decreasing

ratio _q, for _a

given

concentration and

density.

0.3

~ ~ ~~

40

~~ 35

30

Tc/T~-1 25

20

0.1 ~~

io 5

0.00 0,01 0.02 0,03 0,04 0.05 0.06

c

Fig.

5. Relative variation of the critical

Taylor

number with _q in _a semi-concentrated

suspension,

for diverse values of the

density,

and for _an

assigned

value of the concentration

(c

=

0.06).

C=O.06

O.049

O.037 Tc /Tco i

O.025

°'~ O.O13

o.ooi

i , i i 1

0 lo 20 30 40 50

nL3

Fig.

6. -Relative variation of the critical

Taylor

number with the

density

in _a semi-concentrated

suspension,

for diverse values of the concentration, and for _an

assigned

value of _q

in

=

0.2).

The latter results

(semi-concentrated case)

_are in _a

qualitative agreement

with the electro- chemical measurements

performed by

Skhli

[34]

ⁱⁿ

suspensions

^of calibrated

synthetic

fibers.

5 Conclusion.

A

suspension

of stiff fibers of

large aspect

^ratio has been

considered,

in flow between two coaxial

cylinders,

with the inner

rotating

and the outer

being kept

at rest.

In the dilute _range of

concentration,

the

suspension

is known to

obey

the Ericksen

anisotropic

fluid

equation

of state. The

rheological parameters

which appear in this law _are related to the

dimensions of the fibers.

Furthermore,

due to the

macroscopic

nature of the solid

particles,

it _{was necessary} to consider the situation of _a

wide-gap Couette-Taylor problem. Thus,

the transition from circular Couette flow to

Taylor

vortex flow in the

suspension

has been char-

acterized and

compared

to the _case of _pure Newtonian

liquid.

We have found that in all the situations which have been considered

(dimensions

of the

particles

and ratio q:

the _presence of the fibers does _not alter the critical wavenumber;

the critical

Taylor

number increases with ratio q;

the critical

Taylor

number increases with the

aspect

^{ratio of the}

fibers;

and

the relative variation of the critical

Taylor

number increases with

decreasing

ratio q.

In the semi-concentrated _case, when the "fiber-fiber" interactions summation is taken into

account,

the Ericksen law remains valid but the

rheological parameters depend

_on the volumic

concentration of the solid

particles

and _on their

density. Then,

_we have found that:

again,

the _presence of the fibers does not alter the critical

wavenumber;

the critical

Taylor

number increases with the

particle

volumic

concentration,

for _a

given

ratio _q and

density;

the critical

Taylor

number increases with the

particle density,

for _a

given

ratio q and

concentration;

the critical

Taylor

number increases with

decreasing

ratio _q, for _a

given

concentration and

density.

The latter results

(semi-concentrated case)

_are in _a

qualitative agreement

with

experimental

electrochemical measurements

reported

in the literature.

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