16666161

Drawing of annular liquid jets at low Reynolds numbers

J.I. Ramos*

Room I-325, E.T.S. Ingenieros Industriales, Departamento Lenguajes y Ciencias de la Computacion, Universidad de MaÂlaga, Plaza El Ejido, s/n, 29013 MaÂlaga, Spain

Received 18 July 2000;revised 25 January 2001;accepted 25 January 2001

Abstract

Asymptotic methods based on the slenderness ratio are used to obtain the leading-order equations that govern the ¯uid dynamics of axisymmetric, isothermal, Newtonian, annular liquid jets such as those employed in the manufacture of textile ®bres, annular membranes, composite ®bres and optical ®bres, at low Reynolds numbers. It is shown that the leading-order equations are one-dimensional, and analytical solutions are obtained for steady ¯ows at zero Reynolds numbers, zero gravitational pull, and inertialess jets. A linear stability analysis of the viscous ¯ow regime indicates that the stability of annular jets is governed by the same eigenvalue equation as that for the spinning of round ®bres. Numerical studies of the time-dependent equations subject to axial velocity perturbations at the nozzle exit and/or the take-up point indicate that the annular jet dynamics evolves from periodic to chaotic motions as the extension or draw ratio is increased. The power spectrum of the annular jet's radius at the take-up point broadens and the phase diagrams exhibit holes at large draw ratios. The number of holes increases as the draw ratio is increased, thus indicating the presence of strange attractors and chaotic motions.q2001 Elsevier Science Ltd. All rights reserved.

Keywords: Perturbation methods;Annular jets;Isothermal;Newtonian;Linear stability;Nonlinear dynamics;Chaos

1. Introduction

Although there has been quite a lot research on the devel-opment of one-dimensional, mathematical models for the analysis of spinning processes of round jets at low Reynolds numbers under both isothermal and non-isothermal condi-tions [1,2,3], the ¯uid dynamics of hollow or annular liquid jets at low Reynolds numbers (cf. Fig. 1) has received little attention in the past, despite the fact that annular liquid jets are used in ®lm blowing processes [4], which are methods of producing thin sheets of thermoplastics rather more rapidly and economically than is possible with casting processes, and in hollow ®bre drawing [5]. Annular liquid jets are also used in spray applications, in protection systems of inertial-con®nement laser fusion reactors [6,7], and in ink-jet print-ing and particle sortprint-ing where the Reynolds numbers involved are so large that the ¯ows may be considered as inviscid. When surface tension effects are important, annu-lar liquid jets at high Reynolds numbers may acquire closed tulip shapes and may be used as chemical reactors [8].

Pearson and Petrie [9] studied the steady axisymmetric ¯ow of a thin tubular liquid ®lm at low Reynolds numbers in streamline coordinates using as a perturbation parameter the

ratio between a characteristic thickness and a characteristic mean radius. They also performed a phase plane analysis for very low Reynolds numbers in the absence of gravity and surface tension [10], whereas Yeow [11] analyzed the linear stability of the model proposed by Pearson and Petrie [9] when the disturbances are axisymmetric and the ¯ow is isothermal.

Yarin et al. [5] studied the stability loss in isothermal, hollow ®bre drawing by means of a one-dimensional model based on the Cosserat theory of ¯uids, and analyzed the linear stability of slender hollow ®bres at zero Reynolds numbers. They also showed the existence of a root with a positive real part by means of an argument principle and found a critical draw ratio, i.e. the ratio of the axial velocity at the downstream boundary to that at the upstream one, which is identical to that found by Pearson and Matovich [2] and Schultz and Davis [12]. In addition, Yarin et al. [5] analyzed numerically the non-linear dynamics of iso-thermal, hollow ®bre drawing at low Reynolds numbers by solving the discretized forms of their one-dimensional equations subject to perturbations on the hollow jet's axial velocity component and radii at the upstream and down-stream boundaries, and showed the existence of a draw resonance instability. Gospodinov and Yarin [13] studied numerically the draw resonance of hollow microcapillaries under non-isothermal conditions and showed that these 1089-3156/01/$ - see front matterq2001 Elsevier Science Ltd. All rights reserved.

PII: S1089-3156(01)00016-2

www.elsevier.com/locate/ctps

* Tel.:134-95-2131402;fax:134-95-2132816. E-mail address:[email protected] (J.I. Ramos).

conditions result in a quasiperiodic phenomenon with no tendency to chaos.

Previous studies of isothermal, Newtonian, annular jets at low Reynolds numbers have considered iso-thermal, steady-state ¯ows or determined the linear stability of these ¯ows;however, none of these studies has considered the possible steady-state solutions of isothermal, steady, annular ®bres, and determined the effects of the downstream boundary conditions, time-dependent perturbations on the jet's radii, axial velocity and traction and ¯uid dynamics parameters on the non-linear dynamics of these ®bres for draw or extension ratios beyond that determined from linear stability analyses.

The objective of this paper is manifold. First, the leading-order ¯uid dynamics equations of annular, isothermal, Newtonian jets at low Reynolds numbers are derived by means of perturbation methods based on the slenderness ratio [3,14]. These jets have two interfaces and are used in the manufacture of optical and textile ®bres. Moreover, the assumption of Newtonian rheology is in some cases justi-®ed;for example, certain polymer melts, such as those of poly(ethylene terephthalate) (PET) and polyamides (PA), are indeed Newtonian in behaviour at deformation rates and temperatures characteristic of many industrial processes involving them, e.g. injection moulding [4]. Second, analy-tical solutions to the steady-state solutions are obtained for zero Reynolds numbers, zero gravitational pull, and inertia-less jets. Third, a linear stability analysis of the viscous ¯ow regime corresponding to zero Reynolds number is performed analytically and shown to be governed by the same eigenvalue equation as that for isothermal, round jets at low Reynolds numbers [12]. Fourth, numerical studies of the time-dependent equations are performed in order to determine the non-linear dynamics of annular jets

as a function of the non-dimensional parameters that govern the ¯ow, and the location, amplitude and frequency of the applied velocity perturbations. These studies are performed for axial velocity components at the upstream boundary or at the take-up point higher and lower than those determined from the linear stability analysis.

2. Formulation

Consider an axisymmetric, annular liquid jet such as the one shown schematically in Fig. 1, consisting of an incom-pressible (constant density) ¯uid that is isothermal and Newtonian. The interfaces of the annular jet correspond to R1…t;x† and R2…t;x†; and R1…t;x†#r#R2…t;x† with

R1…t;x†±0: The ¯uid dynamics of the annular jet are

governed by the following conservation equations of mass and linear momentum

2u 2x 1

1 r

2…vr†

2r ˆ0; …1†

r 2u_2t 1u2u 2x 1v

2u 2r

!

ˆ22p_2x 1m _2x22u₂ 1 1_{r 2r}2 r2u_2r

!!

1rg;

…2†

r 2v_2t 1u2v_2x 1v2v_2r

ˆ22p_2r 1m _2x22v₂ 1 _2r2 1_r 2…rv†_2r

!

;

…3†

where t is time;u andv are the axial and radial velocity components, respectively;xandr are the axial and radial

coordinates, respectively; r and m are the ¯uid's density

and dynamic viscosity, respectively;p is the pressure;and gis the gravitational acceleration.

Eqs. (1)±(3) are subjected to kinematic and dynamic bound-ary conditions at the jet's interfaces,R₁…t;x†andR₂…t;x†:The kinematic conditions establish that the annular jet's interfaces are material surfaces where the shear stress is continuous, and the jump in normal stresses across the interface is balanced by surface tension. The kinematic and dynamic boundary conditions at the jet's interfaces may be written as

v…R_i;x;t† ˆ 2Ri

2t 1u…Ri;x;t† 2Ri

2x ; iˆ1;2; …4†

2m 2v_2r 2 2u_2x

_2R

i 2x 1m

2u 2r 1

2v 2x

12 2R_2xi

₂!

ˆ0;

rˆRi; iˆ1;2; (5)

2m2u_2x 2R_2x1

₂

12m2v_2r 22m 2u_2r 1 2v_2x

_2R

1

2x

1…pi2p† 11 2R_2x1

₂!

ˆsJ1; rˆR1; …6†

2m2u_2x 2R_2x2

₂

12m2v_2r 22m 2u_2r 1 2v_2x

_2R

2

2x

1…pe2p† 11 2R_2x2

₂!

ˆ2s2J2; rˆR2; …7†

where

Jiˆ

11 2R_2xi

₂!1=2

R_i 2

22_R

i 2x2

11 2R_2xi

₂!1=2; iˆ1;2;

…8†

where si; iˆ1; 2, denotes the surface tension at the

annular jet's inner and outer interfaces, respectively,

and pe and pi are, respectively, the pressures of the

gases surrounding and enclosed by the annular liquid jet. These gases have been assumed to be dynamically passive, i.e. pi and pe are functions only of time, since, in general, they have smaller densities and dynamic vis-cosities than those of liquids. This implies that the gases surrounding the liquid jet may not introduce strong velocity variations along each cross-section of the jet, although they may affect its dynamics.

In addition to the above boundary conditions in the radial direction, conditions in the axial direction must also be provided. If the annular jet emerges from a nozzle, there is a stress singularity at the nozzle±jet interfaces due to the relaxation of the velocity pro®le from no-slip conditions at the nozzle walls to free-surface ¯ow away from the nozzle. This relaxation may result in jet contraction or swelling, implying that the radial velocity component is of importance near the nozzle. Moreover, the stress singularity at the nozzle exit and the jet contraction or swelling near the nozzle may result in a relatively important radial pres-sure gradient near the nozzle;therefore, an accurate analysis of the ¯ow near the nozzle requires a full solution of the Navier±Stokes equations within the nozzle and in the free-surface ¯ow, and the use of numerical methods. In this paper, a long wavelength approximation is used to reduce Eqs. (1)±(8) to a more manageable (and easier to solve) one-dimensional set of equations.

3. Asymptotic analysis of annular liquid jets at low Reynolds numbers

For slender annular jets at low Reynolds number, i.e.eˆ R0=lp1;it is convenient to non-dimensionalizer,x,t,u,v

andpwith respect toR0,l,l=u0;u0,v0andmu0=l;

respec-tively, whereR0andl denote a characteristic radius and a characteristic wavelength in the axial direction, respec-tively,u0is a characteristic (constant) axial velocity

compo-nent, and v₀ˆR₀u₀=l; so that the non-dimensional

governing equations and boundary conditions can be written as 2u 2x 1 1 r 2…vr†

2r ˆ0; …9†

eRe2L…u† ˆ2e22p_2x 1e22 2_u

2x2 1

1 r 2 2r r 2u 2r

1 Re_Fr;…10†

eRe2L…v† ˆ22p_2r 1 2 2_v

2x2 1

2 2r 1 r 2…rv† 2r ; …11†

v…Ri;x;t† ˆ 2R_2ti 1u…Ri;x;t†2R_2xi; iˆ1;2; …12†

2e2 2v

2r 2 2u 2x _2R i 2x 1 2u 2r 1e2

2v 2x

12e2 2Ri 2x

₂!

ˆ0; rˆRi; iˆ1;2; …13†

2e22u

2x 2R₁

2x

₂

122v

2r 22 2u 2r 1e2

2v 2x

_2R

1

2x 1…pi2p†

11 2R_2x1

₂!

ˆ s1

s2

1

eCa₂J1; rˆR1; (14)

2e22u

2x 2R2

2x

₂

122v_2r 22 2u_2r 1e22v

2x

_2R

2

2x 1…pe2p†

11e2 2R2

2x

₂!

ˆ2 1

where

Jiˆ

11e2 2Ri 2x

₂!1=2

Ri 2e

2

22_R

i 2x2

11e2 2Ri 2x

₂!1=2;

iˆ1;2; (16)

L…u† ˆ 2u_2t 1u2u_2x 1v2u_2r; …17† and

Re2 ˆ ru_m0R0; Frˆ u 2 0

gR0; Ca2ˆ

mu0

s2 ; …18†

denote the Reynolds, Froude and capillary numbers, respec-tively, and the same symbols have been used for dimen-sional and dimensionless quantities for the sake of brevity. Eqs. (9)±(17) depend one2_{and the Reynolds, Froude and}

capillary numbers. For small Reynolds numbersRe2ˆeRe

with ReˆO…1†; and, depending on the magnitude of the

Froude and capillary numbers, several ¯ow regimes can be identi®ed. Here, we considerFrˆF=eandCa2ˆCa=e

whereFˆO…1†andCaˆO…1†;which correspond to small

gravitational ®elds and small surface tension. Substitution of these values and expansion of the dependent variables as

fˆf01e2f21O…e4†; …19†

wherefdenotes the dependent variablesu,v,p,R1andR2, in the governing equations and boundary conditions, together with the expansion of the boundary conditions atR₁…t;x†and R₂…t;x†aboutR₁₀…t;x†andR₂₀…t;x†yield asymptotic expan-sions which at leading order, i.e. atO…e0_{†;correspond to}

2u₀ 2x 1

1 r

2…v₀r†

2r ˆ0; …20† 1 r 2 2r r 2u0 2r

ˆ0; …21†

22p0

2r 1 2 2r 1 r 2…rv† 2r

ˆ0; iˆ1;2; …22†

2u0

2r …Ri0;x;t† ˆ0; iˆ1;2; …23† for Eqs. (9)±(11) and (13), respectively. The solutions of Eqs. (20)±(23) can be written as

u₀ˆB…t;x†; p₀ˆD…t;x†; …24†

v0ˆ C_r 2 2B_2x r₂; …25†

whereC…t;x†can be determined from Eq. (12) as

Cˆ _2t2 R₂2i0

!

1 _2x2 BiR

2

i0

2

!

; iˆ1;2; …26†

From Eqs. (14) and (15), one can obtain easily the following equations

22 C

R2 10 1 1 2 2B 2x !

1…p_i2p₀† ˆ s1

s2

1

Ca R10; …27†

22 _RC₂

20 1 1 2 2B 2x !

1pe2p0ˆ2_{Ca R}1

20; …28†

which imply that

p0 ˆDˆpe1 _Ca1 _R1

20 22

C R2 20 1 1 2 2B 2x ! ; …29†

p0 ˆDˆpi2 s_s1 2

1 Ca

1 R10 22

C R2 10 1 1 2 2B 2x ! : …30†

Moreover, Eq. (26) can be written as 2

2t R2

202R210

2

!

1 2

2x B R2

202R210

2

!

ˆ0; …31†

corresponding to the conservation of mass at leading order. Eqs. (26) and (29)±(31) contain ®ve unknowns, i.e.B,C, D,R10andR20;therefore, these equations are not a closed system. In order to close this system of equations, it is necessary to go to higher orders in the asymptotic expan-sion. AtO…e2_{†;Eq. (9) yields}

u₂ ˆ Q

4 r21Mlnr1N; …32†

whereMandNare functions of…x;t†;and

QˆRe 2B_2t 1B2B_2x

1 2D_2x 2 22B

2x2 2

Re

F ; …33†

The shear stress conditions at the jet's interfaces (cf. Eq. (13)) yield toO…e2_†

Mˆ2 _RC

20 1

3 2R20

2B 2x _2R 20 2x 2 2C 2x 1 R220

2 22B 2x2

2QR₂220; …34†

Mˆ2 _RC

10 1

3 2R10

2B 2x _2R 10 2x 2 2C 2x 1 R210

2 22B 2x2

2QR₂210; …35†

which provide two more equations for an additional

unknown, i.e. M. These two equations together with

Eqs. (26), (29) and (30) constitute a closed system of one-dimensional equations for the ¯uid dynamics of isothermal, incompressible, annular liquid jets.

By substitutingMfrom Eq. (34) into Eq. (35) and using

Eqs. (29), (30) and (33) in the resulting equation, one can obtain easily the following equation for the leading-order

axial velocity

A Re 2B_2t 1B2B_2x

ˆARe_F 1 _2x2 3A2B_2x

1 _Ca1 _RA₂ 20

2R₂₀ 2x

12C R_R210₃

20 2R₂₀ 2x 2 1 R0 2R₁₀ 2x !

1 12 R210

R2 20

!

2C

2x; …36† where

Aˆ R2202₂ R210; …37†

and (cf. Eq. (44))

Cˆ pi2pe2 1 Ca

s1

s2

1 R10 1

1 R20 2 1 R2 10 2 1 R2 20

! : …38†

Eq. (26) together with (cf. Eq. (31)), 2A

2t 1 2…AB†

2x ˆ0; …39† whereCis given by Eq. (38), constitute a system of three one-dimensional equations forB,R10andR20that is much easier to solve than Eqs. (1)±(8).

The (dimensional) axial stress on the ®bre is equal to 2m…2u=2x†2p;and can be integrated in the radial direction to determine the (local) axial traction on the annular jet. This traction can be non-dimensionalized with respect to

epmu0R0 and its leading-order values can be written in

dimensionless form as

F_xˆA 22B 2x 2D

; …40†

and using the results previously obtained

FxˆA 32B_2x 2pe2 _{Ca R}1

20 12

C R2 20

!

: …41†

4. Analytical solutions for steady jets

For steady ¯ows, Eqs. (26) and (39) have the following solutions

ABˆQˆ1; BR210ˆQ1; BR220 ˆQ2; …42†

whereQ1andQ2are constants andQ22Q1ˆ2:

Unfortu-nately, due to the coupling between the leading-order

momentum equation Eq. (36)) andC(Eq. (38)), it has not

been possible to obtain analytical solutions to Eqs. (26), (36), (38) and (39) for steady jets and for arbitrary values

ofpi,peandCa. However, forpiˆpeandCaˆ1;i.e.

non-pressurized annular liquid jets with zero surface tension, it is possible to obtain analytical solutions as indicated in the next paragraphs sinceCˆ0 for these conditions.

Viscous regime.This regime is characterized by Reˆ0 and ®nite values ofF, i.e. it corresponds to a creeping ¯ow where the effects of inertia and surface tension are nil and, therefore, Eq. (36) becomes

d

dx 3A

dB dx

ˆ0; …43†

and has the following solution

B…x† ˆbexp…ax†; …44†

wherea andb are integration constants. Note thatbˆ1

because B…0† ˆ1 and aˆlnB…1†: The values ofR10and

R20can be determined easily from Eq. (42) as

R10 ˆQ11=2exp 2a₂x

; R20ˆQ12=2exp 2a₂x

: …45†

Gravitationless regime.This regime corresponds toFˆ

1andRe±0:The solutions of Eqs. (26) and (39) are given in Eq. (42), whereas the leading-order axial momentum equation becomes

RedB_dx ˆ _dxd 3AdB_dx

; …46†

which has the following solution

B…x† ˆ 3_Rea ₁₂bexp…ax†

bexp…ax††; …47† wherea andb are integration constants that can be deter-mined easily from the conditionsB…0† ˆ1 andB…1†;in fact,

bˆ ₃ Re

a1Re: …48†

Gravitational-viscous regime. This regime corresponds

to Reˆ0 and ®nite values of Re=F: The solutions of

Eqs. (26) and (39) are given in Eq. (42), and the leading-order axial momentum equation becomes

Re F A1

d

dx 3A

dB dx

ˆ0; …49†

which has the following solution

2 B2 P

b

12 B B22P

b

₁₌₂

ˆaexp…^b1=2_{x†; …50†}

wherea andb are integration constants, and

Pˆ2_3FRe: …51†

5. Linear stability

The linear stability of the analytical solutions presented in the previous section requires, in general, the use of

numerical techniques due to the coupling between the axial momentum equation,CandR10, as indicated by Eqs. (26),

(36) and (38), because even for Caˆ1; perturbations in

the jet's geometry will introduce perturbations inpiandpe due to the compression and expansion of the gases enclosed

by and surrounding the annular jet. Although pe may be

assumed to be constant, this is not the case forpi. However, for the viscous regime analyzed in the previous section, it is possible to perform a linear stability analysis analytically

for piˆpeˆ0 and Caˆ1; i.e. Cˆ0; because this

regime is governed by Eqs. (26) and (38) and (cf. Eq. (36)) 2

2x 3A 2B1

2x

ˆ0: …52†

Let us de®ne the steady-state solutions obtained in the previous section with the subscript ss, so that these solutions may be written as

Bss…x† ˆexp…ax†; Ass…x† ˆexp…2ax†; …53†

and perturb this steady solution as

B…t;x† ˆB_ss…11b†; A…t;x† ˆA_ss…11a†; …54†

where b and a are functions of t and x. Substitution of

Eqs. (53) and (54) into Eqs. (36) and (39) and neglecting non-linear terms in perturbations yield the following linear partial differential equations

Ass2a_2t 1 2a_2x 1 2b_2x ˆ0; …55†

a2a_2x 1a2b_2x 1 22b

2x2 ˆ0; …56†

which, upon using

aˆa…x†^ exp…st†; bˆb…x†^ exp…st†; …57†

wheres is a complex number, integration of the perturbed

axial momentum equation and use ofa…0† ˆ^ b…0† ˆ^ b…1† ˆ^ 0;can be written as

sexp…2ax†a^1 d_dxa^ 1 d_dxb^ ˆ0; …58†

aa^1ab^1 d_dxb^ ˆd; …59†

whered is an integration constant. Eqs. (58) and (59) imply that db^

dx…1† ˆ2sexp…2a†a…1†^ 2 da^

dx…1†; …60†

dˆaa…1†^ 1ab…1†^ 1 d_dxb^…1†; …61†

db^

dx…0† ˆ2 da^

dx…0†; dˆ

db^

dx…0†; …62†

which, together with Eqs. (58) and (59), yield

…a2sexp…2a††a…1† ˆ^ _dxda^…1†2 d_dxa^…0†; …63†

d2a^

dx2 1sexp…2ax†d ^

a

dx ˆ0; …64†

Integration of Eq. (64) and use of the boundary conditions given by Eqs. (60)±(62) yield the following eigenvalue problem

Zs=a s=aea

ez z dzˆ

es=aea

2es=a

12es=aea ; …65†

which coincides exactly with that derived by Schultz and Davis [12] for round jets at low Reynolds numbers. There-fore, the eigenvalue whose real part is zero corresponds to

acˆ3:00650; B…1† ˆ20:21 and siˆ14:011; where si

denotes the imaginary part of s. Eq. (65) clearly shows

that the eigenvalue s only depends on a which, in turn,

depends on the axial velocity at the take-up point, and the results obtained by Schultz and Davis [12] for round jets at low Reynolds numbers apply to annular jets. This is not surprising, for, in the viscous regime, gravitational and inertia effects (which depend on the density ratio) are absent.

The evolution of the perturbations to the leading-order radiiR10andR20can be determined in an analogous manner to the one used forAandBby linearizing Eqs. (26) and (38), which can be combined to obtained an evolution equation forR10. However, this requires us to take into consideration the perturbations topi;pemay be assumed to be constant and set to zero.

In this paper, the gases enclosed by the inner jet are assumed to behave polytropically with a polytropic expo-nent equal tok, so that

p_iVk

i ˆQ4; …66†

whereQ4is a constant, andVidenotes the volume enclosed by the jet's inner surface, i.e.

Viˆ Z1

0R

2

1dxˆ

Z1

0R

2

10dx1O…e2†; …67†

clearly indicating that perturbations in the jet's geometry produce perturbations inpi.

Stability analyses similar to the one performed in this section for the viscous regime also can be performed analy-tically for the gravitationless and gravitational-viscous regimes for which analytical solutions were obtained in the previous section under steady conditions. The stability of these regimes as well as any other steady solution requires the use of numerical techniques based upon the discretization of the equations for the perturbed quantities. Although not shown here, a weakly non-linear stability analysis of the viscous ¯ow regime indicates that there is a supercritical (stable) Hopf bifurcation, i.e. the steady state bifurcates into a periodic motion, close to the draw ratio at

which the real part of the ®rst eigenvalue of the linear stability analysis is zero. The possible bifurcations of the other ¯ow regimes whose steady-state solutions have been obtained in this paper require the use of numerical techni-ques and have not been studied. The dynamic behaviour of the viscous ¯ow regime for draw ratios larger than that associated with the Hopf bifurcation, i.e. for draw ratios for which a weakly non-linear analysis is not valid, has been determined numerically as indicated in the next section.

6. Presentation of results

As shown in previous sections, the non-linear dynamics of steady, isothermal, Newtonian, annular liquid jets at low Reynolds numbers depends onRe,Re=F;Ca,pi2pe;s1=s2;

and Bss(1). Under transient conditions, it depends on the

above parameters, the polytropic exponent kand the

loca-tion, amplitude and frequency of the applied perturbations to the radius, velocity or axial traction. In this paper, axial velocity perturbations at either the nozzle exit or the take-up point have been employed, i.e.

B…t;xj† ˆBss…xj†…11ajsinSjt†; …68†

where aj and Sj denote the (non-dimensional) amplitude

and frequency of the imposed perturbations at xj; and

…j;xj† ˆ …i;0†and…e;1†denote the nozzle exit and the take-up point, respectively. It should be mentioned that the non-linear dynamics of slender, annular liquid jets also depend on perturbations on the jet's radii at the nozzle exit and at the take-up point, although in this paper only perturbations on the axial velocity component at either the nozzle exit or the take-up point are considered.

Some sample results illustrating the steady-state annular jet's geometry and axial velocity component are presented Fig. 2. Annular jet's geometry (a) and axial velocity component (b).…Re=Frˆ1;Caˆ1;s1=s2ˆ1;peˆ0;piˆ0;Qˆ1;R1…0† ˆ1;B…0† ˆ1;Bss…1† ˆ

100:Solid lines:Reˆ1;dashed lines:Reˆ0:1;dashed-dotted lines:Reˆ2†:

Fig. 3. Annular jet's geometry (a) and axial velocity component (b).…Reˆ1;Re=Frˆ1;Caˆ1;s1=s2ˆ1;Qˆ1;R1…0† ˆ1;B…0† ˆ1;Bss…1† ˆ100:Solid

in Figs. 2 and 3. These ®gures were obtained by solving numerically the steady-state equations presented in previous sections by means of a second-order accurate ®nite differ-ence method;the number of grid points was at least 2001, B…0† ˆ1 andR₁…0† ˆ1:Fig. 2 indicates that the axial velo-city increases rapidly near the downstream or take-up point where it exhibits a boundary layer structure. The thickness of this boundary layer increases as the Reynolds number is decreased. The axial traction on the annular jet also increases quite rapidly near the take-up point, except at low Reynolds numbers for which it increases rather smoothly from the upstream to the downstream boundaries. The annular jet's geometry illustrated in Fig. 2 clearly shows the jet's contraction near the nozzle exit at low Reynolds numbers;the contraction at higher Reynolds numbers is large at the take-up point where the axial velocity component is largest.

Although not shown here, the steady-state annular jet's radii and axial traction force were found to increase slightly asRe=Fwas decreased due to the gravitational pull;the axial velocity component and axial traction on the inner annular

jet increased slightly asRe=Fwas increased for the values of the parameters shown in Fig. 2 and…Re=F† ˆ1;10 and 0.1. The annular jet's geometry was found to be very sensitive to the capillary number,Ca, for the parameters shown in Fig. 2; in fact, the annular jet's radii were found to increase as the capillary number was increased. However, the leading-order axial velocity and axial traction force are not very sensitive toCa.

The steady-state jet's radii were found to increase as

s1=s2 was decreased and aspi2pe was increased. Fig. 3

shows that the jet's geometry is very sensitive topi2pe;for

externally pressurized annular jets, there is a large contrac-tion near the nozzle exit or die, whereas, for internally pres-surized ones, most of the contraction occurs near the downstream boundary. In either case, the axial velocity component was not found to be very sensitive topi2pe:

The initial jet's contraction was found to increase as Q was decreased. Since the leading-order ¯uid dynamics equa-tions imply that the leading-order axial velocity component is uniform across the annular jet cross-section, an increase in Qcorresponds to an increase inR2…x†:

Fig. 4.R1…t;1†(a),R2…t;1†2R1…t;1†(b),B…t;1†(c) and axial traction (d) on the annular jet at the take-up point as functions of time.…Reˆ1024;Re=Fˆ0;

Caˆ1030_;s

Some sample results which illustrate the non-linear dynamics of annular liquid jets when subject to axial velocity perturbations at the nozzle exit, i.e. Eq. (68), are presented in Figs. 4±6 and Tables 1±3. Unless otherwise

stated, the time-dependent results were obtained numerically by discretizing the time derivatives by means of ®rst- or second-order ®nite differences;the advective derivatives were discretized by means of either two-point, ®rst-order Fig. 5. Power spectrum (a) and phase diagram (b) ofR1…t;1†:…Rdot;dR1…t;1†=dt;Reˆ1024;Re=Fˆ0;Caˆ1030;s1=s2ˆ1;peˆpiˆ0;Qˆ1;Radˆ

0:20;Bss…0† ˆ1;Bss…1† ˆ1=Rad2;aiˆ0:05;aeˆ0;Siˆ1;Seˆ1†:

Fig. 6.R1…t;1†(a),R2…t;1†2R1…t;1†(b),B…t;1†(c) and axial traction (d) on the annular jet at the take-up point as functions of time.…Reˆ1024;Re=Fˆ0;

Caˆ1030_;s

accurate or three-point, second-order accurate formulae, whereas the diffusion terms were discretized by means of second-order accurate ®nite difference expressions. The computations were performed using double precision arith-metic, at least 2001 grid points, and a time-step equal to at most 1024_{. When chaotic behaviour or holes in the phase} diagrams were observed, the computations were repeated and performed on quadruple precision with a time-step equal to 1028_{. The results presented in Figs. 4±6 and Tables} 1±3 correspond toB_ss…0† ˆ1;R₁…t;0† ˆRad andB_ss…1† ˆ 1=Rad2_{: In Tables 1±3, the parameters that appear in the}

captions correspond to the basic set employed as a reference in the calculations, and only the parameter under the head-ing was varied while maintainhead-ing the other parameters equal to those of the basic state.

Fig. 4 indicates that the annular jet's radii and axial trac-tion force at the take-up point are periodic functrac-tions of time that have the same frequency as that of the imposed axial velocity perturbation at the nozzle exit or die, i.e. atxˆ0; forBssˆ25:The amplitude of the jet's inner radius is about

0.002, which is much smaller than the mean value of the

jet's inner radius. This periodic behaviour is also observed in both the power spectrum and phase diagram ofR1…t;1†;

i.e.Rdot;dR1…t;1†=dtvs.R1…t;1†;exhibited in Fig. 5. The

power spectrum is characterized by a single peak at a frequency equal to that of the imposed velocity perturba-tions, i.e. 1=2p;and the phase diagram is a circumference.

For Bss<44:5; the results presented in Fig. 6 indicate

that the annular jet's radii are spiky, and their maximum values seem to be modulated with a frequency of the order of that of the imposed velocity perturbations;an analogous comment applies to the axial traction force on the annular jet at the take-up point. The rapid variations in the jet's geometry and axial traction might cause fatigue problems if they were present in the presence of solidi®ca-tion. The power spectrum corresponding to Fig. 6 is char-acterized by several peaks, and the frequency associated with these peaks is not related to that of the applied velocity perturbations as indicated in Fig. 7. Moreover, the phase diagram presented in Fig. 7 has a duck's beak shape, presents corrugations on its periphery, is thick, and contains some holes, i.e. regions that are never visited. In addition, Table 1

Maximum (max) and minimum (min) values of the jet's inner radius,R1…t;1†;and axial traction force,F, on the annular jet at the take-up point, and maximum

spectral power,P, and frequency,f, associated with the maximum power ofR1…t;1†forBss…1† ˆ1=Rad2:effects of the steady-state take-up downstream

boundary conditions

Rad (R1)max(t,1) (R1)min(t,1) Fmax Fmin P f 0.10a,1,4 _{0.4527 0.0155 117.7195 0.6773 38.0239 1.8299} 0.15a,1,5 _{0.4309 0.0419 46.7717 1.6501 60.7774 2.0099} 0.20a,1,6 _{0.2019 0.1980 9.7807 9.5242 0.0365 0.16} 0.10b,2,4 _{0.4520 0.0156 118.9113 0.6956 39.2607 1.8299} 0.15b,2,5 _{0.4287 0.0423 47.1145 1.7044 61.3542 2.0099} 0.20b,2,6 _{0.2020 0.1982 9.7078 9.5973 0.0365 0.16} 0.10c,3,4 _{0.4532 0.0158 117.0654 0.6875 38.8697 1.8299} 0.15c,3,5 _{0.4287 0.0430 46.1325 1.6941 58.1446 2.0099} 0.20c,3,6 _{0.2000 0.2000 9.8311 9.4652 6.467£10}222 _0.16

a _{The upstream axial velocity is sinusoidally excited with an amplitude and frequency equal toa}

iˆ0:01 andSiˆ1 (cf. Eq. (68)), respectively;1identical

results foraiˆ0:05 and 0.10;4phase diagram with holes;5®lled phase diagram;6periodic.

b_{The downstream axial velocity is sinusoidally excited with an amplitude and frequency equal to 0.01 and 1, respectively;}2_{identical results fora}

eˆ0:05

and 0.10;4_{phase diagram with holes;}5_{®lled phase diagram;}6_periodic.

c _{The upstream and downstream axial velocities are sinusoidally excited with an amplitude and frequency equal to 0.01 and 1, respectively;}3_{identical results} foraiˆaeˆ0:05 and 0.10;4phase diagram with holes;5®lled phase diagram;6periodic. The basic set of parameters isReˆ1024;Re=Fˆ0;Caˆ1030;

s1=s2ˆ1;peˆpiˆ0;Qˆ1;Bss…0† ˆ1:

Table 2

Maximum (max) and minimum (min) values of the jet's inner radius,R1…t;1†;and axial traction force,F, on the annular jet at the take-up point, and maximum

spectral power,P, and frequency,f, associated with the maximum power ofR1…t;1†forBss…1† ˆ1=Rad2:effects of the location and amplitude of the imposed

velocity perturbations

(aI,ae,SI,Se) (R1)max(t,1) (R1)min(t,1) Fmax Fmin P f

(0.01, 0, 1, 1)a _{0.2203 0.0992 26.3841 7.8653 27.7191 2.3749}

(0, 0.01, 1, 1)b _{0.2190 0.1001 26.1234 8.0263 28.5028 2.3749}

(0.01, 0.01, 1, 1)c _{0.2185 0.1005 26.3522 7.9105 27.4912 2.3749}

a _{Identical results for…a}

i;Si† ˆ …0:01;5†;(0.01,9), (0.05,1), (0.01,5), (0.05,9), (0.1,1), (0.1,5) and (0.1,9).

b_{Identical results for…a}

e;Se† ˆ …0:01;5†;(0.01,9), (0.05,1), (0.01,5), (0.05,9), (0.1,1), (0.1,5) and (0.1,9).

c _{Identical results for …a}

i;ae;Si;Se† ˆ …0:01;0:01;5;1†; (0.01,0.01,9,1), (0.01,0.01,1,5), (0.01,0.01,1,9), (0.05,0.05,1,1), (0.05,0.05,5,1), (0.05,0.05,9,1),

(0.05,0.05,1,5), (0.05,0.05,1,9). The basic set of parameters isReˆ1021_{;Re=Fˆ0;Caˆ10}30_;s

the largest and smallest values of R₁…t;1† are larger and smaller, respectively, than those presented in Fig. 5. The values of …dR=dt†…t;1† are larger than those presented in Fig. 5, and the thickness of the phase diagram is related to the modulation of the spikes observed in the time history of R1…t;1†shown in Fig. 6.

ForBssˆ100;the time histories presented in Fig. 8 seem

to be analogous to those of Fig. 6, except that the largest and smallest values of R1…t;1† increase and decrease,

respec-tively, as Bss…1†is increased;the axial traction force also

increases as Bss…1† is increased in accord with the large

gradient of the axial velocity at the take-up point. Moreover, the amplitude of the modulation of the spikes in the jet's radii and axial traction force at the take-up point forB_ssˆ 100 are smaller than forB_ss<44:5:This is also observed in the phase diagram illustrated in Fig. 9, which is much thin-ner than that of Fig. 7, and contains many holes. This diagram also shows that there is a high density of points

near…dR=dt†…t;1† ˆ0 in agreement with the time histories presented in Fig. 8.

The power spectrum presented in Fig. 9 is also broader and the power densities corresponding to its peaks are smaller than those of Fig. 7. The broadening of the power spectrum and the appearance of holes in the phase diagram are indicative of the presence of chaos. However, the transi-tion from the periodic motransi-tion presented in Figs. 4 and 5 to the chaotic ones of Figs. 8 and 9 asBssis increased does not seem to follow a standard scenario such as that of Ruelle-Takens at least for suf®ciently large values ofBss;in fact, it has been observed that the transition to chaos for large values of Bss is rather abrupt or explosive, whereas, for smaller values of Bss, it has been observed that R1…t;1†

evolves from a ®xed point to a periodic motion and then to a quasiperiodic motion and chaos asBssis increased. The results presented in Figs. 7 and 9 also indicate that …dR=dt†…t;1†undergoes a large increase asBssis increased. Table 3

Maximum (max) and minimum (min) values of the jet's inner radius,R1…t;1†;and axial traction force,F, on the annular jet at the take-up point, and maximum

spectral power,P, and frequency,f, associated with the maximum power ofR1…t;1†forBss…1† ˆ1=Rad2:effects of the ¯uid dynamics parameters

Parameter (R1)max(t,1) (R1)min(t,1) Fmax Fmin P f

Reˆ0.1a _{0.2203 0.0992 26.3841 7.8653 27.7191 2.3749}

Reˆ1a _{0.1514 0.1485 50.3875 48.6264 0.0207 0.16}

(Re/F)ˆ1a _{0.4164 0.0441 44.4749 1.7865 105.7746 2.0699}

(Re/F)ˆ0.01a _{0.4296 0.0420 46.6549 1.6529 83.4479 2.0099}

Caˆ(105₎a _{0.4305 0.0419 46.7861 1.6501 60.5842 2.0099}

Caˆ(102₎a _{0.4304 0.0410 47.0544 1.6354 52.1644 2.0049}

s1/s2ˆ10a 0.4309 0.0419 46.7717 1.6501 60.7774 2.0099

s1/s2ˆ0.1a 0.4309 0.0419 46.7717 1.6501 60.7774 2.0099

piˆ0.01a 0.4321 0.0420 46.7729 1.6501 61.0402 2.0099

piˆ1a 0.5887 0.0554 46.9049 1.6517 100.9758 2.0099

peˆ0.01a 0.4298 0.0418 46.7691 1.6501 60.5163 2.0099

peˆ1a 0.3384 0.0326 46.6008 1.6476 41.2189 2.0099

a _{The upstream axial velocity is sinusoidally excited with an amplitude and frequency equal toa}

iˆ0:05 andSiˆ1;respectively;identical results have been

observed when the upstream axial velocity is sinusoidally excited with an amplitude and frequency equal to 0.10 and 1, respectively. The basic set of parameters isReˆ1024_{;Re=Fˆ0;Caˆ10}30_;s

1=s2ˆ1;peˆpiˆ0;Qˆ1;Radˆ0:15;Bss…0† ˆ1;aiˆ0:1;aeˆ0;Siˆ1;Seˆ1:

Fig. 7. Power spectrum (a) and phase diagram (b) ofR1…t;1†:…Rdot;dR1…t;1†=dt;Reˆ1024;Re=Fˆ0;Caˆ1030;s1=s2ˆ1;peˆpiˆ0;Qˆ1;Radˆ

It is worth remarking that the frequencies of the quasi-periodic and chaotic motions observed in previous ®gures as well as in other calculations [15] do not seem to be related to that of the imposed velocity perturbations. Moreover, phase

diagrams and power spectra similar to those presented in Figs. 5, 7 and 9 have been observed for axial velocity pertur-bations of amplitudes and frequencies equal to 0.01 and 1, respectively, at the nozzle exit or die. However, for axial Fig. 8.R1…t;1†(a),R2…t;1†2R1…t;1†(b)B…t;1†(c) and axial traction (d) on the annular jet at the take-up point as functions of time.…Reˆ1024;Re=Fˆ0;

Caˆ1030_;s

1=s2ˆ1;peˆpiˆ0;Qˆ1;Radˆ0:10;Bss…0† ˆ1;Bss…1† ˆ1=Rad2;aiˆ0:05;aeˆ0;Siˆ1;Seˆ1†:

Fig. 9. Power spectrum (a) and phase diagram (b) ofR1…t;1†:…Rdot;dR1…t;1†=dt;Reˆ1024;Re=Fˆ0;Caˆ1030;s1=s2ˆ1;peˆpiˆ0;Qˆ1;Radˆ

velocity perturbations at the downstream boundary and B_ss…1† ˆ100;the phase diagrams had more empty regions and the power spectra were broader than that of Fig. 9, even

for a_eˆ0:01: When axial velocity perturbations were

imposed at both the upstream and the downstream bound-aries, the power spectra were narrower than those for axial velocity perturbations imposed at only the downstream boundary and slightly broader than those of Fig. 9. More-over, the phase diagrams when axial velocity perturbations were imposed at both the upstream and the downstream boundaries were more ®lled than those for axial velocity perturbations imposed at only the downstream boundary but presented more empty regions or holes than those of Fig. 9. When either the axial velocity perturbations were imposed at the downstream or at both the upstream and the downstream boundaries, it was observed that some kinds of tongues emerged from the main beak structure, thus resulting in multiple leaves.

The phase diagrams of Figs. 7 and 9 have a duck's beak shape, and exhibit many holes;the presence of holes may be associated with subcritical (unstable) bifurcations. It must be pointed out that the phase diagrams illustrated in Figs. 5,

7 and 9 were obtained for 1000#t#1200 and that, for a

time-step equal to 1028_{, they contain 2£10}10 _{points. The}

broadening of both the power spectrum and the phase diagrams as well as the increase in the number of holes in the phase diagram indicate the presence of strange attractors and chaotic behaviour.

Table 1 shows that the largest value ofR1…t;1†increases

whereas the smallest value ofR1…t;1†decreases asBss(1) is

increased;a similar behaviour is observed in the axial trac-tion force at the take-up or downstream point. Table 1 also shows thatR₁…t;1†is not very sensitive to the amplitude and frequency of the imposed velocity perturbations, but it is sensitive to the location of these perturbations. In addition, Table 1 shows that the frequency of the maximum power ®rst increases asBss(1) is increased, and then decreases;a ®xed point of R1…t;1† results if the axial velocity is

perturbed at both the upstream and downstream boundaries with the same amplitude and frequency ifBss…1† ˆ50;even

though this value exceeds that determined in the linear stabi-lity analysis performed previously in this paper. There is, however, no contradiction between the results of Table 1 and the stability analysis because the latter corresponds to

Reˆ0 and does not account for the compression and the

expansion of the gases enclosed by the annular jet, whereas the former accounts for non-zero Reynolds numbers and the compression and expansion of these gases withkˆ1:4:

Table 1 also shows that the difference between the largest and the smallest traction forces at the take-up point increases as Bss(1) is increased. Table 2 indicates that the difference between the largest and smallest values ofR1…t;1†

is larger for axial velocity perturbations imposed at the upstream than at the downstream boundaries;however, the frequency corresponding to the maximum power of R1…t;1†is independent of the location of the imposed

velo-city perturbations. Note that the Reynolds number in Table 2 is much higher than that in Table 1.

Although not shown here [15], forReˆ1024_{and Radˆ}

0:30; the largest values of R₁…t;1† and the axial traction force at the take-up point are smaller than those shown in Table 2;in fact, forReˆ1024 _{and Radˆ0:30;R}

1…t;1†is

periodic and has the same frequency as that of the imposed

velocity perturbations. It must be pointed out that Radˆ

0:30 corresponds toBss…1†<11:

The effects of the non-dimensional ¯uid dynamics para-meters onR1…t;1†and the axial traction force at the take-up

point are illustrated in Table 3. This table shows that for Reˆ1;the annular jet behaves in a periodic manner with a frequency equal to that of the imposed velocity ¯uctuations, and there is very little difference between the largest and smallest values ofR1…t;1†:The largest and smallest values

of R₁…t;1† increase and decrease, respectively, as Re is decreased on account of the large gradients of the axial velocity at the downstream boundary;a similar comment applies to the axial traction force at the take-up point. For the Reynolds number of the basic set of parameters of Table 3, the effects of the Froude number are small;however, the largest value of R1…t;1†decreases as Re=F is increased on

account of the increase in the gravitational pull.

Table 3 also shows that the effects of the capillary number are small, although an increase inCaresults in an increase in the largest value ofR1…t;1†:The effects ofs1=s2 on the

non-linear dynamics of annular liquid jets have been found to be small for the values of the parameters employed in Table 3. As should be expected, the largest value ofR1…t;1†

increases aspiis increased and aspeis decreased.

For Radˆ0:15; Reˆ1024_{; Re=Fˆ0; Caˆ10}30_;

s1=s2ˆ1; peˆpiˆ0; Qˆ1; Bss…0† ˆ1 and Bss…1† ˆ

1=Rad2_{;the phase diagram ofR}

1…t;1†was found to be rather

similar to but wider than that of Fig. 7 for aiˆ0:05 and

0.10, and Siˆ1: However, for the same values of the

parameters andReˆ1021_{;it was rather smooth, i.e. it did}

not have the duck's beak shape of Fig. 7, and it did not

present any hole;for Reˆ1; the phase diagram was a

circumference.

For Radˆ0:30; Reˆ1024; Re=Fˆ0; Caˆ1030;

s1=s2ˆ1; peˆpiˆ0; Qˆ1; Bss…0† ˆ1 and Bss…1† ˆ

1=Rad2; R1…t;1† is a periodic function of time which has

the same frequency as that of the applied velocity perturba-tion foraiˆ0:01;0.05 and 0.10,aeˆ0;Siˆ1;andSeˆ

1:The same comment applies for Radˆ0:30;Reˆ1024_;

Re=Fˆ0; Caˆ1030_{; s}

1=s2ˆ1; peˆpiˆ0; Qˆ1;

B_ss…0† ˆ1; B_ss…1† ˆ1=Rad2_{; a}

iˆ0; aeˆ0:01; 0.05 and

0.10, S_iˆ1; andS_eˆ1:However, for Radˆ0:30;Reˆ 1024_{;Re=Fˆ0;Caˆ10}30_{; s}

1=s2ˆ1; peˆpiˆ0;Qˆ

1; Bss…0† ˆ1; Bss…1† ˆ1=Rad2; R1…t;1† exhibited a ®xed

point for aiˆaeˆ0:01;0.05 and 0.10, and SeˆSiˆ1;

whereas the axial traction force at the take-up point is a periodic function with a frequency equal to that of the imposed velocity perturbations.

here [15] indicate thatR1…t;1†evolves from a ®xed point, to

a periodic orbit and then to quasiperiodic and chaotic motions as Bss(1) is increased;however, the transition, if one exists, from quasiperiodic to chaotic responses is rather abrupt, and the phase diagram in the chaotic region may present holes, i.e. regions that are not visited, and corruga-tions for large values ofBss(1). It has been observed also that the power spectrum ofR1…t;1†broadens and presents many

holes asBss(1) is increased. The width of the spectrum and the topology of the phase diagrams have also been found to depend on the ¯uid dynamics parameters that characterize annular liquid jets at low Reynolds numbers, and the loca-tion of the imposed velocity perturbaloca-tions. The amplitude and frequency of these perturbations were found to play a very small role for suf®ciently large values of Bss(1), thus indicating that the non-linear dynamics of annular liquid jets are not affected very much by the amplitude and frequency of the imposed velocity perturbations.

The non-linear studies presented in this paper do not exhibit the routes to chaos discussed by Yarin et al. [16] for round jets at low Reynolds numbers and Gospodinov and Yarin [13]. Yarin et al. [16] claimed that the cross-section of Newtonian isothermal round ®bres may vary aperiodically for draw ratios equal to about 30 and larger when periodic variations of the input cross-section are imposed, and that the route to chaos may be smooth, via period doubling, or explosive, via abrupt disappearance of quasiperiodic solu-tions. On the other hand, Gospodinov and Yarin [13] claimed that even for draw ratios equal to 400, the midradius of annular jets exhibits a multimodal quasiperiodic charac-ter. The results for annular jets presented in this paper indi-cate, however, that the critical draw ratio for the viscous regime can be exceeded without the appearance of chaos if the Reynolds number is suf®ciently large but still much smaller than unity, and that the route to chaos is via quasi-periodic motions for very small amplitudes of the applied perturbations to the axial velocity component at either the upstream or downstream boundaries, but for amplitudes equal to or higher than about 0.01 and draw ratios suf®-ciently large, the transition to chaos, if one exists, is abrupt. Moreover, the phase diagrams employed by Yarin et al. [16] are not really phase diagrams because they considerB…t;xj† versusa…t;xj†whereais the ®bre radius;on the other hand, the phase diagrams presented in this paper refer to the time history ofR1…t;1†:

The asymptotic methods, linear stability analysis and numerical studies presented in this paper are valid for slender, isothermal, annular, Newtonian liquid jets, and show that, under certain conditions, it is possible to obtain steady-state solutions in closed form;they also show that the linear stability analysis of annular liquid jets at low Reynolds numbers is identical to those of (planar) ®lm cast-ing processes and the spinncast-ing of round jets, and that complex draw resonance phenomena may be observed for draw ratios greater than the one determined from linear stability when the annular jet is perturbed sinusoidally at

either the nozzle exit and/or the take-up point. When there is draw resonance, the radius, axial velocity component and axial traction at the take-up point exhibit strong variations in time.

When non-isothermal ¯ows are considered and the rheol-ogy of the ¯uid is Newtonian with a dynamic viscosity that increases sharply, e.g. in an Arrhenius manner, with temperature, the coupling between the ¯uid dynamics and energy equations indicates that, at leading-order, the ¯uid dynamics of annular jets is one-dimensional, whereas the temperature ®eld depends on the axial and radial coordi-nates;however, if the axial diffusion of energy is neglected, then the temperature along the ®bre decreases in an expo-nential manner, while the axial velocity of the ¯uid depends in an exponential manner on the Reynolds, Prandtl and Biot numbers at very low Reynolds numbers. When the rheology is non-Newtonian and the process is isothermal, it is expected that the properties of the rheology, e.g. power law, viscoelasticity, etc., should affect some of the conclu-sions presented in the paper, although the asymptotic meth-ods presented here can be applied in a straightforward manner to non-Newtonian, slender, annular ®bres.

7. Conclusions

Perturbation methods based on the slenderness ratio have been employed to determine the leading-order ¯uid

dynamics equations of isothermal, axisymmetric,

Newtonian, annular liquid jets at low Reynolds numbers. It has been shown that the leading-order equations are one-dimensional and correspond to the conservation of mass, global linear momentum conservation, and kine-matics of the interfaces between the annular jet and its surroundings. These (non-dimensional) equations depend on the Reynolds, Froude and capillary numbers, pressure difference across the annular jet, and surface tension ratio.

Analytical solutions to the leading-order steady-state equations have been obtained for in®nite capillary numbers and zero pressure differences across the annular jet and for zero Reynolds numbers, zero gravitational pull, or inertia-less ¯ows. For the steady viscous regime, a linear stability analysis has been performed and shown to be governed by the same eigenvalue equation as that for the spinning of isothermal, round jets at zero Reynolds numbers.

Numerical results of the time-dependent leading-order equations indicate that as the axial velocity at the down-stream boundary is increased, the radius of the annular jet's inner radius at the take-up point evolves from a ®xed point to a limit cycle to a broad spectrum. The phase diagrams of this radius may exhibit holes, i.e. regions that are not visited, when the axial velocity component at the take-up boundary is suf®ciently large, thus indicating the presence of strange attractors and chaos.

It has also been shown that the annular jet's radius and axial traction force at the take-up point exhibit very sharp

spikes whose separation depends on the Reynolds, Froude and capillary numbers, surface tension ratio, pressure differ-ence across the annular jet, and location, amplitude and frequency of the imposed velocity perturbations. These spikes are somewhat modulated with a lower frequency which is of the order of that of the imposed axial velocity perturbation.

Acknowledgements

The research reported in this paper was supported by Project PB97-1086 from the DGES of Spain.

References

[1] Matovich MA, Pearson JRA. Spinning a molten threadline. Ind Chem Engng Fund 1969;18:512±20.

[2] Pearson JRA, Matovich MA. Spinning a molten threadline. Ind Chem Engng Fund 1969;18:605±9.

[3] Schultz WW, Davis SH. One-dimensional liquid ®bres. J Rheol 1982;26:331±45.

[4] Pearson JRA. Mechanics of polymer processing. New York: Elsevier Applied Science, 1985.

[5] Yarin AL, Gospodinov P, Gottlieb O, Graham MD. Stability loss and sensitivity in hollow ®bre drawing. Phys Fluids 1994;6:1454±63. [6] Ramos JI. Liquid curtains Ð I. Fluid mechanics. Chem Engng Sci

1992;43:3171±84.

[7] Ramos JI. Inviscid, slender, annular liquid jets. Chem Engng Sci 1996;51:981±94.

[8] Ramos JI. Annular liquid jets: formulation and steady state analysis. Z Angew Math Mech (ZAMM) 1992;72:565±89.

[9] Pearson JRA, Petrie CJS. The ¯ow of a tubular ®lm. Part 1. Formal mathematical representation. J Fluid Mech 1970;40:1±19.

[10] Pearson JRA, Petrie CJS. The ¯ow of a tubular ®lm. Part 2. Inter-pretation of the model and discussion of solutions. J Fluid Mech 1970;42:609±25.

[11] Yeow YL. Stability of tubular ®lm ¯ow: a model of the ®lm-blowing process. J Fluid Mech 1976;75:577±91.

[12] Schultz WW, Davis SH. Effects of boundary conditions on the stabi-lity of slender viscous ®bres. ASME J Appl Mech 1984;51:1±5. [13] Gospodinov P, Yarin AL. Draw resonance of optical microcapillaries

in non-isothermal drawing. Int J Multiphase Flow 1997;23:967±76. [14] Ramos JI. Asymptotic analysis of compound liquid jets at low

Reynolds numbers. Appl Math Comput 1999;100:223±40. [15] Ramos JI. Nonlinear dynamics of annular liquid jets at low Reynolds

numbers. Report GTCI-1999-3, Universidad de MaÂlaga, Spain, 1999. [16] Yarin AL, Gospodinov P, Roussinov VI. Newtonian glass ®bre draw-ing: chaotic variation of the cross-sectional radius. Phys Fluids 1999;11:3201±8.