16666902

Nonlinear dynamics of hollow, compound jets at low Reynolds

numbers

J.I. Ramos

Dpto. de Leng. y Cienc. de la Comp., Room I-325 E.T.S. Ingenieros Industriales, Universidad de Malaga, Plaza El Ejido, s/n 29013 Malaga, Spain

Received 17 July 2000; accepted 22 September 2000

(Communicated by E.S. SßUHUBI)_

Abstract

The leading-order ¯uid dynamics equations of isothermal, axisymmetric, Newtonian, hollow, compound ®bers at low Reynolds numbers are derived by means of asymptotic methods based on the slenderness ratio. These ®bers consist of an inner material which is an annular jet surrounded by another annular jet in contact with ambient air. 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 hollow, compound jets is governed by the same eigenvalue equation as that for the spinning of round ®bers. 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 ®ber dynamics evolves from periodic to chaotic motions as the extension or draw ratio is increased. The power spectrum of the interface radius at the take-up point broadens and the phase dia-grams 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. Ó 2001 Elsevier Science Ltd. All rights reserved.

Keywords:Hollow; Compound jets; Asymptotic methods; Low Reynolds numbers; Linear stability; Chaotic motions

1. Introduction

This work deals with the ¯uid dynamics of hollow, compound ®bers such as that shown schematically in Fig. 1. These ®bers consist of two di€erent materials, i.e., the inner and outer www.elsevier.com/locate/ijengsci

*_{Corresponding author. Tel.: +34-95-2131402; fax: +34-95-2132816.} E-mail address:[email protected] (J.I. Ramos).

0020-7225/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S0020-7225(00)00099-9

annular jets, which are stretched in the axial direction under the action of a tensile force at the downstream boundary. Freezing takes place between the nozzle exit/die and the take-up point, and, usually, large extension rates, rapid cooling, and high speeds are involved [1]. Spinning processes of hollow, compound jets are used in the manufacture of reinforced ®bers, microcap-illaries and optical ®bers.

Although there has been quite a lot research on the development of one-dimensional, mathe-matical models for the analysis of single-component round ®laments and round jets under both isothermal and nonisothermal conditions at low Reynolds numbers [2,3], hollow, compound ®-bers such as those used in reinforced materials and optical ®®-bers (which are manufactured in coextrusion processes) have received very little attention despite the fact that the combination of two or more di€erent materials with di€erent properties may result in composite ®bers with highly desireable properties. For example, in the manufacture of optical ®bers, the core is surrounded by a sheath of cladding material.

Compound ®bers consisting of a round jet surrounded by an annular one have, however, received some attention in the past [4±6]. Park [4] used perturbation methods based on the slenderness ratio and the smallness of the Deborah number in his studies of steady, isothermal, two-phase or compound ®bers consisting of a Newtonian core layer surrounded by a sheath of nonNewtonian layer with a Maxwell rheology. His studies resulted in a system of ordinary dif-ferential equations for the steady-state axial velocity component and radii of the two-phase ®ber which is more manageable than the two-dimensional conservation equations of mass and linear

momentum from which it was derived. Lee and Park [5] employed the one-dimensional equations developed by Schultz [7] to study the linear stability of the spinning of bicomponent ®bers charaterized by a Newtonian ¯uid for the core and an upper-convected Maxwell ¯uid for the cladding, and showed that the stability of the ®ber can be maintained at higher draw ratios than obtainable when the same ¯uid is employed for both the core and the cladding. Naboulsi and Bechtel [6] introduced a one-dimensional model of bicomponent ®ber ®laments by integrating the three-dimensional equations over the ®lament cross-section, and examined the in¯uence of den-sity, viscosity and surface tension ratios on the steady-state ¯uid dynamics of compound jets at low Reynolds numbers.

Hollow ®bers consisting of a single material have been previously studied by Pearson and Petrie [8,9], Yeow [10], Ramos [11] and Yarin et al. [12]. Pearson and Petrie [8] studied the steady ax-isymmetric ¯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 [9], while Yeow [10] analyzed the linear stability of the model proposed by Pearson and Petrie [8] when the disturbances are axisymmetric and the ¯ow is isothermal, for the case of ®lm blowing. Ramos [11] employed asymptotic methods based on the slenderness ratio for hollow, compound annular jets at low Reynolds numbers and determined the leading-order equations for isothermal ¯ow, whereas Yarin et al. [12] employed the theory of Cosserat ¯uids, studied the linear stability of annular jets or hollow ®bers at low Reynolds numbers, and showed the existence of a critical draw ratio beyond which the isothermal drawing process is unstable; this critical draw ratio was found to be identical to that for the spinning of isothermal round jets at low Reynolds numbers [13].

Although there has not been any previous studies on the ¯uid dynamics of isothermal, hollow, compound jets at low Reynolds numbers, previous work on either annular or compound, iso-thermal jets at low Reynolds numbers has been mainly concerned with isoiso-thermal, steady-state ¯ows or determined the linear stability of these ¯ows; however, none of these studies has con-sidered the possible steady-state solutions of isothermal, steady, bicomponent ®bers, and deter-mined the e€ects of the downstream boundary conditions, forcing and ¯uid dynamics parameters on the nonlinear dynamics of these ®bers for draw or extension ratios beyond that determined from linear stability analyses.

The objective of this paper is several fold. First, the leading-order ¯uid dynamics equations of hollow, compound, isothermal, Newtonian jets at low Reynolds numbers are derived by means of perturbation methods based on the slenderness ratio [3,11]. These jets have three interfaces instead of the two interfaces of hollow or annular jets and compound or bicomponent ®bers. Second, analytical solutions to the steady-state solutions are obtained for zero Reynolds numbers, zero gravitational pull, and inertialess 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 [13]. Fourth, numerical studies of the time-dependent equations are performed in order to determine the non-linear dynamics of hollow, compound jets as a function of the nondimensional 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 and/or at the take-up point higher and lower than those determined from the linear stability analysis.

2. Formulation

Consider an axisymmetric, hollow, compound liquid jet consisting of two immiscible, incom-pressible (constant density) ¯uids which are isothermal and Newtonian such as the one shown schematically in Fig. 1. The inner and outer annular jets correspond to R1…t;x†6r6R…t;x† and

R…t;x†6r6R2…t;x†, respectively, with R1…t;x† 6ˆ0. The ¯uid dynamics of the hollow, compound jet are governed by the following conservation equations of mass and linear momentum:

oui

ox ‡

1

r o…vir†

or ˆ0; iˆ1;2; …1†

qi ou_oti

‡uiou_oxi‡viou_ori

ˆ op_oxi‡li o

2_u

i

ox2

‡1_{r or}o rou_ori

‡qig; iˆ1;2; …2†

qi ov_oti

‡uiov_oxi‡viov_ori

ˆ op_ori‡li o

2_v

i

ox2

‡_oro 1_r o… †_orrvi

; iˆ1;2; …3†

wheret is time; u andv the axial and radial velocity components, respectively; xand r the axial and radial coordinates, respectively; q and l the ¯uid's density and dynamic viscosity, respec-tively;pthe pressure;gthe gravitational acceleration; and, the subscriptsiˆ1;2 denote the ¯uids of the inner and outer jets, respectively.

Eqs. (1)±(3) are subjected to kinematic and dynamic boundary conditions at the jet's interfaces,

R1…t;x†, R…t;x† and R2…t;x†, where R corresponds to the interface between the inner and outer annular jets. The kinematic conditions establish that the compound 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 in-terfaces for hollow, compound liquid jets may be written as

vj…Ri;x;t† ˆoR_oti‡uj…Ri;x;t†oR_oxi; i;jˆ1;2; …4†

vj…R;x;t† ˆoR

ot ‡uj…R;x;t† oR

ox; jˆ1;2; …5†

2l_i ovi or oui ox oRi

ox ‡li ou_ori

‡ovi

ox

1 oRi

ox

2!

ˆ0; rˆRi; iˆ1;2; …6†

2l2 ov_or2

ou2

ox

oR

ox‡l2 ou_or2

‡ov_ox2

1 oR_ox

2!

ˆ2l1 ov_or1

ou1

ox

oR

ox‡l1 ou_or1

‡ov_ox1

1 oR_ox

2!

2l₁ou1 ox

oR1

ox

2

‡2l₁ov1

or 2l1 ou_or1

‡ov1

ox

oR1

ox

‡…pi p1† 1‡ oR_ox1

2!

ˆrJ1; rˆR1; …8†

2l2ou_ox2 oR_ox2

2

‡2l2ov_or2 2l2 ou_or2

‡ov_ox2

oR2

ox

‡…pe p2† 1‡ oR_ox2

2!

ˆ r2J2; rˆR2; …9†

2l₂ou2 ox

oR ox

2

‡2l₂ov2

or 2l2 ou_or2

‡ov2

ox

oR

ox‡…p1 p2† 1‡ oR ox

2!

2l₁ou1 ox

oR ox

2

2l₁ov1

or ‡2l1 ou_or1

‡ov1

ox

oR

ox ˆrJ; rˆR; …10†

where

Jiˆ

1‡ …oRi=ox†2

1=2

Ri

o2_R

i=ox2 …1‡ …oRi=ox†2†1=2

; iˆ1;2; …11†

J ˆ 1‡ …oR=ox†

2

1=2

R

o2_R=ox2

…1‡ …oR=ox†2†1=2; …12†

whereri,iˆ1;2, denotes the surface tension at the inner jet's inner surface and at the outer jet's

outer surface, respectively,r is the surface tension at the interface between the inner and outer jets, and pe and pi are the pressures of the gases surrounding and enclosed by, respectively, the

hollow, compound liquid jet. These gases have been assumed to be dynamically passive, i.e., pe

and pi are only functions of time, since, in general, they have smaller density and dynamic

viscosity than those of liquids. This implies that the gases surrounding the liquid may not introduce strong velocity variations on each cross-section of the jet, although they may a€ect its dynamics.

Eq. (5) implies that…v2 v1† nˆ0, i.e., that the normal velocity at the outer±inner jet interface is continuous, where n denotes the unit vector normal to rˆR…t;x† and v denotes the velocity vector. If the ¯uids are viscous, then …v2 v1† tˆ0at r ˆR…t;x† where t is the unit vector tangent torˆR…t;x† and the tangential velocity component at this interface is continuous, i.e.,

In addition to the above boundary conditions in the radial direction, conditions in the axial di-rection must also be provided. If the hollow, compound jet emerges from a nozzle, there is a stress singularity at the nozzle±jet's interfaces due to the relaxation of the velocity pro®le from no-slip conditions at the nozzle walls to the free-surface ¯ow away from the nozzle. This relaxation may result in jet contraction or swelling which implies that the radial velocity component is of im-portance 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 pressure 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, therefore, the use of numerical methods.

In this paper, a long wavelength or lubrication approximation is used to reduce Eqs. (1)±(13) to a more manageable (and easier to solve) one-dimensional set of equations as shown in the fol-lowing section.

3. Asymptotic analysis of hollow, compound liquid jets at low Reynolds numbers

For slender jets at low Reynolds number, i.e., ˆR0=k1, it is convenient to nondimen-sionalizer,x,t,u,vandpwith respect toR0,k,k=u0,u0,v0andlu0=k, respectively, whereR0andk denote a characteristic radius and a characteristic wavelength in the axial direction, respectively,

u0 is a characteristic (constant) axial velocity component, v0 ˆR0u0=k, and l is a reference vis-cosity. Here, we will take the reference viscosity as l2, so that the nondimensional governing equations and boundary conditions can be written as

oui

ox ‡

1

r o…vir†

or ˆ0; iˆ1;2; …14†

Re2L…ui† ˆ 2q_q2

i

opi

ox ‡ li

l2

q2

qi

2o2ui

ox2

‡1

r o or r

oui

or

‡Re2

Fr ; iˆ1;2; …15†

Re2L…vi† ˆ q_q2

i

opi

or ‡ li

l2

q2

qi

o2_v

i

ox2

‡_oro 1_r o… †_orrvi

; iˆ1;2; …16†

vj…Ri;x;t† ˆoR_oti‡uj…Ri;x;t†oR_oxi; i;jˆ1;2; …17†

vj…R;x;t† ˆoR_ot ‡uj…R;x;t†oR_ox; jˆ1;2; …18†

22 ovi

or

oui

ox

oRi

ox ‡ oui

or

‡2ovi

ox

1 2 oRi

ox

2!

l2

l₁ 22 ov2 or ou2 ox oR ox ‡ ou2 or

‡2ov2

ox

1 2 oR

ox

2!!

ˆ22 ov1

or ou1 ox oR ox ‡ ou1 or

‡2ov1

ox

1 2 oR

ox

₂!

; rˆR; …20†

22ou1

ox oR1

ox

2

‡2ov_or1 2 ou_or1

‡2ov1

ox

oR1

ox

‡l2

l1…pi p1† 1‡

oR1

ox

2!

ˆl2

l1

r1

r2 1

Ca2J1; rˆR1; …21†

22ou2

ox oR2

ox

2

‡2ov_or2 2 ou_or2

‡2ov2

ox

oR2

ox

‡ …pe p2† 1‡2 oR_ox2

2!

ˆ 1

Ca2J2; rˆR2; …22†

22ou2

ox oR ox

2

‡2ov2

or 2 ou2

or

‡2ov2

ox

oR

ox‡…p1 p2† 1‡2 oR ox

2!

l1

l₂ 22 ou1

ox oR ox

2

‡2ov_or1 2 ou_or1

‡2ov1

ox

oR ox

!

ˆ_rr

2 1

Ca2J; rˆR; …23†

where

Jiˆ

1‡2_oR

i=ox†2

1=2

Ri

2 o2Ri=ox2 …1‡2_oR

i=ox†2†1=2

; iˆ1;2; …24†

J ˆ 1‡

2_oR=ox†2

₁₌₂

R 2

o2_R=ox2

…1‡2_oR=ox†2_†1=2; …25†

L…u† ˆou_ot‡uou_ox‡vou_or …26†

and

Re2 ˆq2u_l0R0

2 ; Frˆ

u2 0

gR0; Ca2 ˆ

l2u0

denote the Reynolds, Froude and capillary numbers, respectively, and the same symbols have been used for dimensional and dimensionless quantities for the sake of brevity.

For small Reynolds numbers,Re2ˆRewithReˆO…1†, and, depending on the magnitude of the Froude and capillary numbers, several ¯ow regimes can be identi®ed. Here, we consider

FrˆF= and Ca2 ˆCa= where F ˆO…1† and CaˆO…1†, which correspond to small gravita-tional ®elds and small surface tension. Substitution of these values and expansion of the depen-dent variables as

/ˆ/0‡2/2‡O…4†; …28†

where/denotes the dependent variables,uj,vj,pj,RandRj,jˆ1;2, in the governing equations

and boundary conditions, together with the expansion of the boundary conditions at R…t;x†,

R1…t;x† and R2…t;x† about R0…t;x†, R10…t;x† and R20…t;x† yield asymptotic expansions which at leading order, i.e., at O…0_{†, correspond to}

oui0

ox ‡

1

r o…vi0r†

or ˆ0; iˆ1;2; …29†

1

r o or r

oui0

or

ˆ0; iˆ1;2; …30†

opi0

or ‡ o or

1

r

o…rvi0†

or

ˆ0; iˆ1;2; …31†

oui0

or …Ri0;x;t† ˆ0; iˆ1;2; …32†

l2

l1

ou20

or …R0;x;t† ˆ ou10

or …R0;x;t†; …33†

for Eqs. (14)±(16), (19) and (20).

The solutions of Eqs. (29)±(33) can be written as

ui0 ˆBi…t;x†; pi0 ˆDi…t;x†; iˆ1;2; …34†

vi0 ˆC_ri oB_oxi ₂r; iˆ1;2; …35†

whereCiˆCi…t;x† can be determined from Eqs. (17) and (18) at leading order as

Ciˆ_oto R

2

i0 2

‡ o

ox Bi R2

i0 2

C1 ˆ_oto R 2 0 2

‡_oxo B1R 2 0 2

; …37†

C2 ˆ_oto R 2 0 2

‡_oxo B2R 2 0 2

: …38†

From Eqs. (21)±(23) at leading order, one can easily obtain the following equations:

2 _RC₂1 10

‡1₂ oB_ox1

‡l2

l₁…pi p10† ˆ r1l2

r2l1 1

CaR10; …39†

2 C2

R2 20 ‡1 2 oB2 ox

‡pe p20ˆ _CaR1

20; …40†

2 C_R2₂ 0

‡1₂ oB_ox2

‡p10 p20‡2l_l1 2

C1

R2 0

‡1₂oB_ox1

ˆ_rr

2 1

CaR0; …41†

which imply that

p20ˆD2 ˆpe‡_Ca1 _R1

20 2 C2 R2 20 ‡1 2 oB2 ox

; …42†

p10ˆD1 ˆpi r_r1

2 1

Ca

1

R10 2

l₁ l2 C1 R2 10

‡1₂oB_ox1

; …43†

pi peˆ_Ca1 _rr

2 1

R0

‡r_r1

2 1

R10‡ 1

R20

‡2C2 _R1₂ 0 1 R2 20

‡2l1

l2C1 1 R2 10 1 R2 0

: …44†

We, therefore, have ®ve equations for nine unknowns, i.e.,D1,D2, B1,B2,C1,C2,R0,R10andR20. By imposing the conditions that the jets are viscous (cf. Eq. (13)), we have

u10…R0;x;t† ˆu20…R0;x;t†; u12…R0;x;t† ˆu22…R0;x;t†; …45†

which imply that (cf. Eqs. (34), (37) and (38))

B1ˆB2; C1ˆC2; …46†

i.e., the leading-order axial velocity component of the inner and outer liquid jets is the same. Moreover, Eqs. (36)±(38) can be written as

o ot

R2 20 R20

2

‡_oxo B1R 2 20 R20

2

ˆ0; …47†

o ot

R2 0 R210

2

‡ o

ox B1 R2

0 R210 2

ˆ0; …48†

which correspond to the conservation of mass at leading order, and

C1ˆC2ˆ_oto R 2 0 2

‡_oxo B1R 2 0 2

: …49†

Eqs. (39)±(41) and (47)±(49) contain seven unknowns; 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 expansion.

At O…2_{†, Eq. (15) yields}

ui2 ˆQ₄ir2‡Milnr‡Ni; iˆ1;2; …50†

whereMi and Ni are functions of …x;t†, and

Q2ˆRe oB_ot2

‡B2oB_ox2

‡oD_ox2 o_ox2B₂2 Re_F ; …51†

Q1ˆl_l2 1

q1

q2 Re

oB1

ot

‡B1oB_ox1

Re F

‡l2

l1

oD1

ox o2_B

1

ox2 : …52†

The shear stress conditions at the jet's interfaces (cf. Eqs. (19) and (20)) yield at O…2_†:

M2ˆ2 _RC2 20

‡3

2R20

oB2 ox oR20 ox oC2 ox ‡ R2 20 2

o2_B 2

ox2 Q2

R2 20

2 ; …53†

M1ˆ2 _RC1 10

‡3₂R10oB_ox1 oR10 ox oC1 ox ‡ R2 10 2

o2_B 1

ox2 Q1

R2 10

2 ; …54†

2 C_R1 0

‡3₂R0oB_ox1 oR0 ox ‡ oC1 ox R2 0 2

o2_B 1

ox2 ‡Q1

R2 0 2 ‡M1

ˆl2

l₁

2 C_R2 0

‡3₂R0oB_ox1 oR0 ox ‡ oC2 ox R2 0 2

o2_B 1

ox2 ‡Q2

R2 0 2 ‡M2

; …55†

which provide three more equations for two additional unknowns, i.e., M1 and M2. These three equations together with Eqs. (39)±(41) and (47)±(49) constitute a closed system of

one-dimen-sional equations for the ¯uid dynamics of hollow, compound, immiscible, isothermal, incom-pressible, liquid jets. Note that, for viscous jets, Eq. (13) yields

u12…R0;x;t† ˆu22…R0;x;t†; …56†

Q1

4 R20‡M1lnR0‡N1 ˆQ₄2R20‡M2lnR0‡N2: …57† Substitution ofM2 andM1 from Eqs. (53) and (54), respectively, into Eq. (55) and use of Eqs. (42), (43), (51) and (52) in the resulting equation yield the following equation for the leading-order axial velocity:

Re A2

‡q1

q2A1

oB1

ot

‡B1oB_ox1

ˆRe_F A2

‡q1

q₂A1

‡_oxo 3 A2

‡l1

l₂A1

oB1

ox

‡_Ca1 _RA₂2

20 oR20 ox r1 r2 A1 R2 10 oR10 ox

‡2C1 l_l1 2 1 1 R0 oR0

ox R20 l_l1 2 1 R3 10 oR10 ox 1 R3 20 oR20 ox

‡ 2_RA₂2 20

‡2l1

l₂ A1 R2 10 oC1 ox ;

…58†

where

A1ˆR 2 0 R210

2 ; A2 ˆ

R2 20 R20

2 ; …59†

and (cf. Eq. (44))

C1 ˆ pi

pe _Ca1 _rr

2 1

R0

‡r_r1

2 1

R10‡ 1

R20

2 _R1₂ 0 1 R2 20‡ l1 l2 1 R2 10 1 R2 0

: …60†

Eq. (58) together with (cf. Eqs. (47)±(49))

oA2

ot ‡

o…A2B1†

ox ˆ0; …61†

oA1

ot ‡

o…A1B1†

ox ˆ0 …62†

and o ot R2 0 2

‡_oxo B1R 2 0 2

ˆC1; …63†

whereC1is given by Eq. (60), constitute a system of four one-dimensional equations forB1,R0,R10 and R20 which is much easier to solve than Eqs. (1)±(13).

The (dimensional) axial stresses on the ®ber are equal to 2l…ou=ox† p, which can be integrated in the radial direction to determine the (local) axial tractions on the hollow, compound jet. These tractions can be nondimensionalized with respect topl2u0R0 and their leading-order values can then be written in dimensionless form as

F…1†

x ˆA1 2l_l1

2

oB1

ox

D1

; …64†

F…2†

x ˆA2 2oB_ox2

D2

; …65†

where the superscripts …1† and …2† denote the inner and outer, respectively, jets, and using the results previously obtained:

F…1†

x ˆA1 3l_l1

2

oB1

ox

pi‡r_r1

2 1

CaR10‡2

l1

l2

C1

R2 10

; …66†

F…2†

x ˆA2 3oB_ox2

pe _CaR1

20‡2

C2

R2 20

: …67†

4. Analytical solutions for steady jets

For steady ¯ows, Eqs. (61) and (62) have the following solutions:

A1BˆQ1; A2BˆQ2; …68†

where Qi, iˆ1;2, are constants, Q1‡Q2ˆ1 andBˆB1 ˆB2. Unfortunately, due to the cou-pling between the leading-order momentum equation (Eq. (58)) andC ˆC1ˆC2(Eq. (60)), it has not been possible to obtain analytical solutions for arbitrary values ofpi, peand Cato Eqs. (58)

and (60)±(63) for steady jets. However, for piˆpe and Caˆ 1, i.e., nonpressurized hollow,

compound liquid jets with zero surface tension, it is possible to obtain analytical solutions as indicated in the next paragraphs since Cˆ0and Eq. (63) has the following solution:

R2

0BˆQ3; …69†

whereQ3 is a constant.

4.1. Viscous regime

This regime is characterized byReˆ0and ®nite values ofF, i.e., Eq. (58) becomes

d

dx 3 A2

‡l1

l2A1

dB

dx

and has the following solution:

B…x† ˆbexp…ax†; …71†

whereaand bare integration constants. Note thatbˆ1 becauseB…0† ˆ1 andaˆlnB…1†. The values ofR0,R10andR20can be easily determined from Eqs. (68) and (69), but they are not shown here.

4.2. Gravitationless regime

This regime corresponds to F ˆ 1 and Re6ˆ0. The solutions of Eqs. (61)±(63) are Eqs. (68) and (69), whereas the leading-order axial momentum equation becomes

Re A2

‡q1

q2A1

Bd_dB_x ˆ_dd_x 3 A2

‡l1

l2A1

dB

dx

; …72†

which has the following solution

B…x† ˆ_Pa ₁ bexp_bexp…ax_ax† _†; …73†

where a and b are integration constants which can be easily determined from the conditions

B…0† ˆ1 andB…1†, i.e.,

P ˆRe 1 Q1‡

q1 q2Q1

3 1 Q1‡l_l1₂Q1

; …74†

and

bˆ_a‡P _P: …75†

4.3. Gravitational-viscous regime

This regime corresponds toReˆ0and ®nite values ofRe=F. The solutions of Eqs. (61)±(63) are Eqs. (68) and (69), and the leading-order axial momentum equation becomes

Re

F A2

‡q1

q2A1

‡_dd_x 3 A2

‡l1

l2A1

dB

dx

ˆ0; …76†

which has the following solution:

2 B

P b

‡2 B B

2P_b 1=2

wherea and bare integration constants, and

P ˆ Re_F 1 Q1‡

q1 q2Q1

3 1 Q1‡l1_l2Q1

: …78†

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, C and R0 as indicated by Eqs. (58) and (60)±(63) because, even for Caˆ 1, pertur-bations in the jet's geometry will introduce perturpertur-bations inpi andpe due to the compression and

expansion of the gases enclosed by and surrounding the hollow, compound jet. Although,pemay

be assumed to be constant, this is not the case for pi. However, for the viscous ¯ow regime

an-alyzed in the previous section, it is possible to perform a linear stability analysis analytically, because this regime is governed (for Caˆ 1andpi ˆpe) by Eqs. (61), (62) and (cf. Eq. (58))

o

ox 3 A2

‡l1

l2A1

oB ox

ˆ0; …79†

together with Eqs. (60) and (63). Fortunately, the linear stability of Eqs. (61), (62) and (79) can be performed without using Eqs. (60) and (63) because they are decoupled from them, as follows. 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†; A1ss…x† ˆQ1exp… ax†; A2ss…x† ˆQ2exp… ax†; …80†

and perturb this steady solution as

B…t;x† ˆBss…1‡b†; A1…t;x† ˆA1ss…1‡a1†; A2…t;x† ˆA2ss…1‡a2†; …81†

whereb,a1 anda2 are functions oftandx. Substitution of Eqs. (80) and (81) into Eqs. (61), (62) and (79) and neglecting nonlinear terms in perturbations yield the following linear partial dif-ferential equations:

A1ss oa1

ot ‡Q1 oa1

ox

‡ob

ox

ˆ0; …82†

A2ss oa2

ot ‡Q2 oa2

ox

‡ob_ox

ˆ0; …83†

aQ2oa_ox2‡l_l1 2aQ1

oa1

ox ‡a Q2

‡l1

l2Q1

ob ox‡ Q2

‡l1

l2Q1

@2_b

which, upon using

a1ˆ^a1…x†exp…rt†; a2 ˆa^2…x†exp…rt†; bˆb^…x†exp…rt†; …85†

whereris a complex number, integration of the perturbed axial momentum equation and use of

^

a1…0† ˆa^2…0† ˆ^b…0† ˆ^b…1† ˆ0, can be written as

rexp… ax†^a1‡d_da^_x1‡d

^

b

dxˆ0; …86†

rexp… ax†^a2‡d_da^_x2‡d

^

b

dxˆ0; …87†

aQ2^a2‡l_l1

2aQ1a^1‡a Q2

‡l1

l₂Q1

^

b‡ Q2

‡l1

l₂Q1

db^

dxˆd; …88†

wheredis an integration constant. Eqs. (86)±(88) imply that

d^b

dx…1† ˆ rexp… a†^a1…1†

da^1 dx …1†

ˆ rexp… a†^a2…1† d_da^_x2…1†; …89†

dˆaQ2^a2…1† ‡l_l1

2aQ1^a1…1† ‡ Q2

‡l1

l2Q1

db^

dx…1†; …90†

d^b

dx…0† ˆ

d^a1 dx …0† ˆ

da^2

dx …0†; dˆ Q2

‡l1

l2Q1

db^

dx…0†; …91†

which together with Eqs. (86)±(88) yield

^

a1ˆ^a2; …a rexp… a††^a1…1† ˆd_dx^a1…1† d_dxa^1…0†; …92†

d2_P

dx2 ‡rexp… ax† dP

whereP ˆQ2a^2‡Q1l_l1₂a^1.

Integration of Eq. (93) and use of the boundary conditions given by Eqs. (89)±(91) yield the following eigenvalue problem

Z r=a

r=aea

ez

z dzˆ

er=aea

er=a

1 er=aea ; …94†

which coincides exactly with that derived by Schultz and Davis [13] for isothermal round jets at low Reynolds numbers. Therefore, the eigenvalue whose real part is 0corresponds to

acˆ3:00650, B…1† ˆ20:21 and ri ˆ14:011, whereri denotes the imaginary part of r. Eq. (94)

clearly shows that the eigenvalueronly depends ona, i.e., the axial velocity at the take-up point, and the results obtained by Schultz and Davis [13] for the viscous regime of round jets at low Reynolds numbers apply to the viscous regime of hollow, compound jets. This is not surprising, for, in the viscous regime, gravitational and inertia e€ects (which depend on the density ratio) are absent.

The evolution of the perturbations to the leading-order radius R0 can be determined in an analogous manner to the one used forA1, A2 and Bby linearizing Eqs. (60)±(63). However, this requires to take into consideration the perturbations topi;pemay be assumed to be constant and

set to 0. The gases enclosed by the the inner jet may be assumed to behave polytropically with a polytropic exponent equal tok, so that

piV_ik ˆQ4; …95†

whereQ4 is a constant, andVi denotes the volume enclosed by the inner jet's inner surface, i.e.,

Vi ˆ

Z 1

0 R 2 1dxˆ

Z 1

0 R 2

10dx‡O…2†; …96†

which clearly indicates that perturbations in the jet's geometry produce perturbations in pi.

Stability analyses similar to the one performed in this section for the viscous regime can also be performed analytically for the gravitationless and gravitational-viscous regimes for which ana-lytical 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.

6. Presentation of results

As shown in previous sections, the nonlinear dynamics of steady, isothermal, hollow, com-pound jets at low Reynolds numbers depends onRe, Re=F,Ca,q₁=q₂,l₁=l₂,pi pe,r=r2,r1=r2,

Q1 and Bss…1†. Under transient conditions, it depends on the above parameters, the polytropic exponentkand the location, amplitude and frequency of the forcing. 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†…1‡ajsinSjt†; …97†

where aj and Sj denote the amplitude and frequency of the imposed forcing at xj, and

…j;xj† ˆ …i;0† and …e;1† denote the nozzle exit and the take-up point, respectively.

Some sample results illustrating the steady-state jet's geometry, axial velocity component and axial traction are presented 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 di€erence method; the number of grid points was at least 2001, B…0† ˆ1 and R1…0† ˆ1. Fig. 2 indicates that the axial velocity 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. Fig. 2 also shows that the axial traction on the inner annular jet also increases quite rapidly near the take-up point, except at low Reynolds numbers, whereas the ratio of the axial traction in the outer jet to that in the inner one is larger than one for Reynolds numbers greater than 1, and smaller than unity for Reynolds numbers smaller than 1. The jet's geometry illustrated in Fig. 2 clearly shows the jet's contraction from 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.

Fig. 2. Jet's geometry (a), axial velocity component (b), axial traction force on the inner annular jet (c) and ratio of axial traction on the outer jet to that on the inner one (d). (Re=Frˆ1, Caˆ1, q1=q2ˆ1, l1=l2ˆ1, r=r2ˆ1,

r1=r2ˆ1, peˆ0, piˆ0, Q1ˆ1, Q2ˆ1, Bss…0† ˆ1, Bss…1† ˆ200Q1. Solid lines: Reˆ1; dashed lines: Reˆ0:1;

Although not shown here, the steady-state jet's radii and ratio of axial tractions were found to increase slightly as Re=F was decreased due to the gravitational pull, and the axial velocity component and axial traction on the inner annular jet were almost independent ofRe=F for the values of the parameters shown in Fig. 2 andRe=F ˆ1, 10and 0.1. The jet's geometry was found to be very sensitive to the capillary number for the parameters shown in Fig. 2; in fact, the jet's radii increases as the capillary number is increased, the initial jet's contraction increases asCa is decreased, and the ratio of axial traction forces increases as the capillary number is decreased. The largest ratio of axial traction forces was found to occur at the nozzle exit for Caˆ0:1 and was about 1.64; this ratio decreases downstream towards the value of 1. The steady-state jet's radii, the axial velocity gradient at the take-up point and the ratio of axial tractions increase as q1=q2 is increased and as l1=l2 is decreased. For q1=q2 ˆ5 and l1=l2 ˆ0:1, the largest ratio of axial traction forces was found to occur at the nozzle exit and was about 3.10and 2.76, respectively, whereas forq₁=q₂ ˆ0:1 andl₁=l₂ˆ10, the largest ratio of axial traction forces was also found to occur at the nozzle exit and was about 1.08 and 0.12, respectively; when the ratio of axial traction forces at the nozzle exit is larger than one, it decreases downstream and tends to an asymptotic value of 1.

The steady-state jet's radii were found to increase slightly whereas the ratio of axial traction forces was found to decrease slightly as r=r2 and r1=r2 were decreased and as pi pe was

Fig. 3. Jet's geometry (a), axial velocity component (b), axial traction force on the inner annular jet (c) and ratio of axial traction on the outer jet to that on the inner one (d). (Reˆ1,Re=Frˆ1,Caˆ1,q1=q2ˆ1,l1=l2ˆ1,r=r2ˆ1,

r1=r2ˆ1,peˆ0,piˆ0,Q2ˆ1,Bss…0† ˆ1,Bss…1† ˆ200Q1. Solid lines:Q1ˆ1; dashed lines:Q1ˆ0:5; dashed±dotted

increased. The initial jet's contraction was found to increase as Q1 was decreased due to the smaller axial velocity component at the take-up point as illustrated in Fig. 3 which also shows that the largest gradients at the downstream boundary correspond toQ1ˆ2. Since the leading-order ¯uid dynamics equations imply that the leading-order axial velocity component is uniform across the hollow, compound jet cross-section, an increase inQ1 corresponds to an increase inR…x†and a larger value of R2…x†. As illustrated in Fig. 3, the cross-sectional area e€ect dominates over the increase in axial velocity when determining the ratio of axial traction forces.

Some sample results which illustrate the nonlinear dynamics of hollow, compound jets when subject to axial velocity perturbations, i.e., Eq. (97), 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 di€erences; the advective derivatives were discretized by means of either two-point, ®rst-order accurate or three-point, second-order accu-rate formulae, whereas the di€usion terms were discretized by means of second-order accuaccu-rate ®nite di€erence expressions. The computations were performed with double precision, at least 2001 grid points, and a time step equal to at most 10 4_{. When strange or chaotic behavior or holes} in the phase diagrams were observed, the computations were repeated and performed in qua-druple precision with a time step equal to 10 8_{. The results presented in Figs. 4±6 and Tables 1±3} correspond to Bss…0† ˆ1, R1…t;0† ˆRad and Bss…1† ˆ1=Rad2. In Table 3, the parameters that appear in the caption correspond to the basic set employed as a reference in the calculations, and

Fig. 4.R…t;1† R1…t;1†(a),R…t;1†(b),R2…t;1† R…t;1†(c) and axial traction on the inner jet (d) at the take-up point as

functions of time. (Reˆ10 4_{, Re=Fˆ0, Caˆ10}30_{, q}

1=q2ˆ10, l1=l2ˆ1, r=r2ˆr1=r2ˆ0:1, peˆpiˆ0,

only the parameter under the heading was varied while maintaining the other parameters equal to those of the basic state.

Table 1 shows that the nonlinear dynamics of hollow, compound jets is very sensitive to the steady-state axial velocity component at the downstream boundary, i.e., the draw ratio or the extension of the jet, and the location where the axial velocity perturbation is applied. If this perturbation is applied at the nozzle exit, it is observed that the radius of the inner±outer jet interface at the take-up point exhibits a periodic motion with a frequency equal to that of the Fig. 6.R…t;1† R1…t;1†(a),R…t;1†(b),R2…t;1† R…t;1†(c) and axial traction on the inner jet at the take-up point (d) as

functions of time). (Reˆ10 4_{, Re=F ˆ0, Caˆ10}30_{, q}

1=q2ˆ10, l1=l2ˆ1, r=r2ˆr1=r2ˆ0:1, peˆpiˆ0,

Q1ˆQ2ˆ0:5, Radˆ0:15,Bss…1† ˆ1=Rad2,aiˆ0:1,aeˆ0,Siˆ1,Seˆ1).

Fig. 5. Power spectrum (a) and phase diagram (b) ofR…t;1†. (Reˆ10 4_{,Re=F ˆ0,Caˆ10}30_,q

1=q2ˆ10,l1=l2ˆ1,

applied velocity perturbation for Radˆ0:20, i.e., Bss…1† ˆ50; the frequency at which the power spectrum shows its largest value ®rst increases asBss…1†is increased, and then decreases; and, the di€erence between the largest and the smallest values ofR, the di€erence between the largest and the smallest values of the axial traction forces on the inner annular jet, and the largest axial force increase whereas the smallest axial traction force decreases asBss…1† is increased. Similar results were found when the axial velocity is perturbed sinusoidally at both the upstream and down-stream boundaries or at the downdown-stream boundary.

Fig. 4 shows the jet's radii and the axial traction force on the inner annular jet at the down-stream point for the values of the parameters exhibited in Table 1, aiˆ0:1, aeˆ0and

Table 1

Maximum and minimum values ofR…t;1†and axial traction forces on the inner and outer annular jets at the take-up point, and maximum spectral power and frequency associated with the maximum power ofR…t;1†: e€ects of the steady-state take-up downstream boundary conditionsA

Rad Rmax…t;1† Rmin…t;1† Fmax…1† Fmin…1† Fmax…2† Fmin…2† P f

0:10a;1;4 _{0.6393 0.0220 58.9174 0.3418 58.9174 0.3418 60.1270 1.83}

0:15a;1;5 _{0.6085 0.0594 23.4121 0.8298 23.4121 0.8298 199.3667 2.01}

0:20a;1;6 _{0.2855 0.2800 4.8944 4.7660 4.8944 4.7660 0.0728 0.16}

0:10c;3;4 _{0.6406 0.0223 58.6326 0.3436 58.6326 0.3436 64.2225 1.83}

0:15c;3;5 _{0.6049 0.0610 23.2065 0.8520 23.2065 0.8520 197.0797 2.01}

0:20c;3;6 _{0.2828 0.2828 4.9196 4.7364 4.9196 4.7364 3.8537e)100.16} A a_{The upstream axial velocity is sinusoidally excited with an amplitude and frequency equal to 0.01 and 1, respectively;} 1_{identical results fora}

iˆ0:05 and 0.10;cthe upstream and downstream axial velocities are sinusoidally excited with an

amplitude and frequency equal to 0.01 and 1, respectively;3_{identical results fora}

iˆaeˆ0:05 and 0.10;4phase diagram

with holes;5_{®lled phase diagram;}6_{periodic motion. The basic set of parameters isReˆ10} 4_{,Re=Fˆ0,Caˆ10}30_,

q1=q2ˆ10, l1=l2ˆ1, r=r2ˆr1=r2ˆ0:1, peˆpiˆ0, Q1ˆQ2ˆ0:5, Bss…1† ˆ1=Rad2, aiˆ0:1, aeˆ0, Siˆ1,

Seˆ1.

Table 2

Maximum and minimum values ofR…t;1†and axial traction forces on the inner and outer annular jets at the take-up point, and maximum spectral power and frequency associated with the maximum power ofR…t;1†: e€ects of the steady-state take-up downstream boundary conditionsA

Rad Rmax…t;1† Rmin…t;1† Fmax…1† Fmin…1† Fmax…2† Fmin…2† P f

0:30a;1;4 _{0.4283 0.4200 3.6532 3.5710 3.6532 3.5710 0.1641 0.16}

0:30b;2;4 _{0.4286 0.4203 3.6386 3.5832 3.6386 3.5832 0.1641 0.16}

0:30c;3;4 _{0.4243 0.4243 3.6785 3.5415 3.6785 3.5415 3:3814e 100.16}

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

iˆ0:05 and 0.10;bthe 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;cthe 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;4periodic motion. The basic set of parameters isReˆ10 4,Re=F ˆ0,Caˆ1030,

q1=q2ˆ10, l1=l2ˆ1, r=r2ˆr1=r2ˆ0:1, peˆpiˆ0, Q1ˆQ2ˆ0:5, Bss…1† ˆ1=Rad2, aiˆ0:1, aeˆ0, Siˆ1,

Table 3

Maximum and minimum values ofR…t;1†and axial traction forces on the inner and outer jets at the take-up point, and maximum spectral power and frequency associated with the maximum power ofR…t;1†: e€ects of the nondimensional ¯uid dynamics parametersA

Parameter Rmax…t;1† Rmin…t;1† Fmax…1† Fmin…1† Fmax…2† Fmin…2† P f

Reˆ …10 6_†a;1 _{0.6395 0.0221 59.4526 0.3471 59.4526 0.3471 81.3482 1.83}

Reˆ …10 3_†a;1 _{0.6364 0.0222 60.0142 0.3528 60.0142 0.3528 55.7488 3.665}

Reˆ1a;2 _{0.1429 0.1402 53.5699 51.6780 53.5699 51.6780 0.0181 0.16}

Re=F ˆ0:1a;1 _{0.6359 0.0223 59.6028 0.3556 59.6028 0.3556 53.8896 3.675}

Re=F ˆ1a;1 _{0.6207 0.0232 57.0894 0.3789 57.0894 0.3789 63.0237 1.885}

Caˆ …102_†a;1 _{0.6376 0.0216 59.8811 0.3497 59.8811 0.3497 81.8096 1.825}

Caˆ …10†a;1 0.6403 0.0174 62.5065 0.3131 62.5065 0.3131 73.6339 1.76

q1=q2ˆ0:1a;1 0.6380 0.0222 59.4961 0.3506 59.4961 0.3506 56.9666 1.83 q1=q2ˆ10a;1 0.6242 0.0230 61.4767 0.3811 61.4767 0.3811 61.4203 1.845 l1=l2ˆ0:1a;1 0.6328 0.0224 6.0213 0.0359 60.2132 0.3587 78.3058 1.835 l1=l2ˆ10a;1 0.6328 0.0221 592.9113 3.4928 59.2911 0.3493 79.5677 1.83 r=r2;r1=r2ˆ10;0:1a;1 0.6364 0.0222 60.0142 0.3528 60.0142 0.3528 55.7488 3.665

piˆ0:1a;1 0.6455 0.0225 60.0181 0.3528 60.0181 0.3528 56.9800 3.665

piˆ0:5a;1 0.6896 0.0240 60.0401 0.3529 60.0401 0.3529 63.0127 3.665

peˆ0:1a;1 0.6279 0.0219 60.0035 0.3528 60.0035 0.3528 54.6035 3.665

peˆ0:5a;1 0.5986 0.0209 59.9642 0.3527 59.9642 0.3527 50.7189 3.665

Q1ˆ0:7a;1 0.6971 0.0243 84.0199 0.4940 60.0142 0.3528 66.8986 3.665

Q1ˆ0:3a;1 0.5692 0.0199 36.0085 0.2117 60.0142 0.3528 44.5990 3.665

Q2ˆ0:7a;1 0.6364 0.0222 60.0142 0.3528 84.0199 0.4940 55.7488 3.665

Q2ˆ0:3a;1 0.6364 0.0222 60.0142 0.3528 36.0085 0.2117 55.7488 3.665

A a_{The upstream axial velocity is sinusoidally excited with an amplitude and frequency equal to 0.01 and 1, respectively;}1_{phase diagram with holes;} 2_{periodic. The basic set of parameters is Reˆ10} 3_{, Caˆ10}30_{, Re=F ˆ0, q}

1=q2ˆ1, l1=l2ˆ1, r=r2ˆr1=r2ˆ1, piˆpeˆ0, Q1ˆ0:5,

Q2ˆ1 Q1, Radˆ0:1,Bss…0† ˆ1,Bss…1† ˆ1=Rad2.

J.I.

Ramos

/International

Journal

of

Engineering

Science

39

(2001)

SiˆSeˆ1. For Bss…1† ˆ50(cf. Fig. 4), R…t;1†, R…t;1† R1…t;1†, R2…t;1† R…t;1† and Fx…1† are

periodic functions of time which have a frequency equal to that of the imposed axial velocity at the nozzle exit, i.e.,Si=2p. This can also be observed in the power spectrum and phase diagram of

R…t;1† presented in Fig. 5; the phase diagram is a circumference, the power spectrum shows a single peak at the excitation frequency, i.e., Si=2p, and the motion is periodic.

For Radˆ0:15, the results presented in Fig. 6 show that the jet's radii and axial traction force on the inner annular jet exhibit large amplitude peaks. The amplitude of these peaks is not constant, and the peaks seem to be modulated with a lower frequency. On the other hand, Fig. 7 indicates that the power spectrum is broader than that of Fig. 5 and exhibits several peaks, while the phase diagram is thicker and shows regions which are never visited.

For Bss…1† ˆ200 (cf. Fig. 8), the largest and the smallest values of R…t;1†, R…t;1† R1…t;1†,

R2…t;1† R…t;1† and Fx…1† are larger and smaller, respectively, and the thickness of the peaks is

smaller than those of Fig. 6, thus indicating that the motion becomes more spiky as Bss…1† is increased. Fig. 8 also shows that the spikes do not have the same amplitude. The spikiness of the time histories presented in Fig. 8 is also present in the power spectrum and phase diagram il-lustrated in Fig. 9. This ®gure clearly indicates that there are regions which are not visited, i.e., the phase diagram exhibits holes, and regions which are frequently visited, i.e., those corresponding to dR=dt…t;1† ˆ0. The power spectrum shown in Fig. 9 is broader than that of Fig. 7.

The phase diagrams of Figs. 7 and 9 have a duck's beak shape, and exhibit many holes. It must be pointed out that the phase diagrams illustrated in Figs. 5, 7 and 9 were obtained for 10006t61200 and that, for a time step equal to 10 8_{, they contain 210}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 behavior.

Table 2 shows that, for Bss…1† 11, R…t;1† is a periodic function of time which has the same frequency as that of the imposed velocity perturbations. Table 2 also shows that this periodic motion has very small amplitude. Moreover, the results presented in Tables 1 and 2 are consistent with the linear stability analysis presented in this paper. Such an analysis is valid for

ReˆRe=F ˆpi peˆ0and Caˆ 1, whereas the results presented in Tables 1 and 2 are for

Re6ˆ0. Moreover, the linear stability analysis was based on the assumptions that Caˆ 1 and

pi peˆ0, whereas the time-dependent studies are based on polytropic compression and

ex-pansion of the gases enclosed by the inner annular jet with kˆ1:4.

Fig. 7. Power spectrum (a) and phase diagram (b) ofR…t;1†. (Reˆ10 4_{,Re=F ˆ0,Caˆ10}30_,q

1=q2ˆ10,l1=l2ˆ1,

A detailed summary of the e€ects of the ¯uid dynamics parameters on the nonlinear dynamics of hollow, compound jets subject to axial velocity perturbations at the nozzle exit is presented in Table 3.

This table indicates that the dynamics ofR…t;1†is periodic with a frequency equal to that of the imposed velocity forReˆ1, that the power spectrum broadens asRe is decreased, and that the frequency corresponding to the largest power ®rst increases and then decreases asReis decreased. The value ofRmax…t;1†decreases slightly asRe=F is increased on account of the gravitational pull; it also decreases as the capillary number is increased.

Fig. 8.R…t;1† R1…t;1†(a),R…t;1†(b),R2…t;1† R…t;1†(c) and axial traction on the inner jet at the take-up point (d) as

functions of time). (Reˆ10 4_{, Re=F ˆ0, Caˆ10}30_{, q}

1=q2ˆ10, l1=l2ˆ1, r=r2ˆr1=r2ˆ0:1, peˆpiˆ0,

Q1ˆQ2ˆ0:5, Radˆ0:10,Bss…1† ˆ1=Rad2,aiˆ0:1,aeˆ0,Siˆ1,Seˆ1).

Fig. 9. Power spectrum (a) and phase diagram (b) ofR…t;1†. (Reˆ10 4_{.Re=F ˆ0,Caˆ10}30_,q

1=q2ˆ10,l1=l2ˆ1,

For the values of the parameters presented in Table 3, i.e.,Reˆ10 3_{, the e€ects ofq}

1=q2,l1=l2,

r=r2 and r1=r2 on the jet's geometry were found to be rather small; however, the value of l1=l2 plays a paramount role in determining the axial traction forces on the inner and outer annular jets. The value of Rmax…t;1† was found to increase as pi was increased and as pe was decreased.

Finally, the value ofQ1 plays a more important e€ect on the jet's geometry thanQ2.

It is interesting to notice that the excitation frequency in Table 3 is 0.16, and that the results presented in this table indicate that the power peak is associated with a frequency of about 1.83 or about twice this value. It is also worth stating that the number of holes or regions of the phase diagram which are not visited increases as the Reynolds number is decreased. Furthermore, the phase diagrams corresponding to the low Reynolds numbers of Table 3 resemble those of Figs. 5, 7 and 9, but exhibit interleaving, i.e., the phase diagrams seem to be composed of tongues that interleave with the beak structure exhibited in Figs. 5, 7 and 9, while the power spectrum broadens as the number of interleaves increases.

The nonlinear studies presented in previous paragraphs and others not shown here [14] indicate that, for small amplitudes of the velocity perturbation at either the upstream or downstream boundaries, the radiusR…t;1†evolves from a ®xed point to a limit cycle and to quasiperiodic and then chaotic motions as the draw ratio, i.e.,Bss…1†, is increased. However, if the amplitude of the velocity perturbation is equal to or larger than 0.01 and the draw ratio is suciently large, then the transition to chaos, if one exists, is abrupt or explosive. In order to verify that these results are physically plausible and do not su€er from numerical truncation errors, calculations performed with quadruple precision, 10001 grid points, and a time step equal to 10 8 _{showed di€erences of} less than 10 2 _{percent with the results presented here.}

The nonlinear studies presented in this paper do not exhibit the route to chaos discussed by Yarin et al. [15] for round jets at low Reynolds numbers. These authors claimed that the cross-section of Newtonian, isothermal round ®bers may vary aperiodically for draw ratios equal to about 30and 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 solutions. The results for hollow, compound jets presented in this paper indicate, however, that the critical draw ratio for the viscous regime can be exceeded without the ap-pearance of chaos if the Reynolds number is suciently large but still much smaller than unity, and that the route to chaos is via quasiperiodic motions for very small amplitude of the forcing, but for amplitudes equal to or higher than about 0.01 and draw ratios suciently large, the transition to chaos, if one exists, is abrupt. Moreover, the phase diagrams employed by Yarin et al. [15] are not really phase diagrams because they considerB…t;xj†versusa…t;xj†whereais the round ®ber's radius; on the other hand, the phase diagrams presented in this paper refer to the time history ofR…t;1†.

7. Conclusions

Perturbation methods based on the slenderness ratio have been employed to determine the leading-order ¯uid dynamics equations of isothermal, axisymmetric, Newtonian, hollow, compound jets at low Reynolds numbers. It has been shown that the leading-order equations are one-dimensional and correspond to the conservation of mass for the inner and outer annular jets,

global linear momentum conservation, and kinematics of the interface between the inner and outer annular jets. These (nondimensional) equations depend on the Reynolds, Froude and capillary numbers, pressure di€erence across the hollow compound jet, and density, dynamic viscosity and (two) surface tension ratios.

Analytical solutions to the leading-order steady-state equations have been obtained for in®nite capillary numbers and zero pressure di€erences across the hollow, compound jet for zero Rey-nolds numbers, zero gravitational pull, or inertialess ¯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 downstream boundary is increased, the radius of the interface between the inner and outer annular jets 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 which are not visited, when the axial velocity component at the take-up boundary is suciently large, thus indicating the presence of strange attractors and chaos.

It has also been shown that the radius of the interface between the inner and outer annular jets at the take-up point exhibits very sharp spikes whose separation depends on the Reynolds, Froude and capillary numbers, density, viscosity and surface tension ratios, pressure di€erence across the hollow, compound jet, and location, amplitude and frequency of the imposed velocity pertur-bations. 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 D.G.E.S. of Spain.

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