16666240

Periodically forced Hopf bifurcation in

annular liquid jets with mass transfer

J.I. Ramos

Departamento de Lenguajes y Ciencias de la Computacion, E.T.S. Ingenieros Industriales, Universidad de Malaga, Plaza El Ejido, s/n, 29013 Malaga, Spain

Abstract

The leading-order ¯uid dynamics equations of thin,annular liquid jets at high Reynolds numbers are obtained by means of perturbation methods and employed to determine numerically the nonlinear dynamics of these jets with mass transfer,when these jets are forced by periodic body forces near the solubility ratio at which a Hopf bifurcation occurs in the absence of forcing,for both Henry's and Sievert's solubility laws. It is shown that the nonlinear dynamics associated with Sievert's law is much richer than that with Henry's law. It is also shown that,for values of the bifurcation parameter smaller than the critical one and Henry's law,the jet's dynamics are periodic with a frequency equal to that of the excitation and undergo period-doubling as the amplitude of the forcing is increased,whereas,for values of the bifurcation parameter larger than the critical one,the dynamics undergo a ®rst transition to quasiperiodic motion followed by a still further transition to chaos as the amplitude of the excitation is increased. For Sievert's law,it is found that,if the value of the bifurcation parameter is equal to that at which the Hopf bifurcation occurs,an increase in the forcing amplitude results in quasiperiodic motions whose amplitude increases as the forcing amplitude is increased, whereas,for values of the bifurcation parameter larger than the critical one,the ¯ow bifurcates to period-four solutions and quasiperiodic motions upon increasing both the amplitude and frequency of the excitation. For Sievert's law and values of the bifurcation parameter smaller than the critical one,it is shown that an increase in the forcing fre-quency ®rst results in ¯ows characterized by several frequencies and broad spectra and that further increases in the forcing amplitude yield broader spectra,whereas increases in the forcing frequency result ®rst in quasiperiodic motions and then frequency locking phenomena. It is also shown that periodically forced Hopf bifurcations in annular liquid

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jets may result in peak transfer rates which are about twice as large as those that occur at the Hopf bifurcation without forcing,even for very small excitation amplitudes. Ó 2001 Elsevier Science Inc. All rights reserved.

Keywords: Annular liquid jets; Periodically forced Hopf bifurcation; Mass transfer; Henry's solubility law; Sievert's solubility law

1. Introduction

Annular liquid jets are sheets of liquid falling from an annular nozzle or ori®ce under gravity which merge onto the symmetry axis to become round jets due to surface tension (Fig. 1). Because of their simple geometry,annular liquid jets have been employed to determine the dynamic surface tension of liquids [1] and to measure the di€usivities of sparingly soluble gases in water [2]. Baird and Davidson [2] developed an equation for the rate of steady liquid phase-controlled gas absorption by assuming equilibrium at the annular liquid jet's inner interface. Their formulation is valid for steady annular liquid jets,em-ploys Henry's solubility law,i.e.,the equilibrium concentration at the annular jet's inner interface is proportional to the (partial) pressure of the gases en-closed by the annular jet,uses the simpli®ed ¯uid dynamics equations em-ployed by Baird and Davidson [1],and assumes that the penetration depth of the gases absorbed by the liquid is much smaller than the jet's thickness.

The ¯uid dynamics equations employed by Baird and Davidson [1,2] are based on Boussinesq's model [3] which is strictly valid for steady annular liquid membranes,i.e.,annular liquid jets of zero thickness,do not account for gravity and viscosity,neglect the curvature of the annular jet in the vertical plane,and assume that the annular jet's slope is small so that the liquid sheet moves almost vertically. All these e€ects except for the viscous ones were in-cluded in the formulation of Ramos [4] who used the Boussinesq equations and Lagrangian coordinates to analyze the ¯uid dynamics of steady,inviscid, annular liquid jets,and obtained analytical solutions for long and thin,over-, under- and non-pressurized annular liquid jets. Ramos and Pitchumani [5] used the Boussinesq's equations [3,4] to determine the steady-state mass absorption rate of the gases enclosed by the annular jet. They also obtained analytical solutions for the steady mass absorption rate by assuming that the gas pene-tration depth in the liquid is smaller than the jet's thickness. Their mass ab-sorption rates were based on equilibrium conditions at the annular jet's inner interface and on Henry's solubility law.

The Boussinesq equations were derived for annular liquid membranes by establishing dynamic equilibrium of ¯uid elements along and normal to the annular liquid membrane under steady-state conditions. The same approach was followed by Lee and Wang [6] to derive the equations for unsteady,an-nular liquid membranes. On the other hand,Ramos [7] considered the Euler equations,integrated them across the annular liquid jet,applied the kinematic and dynamic boundary conditions at the jet's interfaces,and assumed that the jet is slender and thin in his derivation of the equations for the ¯uid dynamics of long annular jets. These assumptions require that the jet's mean radius be much larger and much smaller,respectively,than the jet's thickness and the convergence length,respectively,where the convergence length is the axial distance measured from the nozzle exit at which the annular jet becomes an ordinary,round one. The equations derived by Ramos [7] for annular liquid jets are also valid for annular liquid membranes and are asymptotically iden-tical to those developed by Boussinesq if the jet is thin and slender. Further-more,these equations have been used to study numerically the growth of underpressurized annular liquid jets [8] when the jet's interfaces are clean and mass transfer resistance is neglected,by employing Henry's [9] and Sievert's [10] solubility laws which relate the equilibrium interfacial concentration to the partial pressure and to the square root of the partial pressure,respectively,of the gases enclosed by the annular liquid jet,and a domain-adaptive ®nite di€erence formulation which maps the unsteady,curvilinear geometry of the annular jet into a unit square [11]. More recently,the equations derived by Ramos [7] have been used to determine the e€ects of mass transfer resistance on the growth of under-pressurized,annular liquid jets [12].

Another set of equations for the ¯uid dynamics of inviscid,long and slender, annular liquid jets has been developed by Ramos [13] by means of perturbation

methods based on the slenderness ratio,i.e.,the ratio of the jet's mean radius at the nozzle exit to the convergence length. The leading-order equations obtained by perturbation methods are asymptotically equivalent to those derived by Boussinesq [3] for slender and thin,inviscid,annular liquid jets; they are also asymptotically equivalent to those derived by Ramos [4]. Because of the as-sumptions made in the derivation of the one-dimensional models of inviscid, annular liquid jets [4,7,13], their equations are not valid near the nozzle exit where the ¯ow relaxes from no-slip conditions at the nozzle walls to the free-surface conditions downstream. Moreover,the few available experimental data on annular liquid jets indicate that the Reynolds,Weber and capillary numbers

based on the jet's mean radius (R

0) and mean axial velocity (u0) at the nozzle

exit and the properties of pure water at standard pressure and temperature conditions in the experiments of Baird and Davidson [1] range from 5519,34 and 0.012,respectively,to 16557,305 and 0.037,respectively; the range of these numbers in the experiments of Ho€man et al. [14] is from 3324,7 and 0.0021 to 221 600,30 508 and 0.14,respectively,whereas that of Kihm and Chigier [15] is from 39 012,422 and 0.011,respectively,to 103 000,2954 and 0.029,respec-tively; therefore,in all the experiments mentioned above the Reynolds and capillary numbers are large and small,respectively. If the Reynolds number is

based on the jet's thickness at the nozzle exit,b

0,then it ranges from 258 to 773,

949 to 8707,and 390 to 5150 for the data of Baird and Davidson [1] (b

0=R0ˆ0:047),Ho€man et al. [14] (b0=R0ˆ0:285),and Kihm and Chigier

[15] (b

0=R0ˆ0:01±0:05),respectively. Therefore,for these three sets of

exper-iments,the viscous terms in the linear momentum equations are smaller than the inertia ones,and the surface tension contribution is larger than that due to viscosity but smaller than that of inertia. However,the Reynolds numbers

based onb

0suggest that viscous e€ects may be of some importance,especially

for the lowest values of this number. On the other hand,if the gas enclosed by

the annular liquid is assumed to be carbon dioxide,the Peclet numbers (based

on the jet's mean radius at the nozzle exit) in the experiments of Baird and Davidson [1],Ho€man et al. [14] and Kihm and Chigier [15] range from

2:77106_;1:67106 _{and 1:9610}7_{,respectively,to 8:3110}6_;1:11108

and 5:16107_{,respectively. These large values of the Peclet number indicate}

that mass transfer by di€usion is a slower process than viscous e€ects. In addition to their use for measuring the dynamic surface tension of liquids [1] or the binary di€usion coecients of gases in liquids [2],the volume enclosed by an annular liquid jet may be used as a chemical reactor for the reaction and control of toxic wastes,the direct reduction of zirconium from zirconium tetrachloride and sodium,scrubbing of radioactive and non-radioactive materials,etc. [16]. In some of these potential applications, especially those related with the reaction and control of toxic wastes where the liquid is puri®ed and recycled and chemical reactions occur within the volume enclosed by the annular jet,it is highly desirable that the gaseous

combustion products be absorbed by the liquid jet at a high rate so that they do not accumulate within the volume enclosed by the jet and,thus,do not increase the pressure of these gases. Unfortunately,the binary di€usion co-ecients of gases in liquids are so small [2] that mass transfer phenomena are very slow. However,under certain conditions and due to the nonlinear,in-tegrodi€erential coupling between the ¯uid dynamics and mass transfer equations in annular liquid jets,Hopf's bifurcations have been observed when employing the inviscid model of Ramos [7] for both Henry's and Sievert's solubility laws [17,18]; these bifurcations result in periodic mass transfer rates. Moreover,for larger values of the bifurcation parameter,it has been observed that the nonlinear dynamics of annular liquid jets with mass transfer exhibit period-doubling and chaotic phenomena characterized by large mass transfer rates. Unfortunately,these chaotic phenomena are di-cult to control due to the nonlinear coupling between the ¯uid dynamics and mass transport equations.

The appearance of Hopf's bifurcation in annular liquid jets with mass transfer [17,18] and asymptotic analysis of periodically forced bifurcations in ordinary di€erential equations [19±22] suggest the use of periodic forcing near the Hopf bifurcation in order to explore the e€ects of both the amplitude and frequency of the excitation on the mass absorption rate,and the conditions under which the mass absorption is enhanced. Such forcing may be achieved by means of periodic mass injection into the volume enclosed by the annular liquid jet,¯uctuating body forces,time-dependent periodic excitations at the nozzle exit,and/or acoustic excitations.

Analytical studies of periodic forcing of a nonlinear second-order oscillator close to a Hopf bifurcation [19] indicate that,as the forcing amplitude is in-creased,there is a sequence of bifurcations to quasiperiodic motions,transition regions,and periodic motions with a frequency equal to that of the forcing. Other studies of periodically perturbed Hopf bifurcations for systems of two ordinary di€erential equations show that the addition of small periodic para-metric excitation gives rise to secondary bifurcations [20] and quasiperiodic solutions [21,22]. Kath [23] has shown that, when the frequency of the Hopf bifurcation coincides with that of the forcing,the resulting dynamics may ex-hibit frequency or phase pulling and locking. All of these studies considered ordinary di€erential equations and employed perturbation methods such as averaging techniques.

There are quite a few di€erences between the study presented here and those of [19±23]. First of all,the equations considered here are of the integrodi€er-ential type and contain partial derivatives with respect to the axial and radial coordinates and time,rather than ordinary di€erential equations. Second, because of the nonlinear coupling between the ¯uid dynamics and gas con-centration equations,periodically perturbed Hopf bifurcations in annular liquid jets with mass transfer cannot be studied analytically,and recourse has

to be made to numerical methods which replace the continuous space and time variables by discrete ones.

The objectives of this paper are two-fold. First,a model that accounts for large but ®nite Reynolds numbers in annular liquid jets is developed; as indi-cated previously,all the existing models for annular liquid jets are based on the inviscid ¯ow equations. Second,this model and the inviscid one developed by the author [7] are employed to determine the nonlinear dynamics of annular liquid jets with mass transfer close to the Hopf bifurcation as a function of both the amplitude and frequency of the periodic excitation and as a function of the solubility law; both Henry's and Sievert's solubility laws are considered in this paper. This dynamics is quanti®ed by means of time series,phase

dia-grams,Poincare sections and power spectra for the case that the interfaces are

clean and in equilibrium. The third objective of this paper is to investigate the conditions under which the quasiperiodic motions that may arise from peri-odically forced Hopf bifurcations in annular liquid jets,may result in higher mass transfer rates than that corresponding to the Hopf bifurcation as a consequence of the slowness of di€usion and the di€erences between the compression and expansion cycles of the gases enclosed by the jet. The fourth and ®nal objective is to provide a comparison between the nonlinear dynamics of annular liquid jets and those of other systems of ordinary di€erential equations [19±23].

The paper has been organized as follows. In Section 2,a long wavelength analysis of the Navier±Stokes equations at high Reynolds numbers is presented as a function of the capillary number. An inviscid model for the ¯uid dynamics equations is presented in Section 3,whereas the thermodynamics of the gases enclosed by the liquid jet is considered in Section 4. The transport equation for the gas absorbed by the liquid is treated in Section 5 together with the inter-facial boundary conditions and the two solubility laws employed in this paper. A brief description of both the numerical method employed to solve the gov-erning equations and the incremental approach followed for the analysis of periodically forced Hopf bifurcations in annular liquid jets with mass transfer is presented in Section 6. Finally,a rather long section on results concludes the paper.

2. Fluid dynamics of annular liquid jets at high Reynolds numbers

Consider an axisymmetric,immiscible,annular liquid jet,and assume that the ¯uid is incompressible (constant density; this will be justi®ed later),iso-thermal,two-dimensional and Newtonian so that the conservation equations of mass and linear momentum can be written as

ou

ox‡

1

r

o…vr†

q o_ou_t

‡uo_ou_x‡vo_ou_r

ˆ o_op_x‡l o_o2_xu₂

‡1_{r o}o_r ro_ou_r

‡qg; …2†

q o_ov_t

‡uo_ov_x‡vo_ov_r

ˆ o_op_r‡l o_ox2v₂

‡_oo_r 1_r o… †_orv_r

; …3†

wheretis time;u andvare the axial and radial velocity

components,respec-tively;xandrare the axial and radial coordinates,respectively;qandlare the

liquid's density and dynamic viscosity,respectively;pis the pressure; and,gis

the gravitational acceleration.

Eqs. (1)±(3) are subjected to kinematic and dynamic boundary conditions at

the jet's interfaces,R1…x;t†andR2…x;t†,whereR1 andR2denote the inner and

outer radii of the annular liquid jet,respectively. The kinematic conditions establish that the liquid-surroundings 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

condi-tions at the jet's interfaces (iˆ1;2) may be written as

v…Ri;x;t† ˆo_oR_ti‡u…Ri;x;t†o_oR_xi; …4†

2l o_ov_r

ou

ox

oRi

ox ‡l

ou

or

‡o_ov_x

1 o_oR_xi

₂!

ˆ0; …5†

2lo_ou_x o_oR_x1

2

‡2lo_ov_r 2l o_ou_r

‡o_oxv

oR1

ox ‡ …pi p† 1‡

oR1

ox

2!

ˆr 1‡…oR1=ox†

2

₁₌₂

R1

0 B

@ …o2R1=ox2†

1‡…oR1=ox†2

1=2

1 C

A; …6†

2lo_ou_x o_oR_x2

2

‡2lo_ov_r 2l o_ou_r

‡o_oxv _oR

2

ox ‡ …pe p† 1‡

oR2

ox

2!

ˆ r 1‡…oR2=ox†

2

1=2

R2

0 B

@ …o2R2=ox2†

1‡…oR2=ox†2

1=2

1 C

A; …7†

whererdenotes the liquid's surface tension,andpeandpiare the pressures of

These gases have been assumed to be dynamically passive 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 vari-ations at each cross-section of the jet,although they may a€ect its dynamics. In addition,the shear stress resulting from the gradient of interfacial gas con-centration,i.e.,the Marangoni e€ect,which would a€ect the dynamic surface tension of the liquid,has been neglected in the above equations. This ap-proximation is justi®ed because,if the liquid's surface tension is assumed to decrease linearly with the interfacial concentration,the relative change in surface tension due to the gas absorption at the interface is equal to the product of the derivative of the surface tension with respect to the concentration times the interfacial gas concentration divided by the surface tension in the absence of gas absorption,and the numerator of this ratio is small for typical liquids, e.g.,water,and for the annular jets considered here where the di€usion coef-®cient of gases in liquids is very small.

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 an annular 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 importance near the nozzle. Furthermore, at high Reynolds numbers,boundary layers are formed on the jet's interfaces; these boundary layers grow downstream until they merge. After the merging, the ¯ow is essentially parabolic and governed by boundary layer equations until the velocity across the jet becomes uniform and the jet falls according to Torricelli's law. 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 requires the use of nu-merical methods.

In this section,a long wavelength or lubrication approximation is used to reduce Eqs. (1)±(7) to a more manageable (and easier to solve) set of equations. Note that Eqs. (1)±(7) have also been analyzed asymptotically by the author

[13] for inviscid ¯uids,i.e.lˆ0,and that the resulting leading-order equations

for inviscid ¯uids coincide with those derived from an integral balance tech-nique [7] whose analytical and numerical solutions are in good agreement with available experimental data for long annular liquid jets [15].

Ifr;x;t;u;vandpare nondimensionalized with respect tob0;k;k=u0;u0;v0and

qu2

0,whereb0 andkdenote a characteristic jet's thickness and a characteristic

wavelength in the axial direction,respectively,u0 is a characteristic (constant)

ou

ox‡

1

r

o…vr†

or ˆ0; …8†

ou

ot‡u

ou

ox‡v

ou

orˆ

op

ox‡

Re

o2_u

ox2‡

1 Re

1

r

o or r

ou

or

‡1_Fr; …9†

ov

ot‡u

ov

ox‡v

ov

orˆ

1 2

op

or‡

Re

ov2

ox2‡

1 Re o or 1 r o…rv† or

; …10†

v…Ri;x;t† ˆo_oR_ti‡u…Ri;x;t†o_oR_xi; …11†

22 ov

or

ou

ox

oRi

ox ‡

ou

or

‡2ov

ox

1 2 oRi

ox

2!

ˆ0; …12†

22ou

ox

oR1

ox

2

‡2o_ov_r 2 o_ou_r

‡2ov

ox _oR

1

ox ‡ Re

…pi p† 1‡2

oR1

ox

2!

ˆ_Ca1 1‡

2_oR1=ox†2

1=2

R1

0 B

@ 2 …o2R1=ox2†

1‡2_oR1=ox†2

1=2

1 C

A; rˆR1…x;t†;

…13†

22ou

ox

oR2

ox

2

‡2 ou

or

‡2ov

ox

oR2

ox ‡ Re

…pe p† 1‡2

oR2

ox

2!

2ov

or

ˆ _Ca1 1‡

2_oR 2=ox

… †2

1=2

R2

0 B

@ 2 …o2R2=ox2†

1‡2_oR2=ox†2

1=2

1 C

A; rˆR2…x;t†;

…14†

where for the sake of brevity the same symbols have been used to denote

di-mensional and dimensionless variables, Frˆu2

0 =gb0; Reˆ …qu0b0†=l and

Caˆ …l_u

0†=r are the Froude,Reynolds and capillary numbers,respectively,

andˆb

0=k. Note that the asterisks denote dimensional quantities and have

been introduced for the reader's convenience,while the axial wavelength is a typical distance along the annular jet's axis,e.g.,the jet's convergence length. Therefore,the long wavelength approximation employed here is justi®ed

provided that the jet's thickness is much smaller than the jet's convergence length.

In what follows,it will be assumed that 1,i.e.,a long wavelength

analysis will be performed,and ReˆR= and FrˆF=,where R and F are

O…1†. Furthermore,depending on the magnitude of the capillary number,

several ¯ow regimes can be identi®ed as indicated in Sections 2.1. The

assumption1 is justi®ed (cf. Section 1),since,for example,k_=R

0is larger

than 2 andb

0=R0ˆ 0.01±0.05 in the experiments of Kihm and Chigier [15].

2.1. Large capillary numbers

IfCaˆC=,Eqs. (8)±(14) indicate that only terms proportional to2_appear

in these equations; therefore,we look for the following asymptotic expansions:

pˆp0‡2p2‡O…4†; uˆu0‡2u2‡O…4†; …15†

vˆv0‡2v2‡O…4†; R1ˆR10‡2R12‡O…4†; …16†

R2ˆR20‡2R22‡O…4†: …17†

Substitution of Eqs. (15)±(17) into the continuity and linear momentum equations,and expansion of the kinematic and dynamic boundary conditions

aroundR10andR20yield a system of equations in powers of2. By setting the

coecients of each power in 2 _{to zero in each equation,a hierarchy of}

equations results. To0_{,one obtains}

ou0

ox ‡

1

r

o…v0r†

or ˆ0; …18†

ou0

ot ‡u0

ou0

ox ‡v0

ou0

or ˆ

op0

ox ‡

1

R

1

r

o or r

ou0

or

‡_F1; …19†

op0

or ˆ0; …20†

v0…Ri0;x;t† ˆo_oR_ti0‡u0…Ri0;x;t†o_oR_xi0; …21†

ou0…Ri0;x;t†

or ˆ0; …22†

p0…R10;x;t† ˆpi; p0…R20;x;t† ˆpe; …23†

which indicate thatp0ˆA…x;t†,i.e.,the leading-order pressure is not a function

mathemat-ical compatibility,p0ˆpiˆpe,i.e.,since the gases surrounding the liquid were

assumed to be dynamically passive,p0is at most a function of time. Therefore,

annular liquid jets at high Reynolds and capillary numbers are governed,at leading order,by Eqs. (18) and (19) without the pressure gradient term,Eq. (22) and the kinematic conditions given by Eq. (21),i.e.,the jet is governed by the following boundary layer equations:

ou0

ox ‡

1

r

o…v0r†

or ˆ0; …24†

ou0

ot ‡u0

ou0

ox ‡v0

ou0

or ˆ

1

R

1

r

o or r

ou0

or

‡_F1; …25†

v0…Ri0;x;t† ˆo_oR_ti0‡u0…Ri0;x;t†o_oR_xi0; …26†

ou0…Ri0;x;t†

or ˆ0: …27†

Eq. (25) reduces to the Euler equation forRˆ 1. In fact,Eqs. (24)±(26) are

identical to those of inviscid,irrotational,annular liquid jets if Rˆ 1 [13];

moreover,for inviscid,irrotational,annular liquid jets,Eq. (27) is to be

re-placed by…ou0=or†…r;x;t† ˆ0. Therefore,the analysis presented in this section

is also valid for inviscid,irrotational,annular liquid jets if the Reynolds number is in®nite and the ¯ow is irrotational.

SinceRwas assumed to be O…1†; q_u

0b0l,and sinceCawas assumed to

be much larger than one,i.e.,l_r=u

0,it may be concluded that the analysis

presented in this section is valid forWeˆ …q_u2

0R0†=r1,where We is the

Weber number. Furthermore,sinceRˆO…1†,the inertia terms are of the same

order of magnitude as the di€usion ones in Eq. (19).

Eq. (20) implies that the pressure is independent of the radial coordinate at leading order; therefore,Eqs. (24)±(27) are not valid near the nozzle on account of the stress singularity and the radial pressure gradients there. Moreover,the equations derived in this section are not applicable to the experiments of Baird and Davidson [1],Ho€man et al. [14] and Kihm and Chigier [15] because of the

O…1=†assumption on the capillary number (cf. Section 1). The equations for

small capillary numbers are derived in Section 2.2.

2.2. Small capillary numbers

If CaˆC,where CˆO…1†,substitution of Eqs. (15)±(17) into Eqs. (8)±

(14) yields a system of equations in powers of 2_{. Equating the coecients in}

p0…R10;x;t† ˆpi _C1_RR1

10; p0…R20;x;t† ˆpe‡

1

CR

1

R20; …28†

which together withp0ˆA…x;t†imply that

pi peˆ_C1_{R R}1

10

‡_R1

20

; …29†

p0ˆA…x;t† ˆpi _C1_RR1

10ˆpe‡

1

CRR 20: …30†

Therefore,the di€erence between the pressure of the gases enclosed by and that of those surrounding the annular liquid jet is balanced by surface tension,

and the leading-order pressure is a function of bothxandt. Furthermore,the

leading-order equations can be summarized as

ou0

ox ‡

1

r

o…v0r†

or ˆ0; …31†

ou0

ot ‡u0

ou0

ox ‡v0

ou0

or ˆ

1 CR 1 R2 10

oR10

ox ‡

1 R 1 r o or r

ou0

or

‡_F1; …32†

v0…Ri0;x;t† ˆo_oR_ti0‡u0…Ri0;x;t†o_oR_xi0; …33†

ou0…Ri0;x;t†

or ˆ0; …34†

p0ˆA…x;t† ˆpi _C1_RR1

10; pi peˆ

1 CR 1 R10

‡_R1

20

: …35†

Note that Eqs. (31)±(35) reduce to those of Section 2.1 ifCˆ 1.

Further-more,the asymptotic analysis presented in this section indicates that,to leading order,the pressure in the annular liquid jet is uniform in the radial direction and only depends on one of the radii of curvature,whereas Esser and Abdel-Khalik [24] maintained both radii of curvature and considered the pressure as a

function ofr; however,they neglected the axial gradient of the pressure in their

Eq. (3). The inconsistency between their Eq. (3) and their Eqs. (6) and (7) may be the cause for the poor agreement between their theoretical and experimental data [24]. In addition,the leading-order radial momentum equation reduces to

the condition that the leading-order pressure is only a function of x and t,

whereas Esser and Abdel-Khalik [24] maintained the full radial momentum equation (cf. Eq. (3)) in their studies.

As shown above,the pressure is independent of the radial coordinate at leading order; therefore,Eqs. (31)±(35) are not valid near the nozzle on account of the stress singularity and the radial pressure gradients there.

Eqs. (31)±(33) are identical to those of inviscid,irrotational,annular liquid

jets if Rˆ 1 [13]; moreover,for inviscid,irrotational,annular liquid jets,

Eq. (34) is to be replaced by …ou0=or†…r;x;t† ˆ0. Therefore,the analysis

pre-sented in this section is also valid for inviscid,irrotational,capillary,annular liquid jets if the Reynolds number is in®nite and the ¯ow is irrotational.

The perturbation method employed in this section may be easily used to obtain higher-order terms in the asymptotic expansion,and the

leading-or-der equations are asymptotically valid provided that j2_/

2j j/0j,where /

denotes any dependent variable (cf. Eqs. (15)±(17)). This condition also

provides a limit on the value of for which the analysis presented above

holds.

Reverting to the dimensional quantities in Eqs. (31)±(35) and nondimen-sionalizing the resulting equations using the average axial velocity and the jet's mean radius at the nozzle exit,the leading-order equations can be written as

ou

ox‡

1

r

o…vr†

or ˆ0; …36†

ou

ot‡u

ou

ox‡v

ou

orˆ

1

We

1

R2 1

oR1

ox ‡

1

Re

1

r

o or r

ou

or

‡ 1

Fr; …37†

v…Ri;x;t† ˆo_oR_ti‡u…Ri;x;t†o_oR_xi; …38†

ou…Ri;x;t†

or ˆ0; …39†

pi peˆ_We1 _R1

1

‡_R1

2

; …40†

where the subscript 0 has been removed andReˆq_R

0u0=l;Weˆqu02R0=r

andF ˆu2

0 =gR0. In addition to these three parameters,one must consider the

jet's thickness-to-radius ratio at the nozzle exit,i.e.,b

0=R0,the nozzle exit angle

h0,and the velocity distribution at the nozzle exit.

In order to provide a comparison between the high Reynolds number equations deduced above and those of the inviscid model presented in Section 3,it proves convenient to introduce the pressure coecient

CpnˆCp p

i p e 1

and the following boundary conditions at the nozzle exit

u…r;0;t† ˆF…r;t†; v…r;0;t† ˆF…r;t†tanh0: …42†

Eqs. (36)±(42) clearly indicate that periodic forcing of the ¯uid dynamics equations can be achieved through excitations in the mass ¯ow rate at the

nozzle exit,F…r;t†,pressure di€erence,pi pe,and body force,Fr. The latter is

a common phenomenon in microgravity where it is called g-jitter,but may be achieved on earth experiments by oscillating the nozzle in an axial manner without rotation. All these three forcings were considered in the model pre-sented here and produced the same qualitative results; therefore,we shall consider here only excitations produced by periodic modulations in the grav-itational acceleration,i.e.,

g_ˆg

0…1‡asin…xt††; …43†

where the ground gravitational acceleration, g

0,is constant,a denotes the

nondimensional amplitude of the gravitational excitation,andx _{is a}

dimen-sional frequency.

Using the nondimensionalization discussed above,it is easily shown that 1

Frˆ

1

Fr0…1‡asin…2pSt††; …44†

where Sˆx_R

0=2pu0 denotes the Strouhal number or the ratio between a

characteristic residence time and the period of the excitation.

3. Inviscid ¯ow equations

The governing equations for inviscid,annular liquid jets can be derived

from those presented in Section 2 by simply setting Reˆ 1in Eq. (37),and

the resulting equations would be valid for long and thin,inviscid,annular liquid jets. Alternatively,one may use the asymptotic equations for inviscid, annular liquid jets deduced by the author [7] from the integration of the Euler equations across the jet by assuming that the annular liquid jet is thin. For the sake of completeness,these latter equations can be written in non-dimensional form as

om

ot ‡

o

ox…mu† ˆ0; …45†

o ot…mu† ‡

o

ox…muu† ˆ m Fr‡

1

We

oJ

ox

CpnRo_oR_x

o ot…mv† ‡

o

ox…muv† ˆ

1

We CpnR

oJ=ox

oR=ox

; …47†

vˆoR

ot ‡u

oR

ox; J ˆR 1 " ,

‡ oR

ox ₂#1=2

; …48†

where

R2ˆR‡b=2; R2ˆR b=2; …49†

bˆm_R b0

R

0; We

_ˆm

0u

2

0=2rR0; …50†

R;u;bandmdenote the annular jet's mean radius,axial velocity component,

thickness and mass per unit length,respectively. These equations have been obtained from the dimensional ones by nondimensionalizing the length,time,

velocity components and mass per unit length with respect toR

0andR0=u0;u0

andm

0ˆqR0b0,respectively,where the subscript 0 denotes the annular nozzle

exit. Note thatWe_ˆWe…b

0=2R0†.

In this paper,the following boundary conditions at the nozzle exit were employed for Eqs. (45)±(48):

m…t;0† ˆ1; u…t;0† ˆ1; v…t;0† ˆtanh0; R…t;0† ˆ1: …51†

It must be pointed out that a simple asymptotic analysis shows that the equations presented in this section coincide,at leading order,with those of

Section 2 ifReˆ 1and the annular liquid jet is thin and long.

4. Gases enclosed by the annular liquid jet

Since the mass di€usivities of gases are much larger than the binary di€usion coecients of gases in liquids,it may be assumed that the concentration of the gases enclosed by the annular liquid jet is uniform. If,in addition,these gases

are assumed to be ideal and isothermal at the (constant) temperatureT _and

consist of a single species,and the Mach number is small,the pressure of the gases enclosed by the annular liquid jet can be assumed to be uniform and can be determined as

p

i ˆM

iR~T pR3

0

mi

Z _L

0 R 2 1dx

; pi

p

e

ˆm_Vi; …52†

where

miˆmi=Mi; MiˆpR

3

0pe=R~T; V ˆ

Z L

0 R 2

whereR~_{is the speci®c gas constant;m}

i is the mass of the gases enclosed by the

annular liquid jet, L is the nondimensional convergence length,andV is the

nondimensional volume of the gases enclosed by the annular liquid jet. The gases surrounding the liquid jet are also assumed to be ideal and

iso-thermal,and in®nite in extent so thatp

e can be assumed to be constant.

Substitution of Eqs. (51) and (52) into Eq. (41) yields

CpnˆCp mi

Z _L

0 R 2 1dx

1

; …54†

which indicates that the pressure coecient depends on the nondimensional

convergence length,L,which can be determined from the condition that,at the

convergence point,the annular jet's inner radius is zero,i.e.,

R1…L…t†;t† ˆ0: …55†

It must be pointed out that the equations presented in this section are ap-proximations since,in determining the volume enclosed by the jet's inner in-terface,the leading-order radius was employed; the errors incurred by this

approximation are O…2_†.

It is clear from Eqs. (52) and (54) that the ¯uid dynamics of annular liquid

jets depends onmiwhich,in turn,depends on the mass absorbed by the liquid

and on the mass injected into the volume enclosed by the jet. In the absence of mass injection into or mass generation in the volume enclosed by the annular

liquid jet, mi depends on the mass absorbed by the liquid as determined in

Section 5.

5. Gas concentration in the annular liquid jet

Since the binary di€usion coecient of gases in liquids is small,the mass absorption rate is expected to be small; in the experiments of Baird and Davidson [2],the volumetric absorption rate was 0.01 ml/s compared with a liquid volumetric ¯ow rate of 3.35 ml/s. As a consequence,volumetric dis-placement e€ects due to the gas absorbed by the liquid are small,and the liquid may be assumed to be incompressible (cf. Section 2).

The (dimensional) equation for the concentration of the gases absorbed by the liquid can be written as

oc

ot‡u

oc

ox‡v

oc

orˆD 1

r

o or r

oc

or

‡o_o2_xc₂

; …56†

In what follows,it will be assumed that the Peclet number is suciently large (cf. Section 1) so that the thickness of the concentration boundary layer formed at the annular jet's inner interface is smaller than the jet's thickness,and that both interfaces are clean and in equilibrium,i.e.,

c_R

1;x;t† ˆSipin; c…R2;x;t† ˆSepen; …57†

whereS

i and Se denote the solubilities of the gases at the annular jet's inner

and outer,respectively,interfaces,andnˆ1 and 1=2 denote Henry's [9] and

Sievert's [10] solubility laws,respectively.

If lengths,time,and velocity components are nondimensionalized as before,

i.e.,with respect to R

0; R0=u0,andu0,respectively,and the following

nondi-mensional concentration is introduced

cˆ2R~T p

e

…c _c

eqe†; …58†

wherec

eqeˆSependenotes the equilibrium gas concentration at the jet's outer

interface,the nondimensional equation for the gas dissolved in the liquid is

oc

ot‡u

oc

ox‡v

oc

orˆ

1

Pe

1

r

o or r

oc

or

‡o_o2_xc₂

; …59†

wherePeˆu

0R0=D is the mass Peclet number. Note that,using the same

ar-guments as those of Section 2 (cf. Section 1),the axial di€usion in Eq. (59) can

be neglected because it is O…2_†.

With the nondimensionalization introduced above,the boundary conditions at the annular nozzle's exit and at the jet's outer interface are

c…r;0;t† ˆa…b 1†; …60†

c…R1…x;t†;x;t† ˆa c R_Lmi 0 R21dx

!_n

1

!

; …61†

c…R2…x;t†;x;t† ˆ0; …62†

where

aˆ2R~_TS

ep…en 1†; bˆc0=Sepen; cˆSi=Se; …63†

andc

0…r;0;t†denotes the (dimensional) gas concentration in the liquid at the

nozzle exit.

The ¯uid dynamics and gas concentration equations presented above

by the annular liquid jet,in turn,depends on the mass absorption rate which can be written in dimensionless form as:

dmi

dt ˆ

1

Pe Z L

0 R1

oc

or…R1…x;t†;x;t†…1‡tan2h1†dx; …64†

subject to

mi…0† ˆp

i…0†

p

e

Z L

0 R 2

1…x;0†dx; …65†

where Eq. (52) has been used and tanh1ˆoR1=ox.

6. Numerical solution procedure

The equations presented in this and the previous sections are partial dif-ferential equations of the integrodi€erential type and cannot be solved ana-lytically due to their nonlinear coupling even under steady-state conditions because the e€ects of capillarity on the ¯uid dynamics equations. They can, however,be solved numerically to study both the ¯uid dynamics of and mass absorption by annular liquid jets under steady or transient conditions by using a domain-adaptive technique which maps the curvilinear geometry of the annular liquid jet into a unit square [11]. Moreover,the nonlinear dy-namics of annular liquid jets is potentially very rich because it does depend

on the following parameters Re (for the high Reynolds number model

presented in Section 2), Fr0;We, h0;b0=R0;F…r;t†, pi…0†=pe;Pe;a;b;c;n and

c0…r;0;t†,and the amplitude and frequency of the forcing,i.e.,a and S,

respectively.

A thorough study of the e€ects of each of the above parameters on the nonlinear dynamics of and mass absorption by annular liquid jets is a highly demanding task which requires very large computational resources and time because the problem is a free-surface one where the radii of the jet's inter-faces are to be determined; the inner interface,in particular,plays a key role in determining the mass and volume of the gases enclosed by the annular jet and the mass absorption rate,and,therefore,the pressure of the gases en-closed by the jet. Furthermore,since the binary di€usion coecient of gases in liquids is very small,the absorption of the gases by the ¯owing liquid requires the use of very re®ned grids near the jet's interfaces in order to determine accurately the mass boundary layer and the concentration gradi-ents there. In addition,when the ¯ow exhibits a Hopf bifurcation or this bifurcation is periodically forced as in this paper,it is necessary to use suciently small time steps to accurately capture the time response of the

annular jet. In most of the calculations presented below,at least 1000 grid points in the radial direction were employed; usually,1000 points were used in the axial direction; and,the time step did not exceed 0.0001 and 0.001 for Henry's and Sievert's,respectively,solubility laws,when the equations exhibit a Hopf bifurcation. Moreover,we have used ®xed values of all the

param-eters except forcwhich was considered to be the bifurcation parameter,and

a and S which were employed to determine the e€ects of the amplitude and

frequency of forcing near the Hopf bifurcation.

Since the value of cat which a Hopf bifurcation is observed has been

de-termined numerically previously [17,18] for Henry's and Sievert's solubility laws for the inviscid model presented in Section 3,the calculations were started

for these values and employed an incremental approach inc,i.e.,cwas varied

incrementally from the solution at which the Hopf bifurcation occurred,and the calculations for this new value were performed a large number of time steps in order to ensure that the results truly corresponded to the desired value of the bifurcation parameter. This was also veri®ed by starting the calculations with

this value ofcas follows. In order to maintain steady-state conditions,mass

must be injected into the volume enclosed by the annular liquid jet at a rate equal to the mass absorption rate by the liquid. For steady,annular liquid jets,

pi and mi are constant,the boundary conditions are independent of time,

dL=dtˆ0,and the ¯uid dynamics and gas concentration equations are not

coupled. This steady-state solution can be used as initial condition to determine

the growth of underpressurized annular liquid jets as a function ofconce no

gases are injected into the volume enclosed by the jet. For the sake of

conve-nience,tˆ0 corresponds to steady-state conditions,whereas,fortP0‡_,the

mass injection rate into the volume enclosed by the annular liquid jet was set to zero. The di€erences between the results of the incremental continuation method and those started from the steady state described above were less than 10 12_.

Once a valid solution for a value of the bifurcation parameter was achieved,

aandSwere set to their desired values,i.e.,values di€erent from zero,in order

to determine the e€ects of the amplitude and frequency of the periodic forcing on the dynamics of annular liquid jets with mass transfer. Therefore,the in-cremental approach proposed here allows to determine the dynamics near the bifurcation points with and without forcing.

7. Presentation of results

As stated in Section 1,this paper deals with periodically forced Hopf

bifurcation in annular liquid jets with mass transfer when Henry's (nˆ1)

interface gas concentrations. Some sample results that illustrate the non-linear dynamics of annular liquid jets with mass transfer are presented in this section.

7.1. Henry's solubility law

For the study of periodically forced Hopf bifurcations in annular liquid jets with mass transfer and interfacial concentrations governed by Henry's solu-bility law,the parameters Re;Fr0;We,Pe,h0, b0=R0;Cp;pi…0†=pe;a andb were

set to 10 000,10,250,106_{;0;0:05;1,0:75;1 and 1,respectively. For these}

pa-rameters, CaˆWe=Reˆ0:025 which corresponds to the small capillary

number regime analyzed previously. It should be noted that,for uniform axial velocity pro®les at the nozzle exit,Eqs. (32) and (37) indicate that the linear momentum equation is independent of the viscous terms. Moreover,as shown in [17],for these parameters and the inviscid model of the author [7] which is

based on an integral formulation,a Hopf bifurcation occurs forcH4:8,and

the initial (tˆ0 ) convergence length corresponding to these parameters was

Lˆ4:28; therefore,the annular jet is long and thin (cf. Section 2). Nearly the

same value ofc_Hwas found for the above parameters with the high Reynolds

number model of Section 2 when the axial velocity at the nozzle exit is uniform; therefore,the long wavelength or lubrication approximation employed in this paper is justi®ed.

Forcˆ4 andaˆ0,i.e.,without forcing andc<c_H,it was observed that

the initially underpressurized annular liquid jet grew until an asymptotic

steady state was reached at about tˆ2000; this corresponds to no mass

transfer between the gases enclosed by the annular jet and the liquid. For

cˆ4;aˆ0:05 and Sˆ0:50,the results shown in Fig. 2 indicate that the

pressure coecient,the gas concentration at the jet's inner interface and the mass absorption rate are periodic but not sinusoidal functions of time with a period equal to that of the forcing,and the phase diagram for the mean radius of the jet at the convergence point is a closed curve. Although not shown here,the jet's thickness and axial velocity component at the jet's convergence length,and the volume of the gases enclosed by the annular jet are also periodic functions with a period equal to that of the forcing. The ratio of the volume of the gases enclosed by the annular liquid jet to the initial volume enclosed by these gases was at most equal to 0.9945. Similar behaviour to that illustrated in Fig. 2 has also been observed for the same

value of cand di€erent amplitudes and frequencies of forcing; however,the

amplitude of the oscillations in the pressure coecient,gas concentration at the jet's inner interface,mass absorption rate and volume enclosed by the jet increase as the forcing amplitude is increased,and decrease as the excitation

frequency is increased. For the same values of a and S as in Fig. 2 but for

However,for aˆ0:05; Sˆ0:25 and cˆ4:2,the results exhibited in Fig. 3 clearly indicate that the amplitude of the oscillations in the pressure coe-cient has increased; this coecoe-cient is periodic but exhibits several relative extrema in each period. Both the time series and the phase diagram for the jet's mean radius at the jet's convergence point indicate that this radius has two characteristic frequencies. The ratio of the volume of the gases enclosed by the annular liquid jet to the initial volume enclosed by these gases was at most equal to 0.999; therefore,the amplitude of the oscillations in the volume

enclosed by the annular jet increases as cis increased.

A comparison between the mass transfer rates shown in Figs. 2 and 3

clearly shows that the peak mass transfer rate for cˆ4:2 is about 10 times

larger than that for cˆ4,and indicates the great e€ect of the forcing

fre-quency on the nonlinear dynamics of annular liquid jets. It is interesting to

notice that,foraˆ0:05;Sˆ1 andcˆ4:2,the amplitude of the oscillations

in the pressure coecient is similar to that shown in Fig. 2; these oscillations are periodic with a single period and their amplitude is smaller than that shown in Fig. 3.

Fig. 2. Pressure coecient (top left),phase diagram for the jet's mean radius at the convergence point (top right),gas concentration at the jet's inner interface (bottom left) and mass absorption rate at the jet's inner interface (bottom right). (Henry's solubility law;cˆ4;aˆ0:05;Sˆ0:50.)

The results presented in Figs. 2 and 3 are illustrative of the trends observed

forc<cH; those presented below illustrate the nonlinear dynamics forc>cH.

Figs. 4 and 5 correspond to cˆ6;aˆ0:01 and Sˆ0:50 and show that the

oscillations of the pressure coecient and jet's mean radius at the jet's con-vergence point are not longer periodic; they are quasiperiodic as indicated in

the phase diagrams and the Poincare sections; these sections were determined

from the condition that dR…L;t†=dt has the same value upon crossing

R…L;t† ˆconstant. Moreover,the power spectrum of the jet's mean radius at

the convergence point (Fig. 5) indicates the presence of several peaks associated with the frequency of excitation and its subharmonics; the power spectrum

shown in Fig. 5 corresponds to that of the di€erence R…t;L† Rav where Rav

denotes the average value ofR…L;t†. This average value and the phase diagrams

presented in all the ®gures of this paper were usually computed fromtˆ4500

to 5000 in order to remove any possible initial transients and obtain accurate results; in fact,the time window employed to determine the power spectra,

phase diagrams and Poincare sections was varied until the results were nearly

Fig. 3. Pressure coecient (top left),phase diagram for the jet's mean radius at the convergence point (top right),gas concentration at the jet's inner interface (bottom left) and mass absorp-tion rate at the jet's inner interface (bottom right). (Henry's solubility law; cˆ4:2;aˆ0:05; Sˆ0:25.)

independent of the size of the window. The phase diagram presented in Fig. 4 also indicates the presence of torii.

The gas concentration at the jet's inner interface shown in Fig. 5 is analo-gous to the pressure coecient since both are linearly related,whereas the mass absorption rate is of the same order of magnitude as that illustrated in Fig. 3 for a larger amplitude. The volume enclosed by the jet's inner interface also shown in Fig. 5 is oscillatory.

Forcˆ6;aˆ0:01 andSˆ1,the results presented in Fig. 7 indicate that,

beyond the Hopf bifurcation,an increase in the forcing frequency results in quasiperiodic oscillations whose amplitude decreases as the forcing frequency is increased. Moreover,the mass absorption rate is much smoother and has a

smaller amplitude forS ˆ1 than for Sˆ0:50. For still larger values of the

bifurcation parameter,e.g.,Sˆ10,it has been observed that the amplitude

of the oscillations increases as c is increased and the power spectrum is

broad,thus indicating the presence of chaotic motion. In view of the results presented in Figs. 4±7 and others not shown here,it may be stated that the

Fig. 4. Pressure coecient (top left),jet's mean radius at the convergence point (top left),phase diagram for the jet's mean radius at the convergence point (bottom right) and Poincare sections of the jet's mean radius at the convergence point (bottom right). (Henry's solubility law; cˆ6;aˆ0:01;Sˆ0:50.)

periodically forced Hopf bifurcation in annular liquid jets with mass transfer and Henry's solubility law undergoes a ®rst transition to quasiperiodic motion upon increasing the amplitude of the excitation for values of the bifurcation parameter greater than the one at which the Hopf bifurcation occurs in the absence of forcing. This transition is consistent with the ana-lytical results obtained by Gross [22] for a system of two ordinary di€erential equations by means of the method of averaging. Upon a further increase in the amplitude of the excitation,chaotic phenomena characterized by large amplitudes of oscillation and broad-band spectra appear. This second tran-sition was not observed by Gross [22]; instead,he observed a periodic motion at a frequency equal to that of the forcing. The discrepancies observed in the second transition between the results presented here and those of Gross [22] may be due to the larger number of degrees of freedom which result upon the discretization of the ¯uid dynamics and gas concentration equations of an-nular jets,and the two degrees of freedom employed by Gross [22]; they may also be due to the di€erent couplings between the equations. Note that the amplitudes of forcing considered here were chosen to be,at most,0.05,i.e.,

Fig. 5. Power spectrum of the jet's mean radius at the convergence point (top left),mass absorption rate at the jet's inner interface (top right),gas concentration at the jet's inner interface (bottom left) and volume of the gases enclosed by the annular liquid jet (bottom right). (Henry's solubility law; cˆ6;aˆ0:01;Sˆ0:50.)

small,in order to provide a qualitative comparison between the results pre-sented here and those of Gross [22].

For values of the bifurcation parameter smaller than the critical one,it has been observed that the dynamics of annular liquid jets with mass transfer and Henry's solubility law are periodic with a frequency equal to that of the forcing and that the amplitude of the oscillations increases as the amplitude of the forcing is increased.

Although not shown here,the numerical results obtained with the inviscid ¯uid dynamics equations presented in Section 3 are qualitatively and quan-titatively identical to those described above obtained with the high Reynolds number equations derived in Section 2. This is not surprising since,if the axial velocity component at the nozzle exit is assumed to be uniform,then the equations derived in Section 2 reduce to those presented in Section 3 for long and thin,annular liquid jets. Note that for the parameters considered in this

section ˆb

0=L0ˆ0:01 and b0=R0ˆ0:05,i.e.,the jet is thin and long,and

the liquid's axial velocity pro®le at the nozzle exit was assumed to be uniform.

Fig. 6. Pressure coecient (top left),jet's mean radius at the convergence point (top right),phase diagram for the jet's mean radius at the convergence point (bottom left) and Poincare sections of the jet's mean radius at the convergence point (bottom right). (Henry's solubility law; cˆ6;aˆ0:01;Sˆ1:0.)

7.2. Sievert's solubility law

For the study of periodically forced Hopf bifurcations in annular liquid jets with mass transfer and interfacial concentrations governed by Sievert's solu-bility law (nˆ1=2),the parameters Fr;We_;Pe;h0;b

0=R0;Cp;pi…0†=pe;a and b

were set to 10,10,106_{;0;0:05;1,0:75;1 and 1,respectively,and the inviscid}

¯ow model of Section 3 was employed. The phase diagrams and Poincare

sections presented in this section were obtained in the time window

47506t65000 and,therefore,they are not subjected to initial transients and

are truly representative of the nonlinear behavior of annular liquid jets. This

was also veri®ed by performing longer simulations (up totˆ10000) in some

cases.

As shown in Fig. 8,a Hopf bifurcation occurs foraˆ0 andcH 4:4,and

the initial (tˆ0 ) convergence length corresponding to these parameters was

Lˆ4:28. This Hopf bifurcation results in an oscillatory pressure coecient

whose amplitude is about 0.003,and periodic mass transfer rates and inter-facial concentrations. Although not shown here,the jet's thickness and axial

Fig. 7. Power spectrum of the jet's mean radius at the convergence point (top left),mass absorption rate at the jet's inner interface (top right),gas concentration at the jet's inner interface (bottom left) and volume of the gases enclosed by the annular liquid jet (bottom right). (Henry's solubility law; cˆ6;aˆ0:01;Sˆ1:0.)

velocity component at the convergence length,and the volume of the gases enclosed by the annular jet are also periodic functions; and,there is a phase lag between the jet's mean radius and thickness at the convergence point,and between the pressure coecient and the mass absorption rate. The phase di-agram for the jet's mean radius at the convergence point shown in Fig. 8 is a

single closed curve and,therefore,the Poincare section is a single point. Fig. 8

also shows that the largest mass absorption rate is about 0.003.

The results presented in Figs. 9±11 correspond to periodic forcing of the

Hopf bifurcation illustrated in Fig. 8,i.e.,cˆ4:4;aˆ0:01,and di€erent

excitation frequencies. For S ˆ0:25,the results presented in Fig. 9 indicate

that pressure coecient and jet's mean radius at the convergence point are no longer periodic functions; the same has been observed for the jet's axial ve-locity component and thickness at the convergence point. The phase diagram for R…L…t†;t† clearly exhibits the presence of many torii which indicate that

the motion is quasiperiodic,while the power spectrum for R…L…t†;t† Rav

exhibits several peaks at di€erent frequencies; the leftmost peak is associated with the forcing frequency and has a power of about 0.005,whereas the two

Fig. 8. Pressure coecient (top left),phase diagram for the jet's mean radius at the convergence point (top right),gas concentration at the jet's inner interface (bottom left) and mass absorption rate at the jet's inner interface (bottom right). (Sievert's solubility law;cˆ4:4;aˆ0:0.)

main peaks on the right have higher frequencies and powers of about 0.0003 and 0.004.

For S ˆ0:50,the results exhibited in Fig. 10 indicate that neither the

pressure coecient nor the jet's convergence length are periodic functions of time but their behaviour is more periodic than those of Fig. 9,and the am-plitude of the oscillations in the pressure coecient decreases as the forcing frequency is increased. The phase diagram illustrated in Fig. 10 also indicates that the jet's mean radius at the convergence point is quasiperiodic but the

ranges of bothR…L…t†;t†and its time derivative are smaller than those presented

in Fig. 9. Fig. 10 also indicates that the width and the number of frequency peaks of the spectrum decrease,whereas the largest power increases as the forcing frequency is increased. In fact,there are two main peaks in the power spectrum shown in Fig. 10; the one in the left corresponds to the forcing fre-quency and has a power of about 0.01,while the one on the right has a power of about 0.001 and a larger frequency than the corresponding peak on the right in Fig. 9. This indicates that the peak on the right is displaced towards higher frequencies and its magnitude decreases as the forcing frequency is increased.

Fig. 9. Pressure coecient (top left),jet's mean radius at the convergence point (top right),phase diagram for the jet's mean radius at the convergence point (bottom left) and power spectrum of the jet's mean radius at the convergence point (bottom right). (Sievert's solubility law; cˆ4:4;aˆ0:01;Sˆ0:25.)

For still larger forcing frequencies,i.e.,Sˆ1,Fig. 11 indicates that the nonlinear response of the annular liquid jet becomes more periodic as the frequency of excitation is increased. This is also re¯ected in both the phase diagram which becomes less space-®lling and tends towards a single closed curve,and the power spectrum which has less width and fewer peaks than that of Fig. 10; in fact,the largest peak is associated with the forcing frequency and has a power equal to about 0.009,while the smaller peaks,i.e.,those corre-sponding to higher frequencies,have powers smaller by,at least,a factor of ten than that of the largest peak.

The interfacial gas concentrations and mass absorption rates presented in Fig. 12 which corresponds to the dynamics illustrated in Figs. 9 and 10 indicate that the amplitude of the oscillations in these quantities ®rst increases as the forcing frequency is increased (compare Figs. 8 and 12) and then decreases. Fig. 12 also indicates that the interfacial gas concentration exhibits similar trends but it is not proportional to the pressure coecient. Furthermore,the results presented in Figs. 8 and 12 clearly indicate that,even a small forcing amplitude of one percent increases the peak mass transfer rate by more than a

Fig. 10. Pressure coecient (top left),jet's mean radius at the convergence point (top right),phase diagram for the jet's mean radius at the convergence point (bottom left) and power spectrum of the jet's mean radius at the convergence point (bottom right). (Sievert's solubility law; cˆ4:4;aˆ0:01;Sˆ0:50.)

factor of two. Although not shown here,the volume enclosed by the annular liquid jet also exhibits oscillations similar to those of Figs. 8±11,whereas the mean value and the amplitude of the oscillations in the mass transfer rate at the

jet's outer interface were found to be always smaller than 10 40_{; therefore,it}

may be stated that,for the conditions considered here,the mass transfer rate increases as the amplitude of the periodic forcing is increased,and there is nearly zero mass transfer rates at the jet's outer interface. Moreover,the mass transfer rates ®rst increase as the forcing frequency is increased and then de-crease.

For the values of the parameters indicated in Fig. 11,the gas concentration at the jet's inner interface and the mass absorption rate are more periodic than those shown in Fig. 12,in agreement with the results presented in Figs. 9±11.

Foraˆ0 andcˆ4:5,i.e.,for a bifurcation parameter larger than that at

which the Hopf bifurcation occurs in the absence of forcing,it has been ob-served that the jet response is periodic,and the amplitude of these periodic

oscillations increases ascis increased. Moreover,the peak mass transfer rate

was about 0.004. For cˆ4:5;aˆ0:01 and Sˆ0:25,the results presented in

Fig. 11. Pressure coecient (top left),jet's mean radius at the convergence point (top right),phase diagram for the jet's mean radius at the convergence point (bottom left) and power spectrum of the jet's mean radius at the convergence point (bottom right). (Sievert's solubility law; cˆ4:4;aˆ0:01;Sˆ1:00.)

Fig. 13 indicate that the response of the annular liquid jet is characterized by

four frequencies as illustrated in the phase diagram and Poincare section (not

shown here). Note that what appears to be a cusp point in this diagram is really a small loop. The four frequencies identi®ed by the number of closed loops in Fig. 13 can be observed in the power spectrum which shows that the four peaks of largest amplitude have powers equal to about 0.0002,0.012,0.0006 and 0.0004; the peak with largest power is associated with the forcing frequency and that with the smallest one is associated with the ``cusp'' points in the phase

diagram. Forcˆ4:5,aˆ0:01 andSˆ0:50,the results illustrated in Fig. 14

show a quasiperiodic motion with three frequencies and the presence of torii,a more periodic pressure coecient and a smaller area of the phase diagram than Fig. 13. Fig. 14 also shows that there are three main frequencies whose powers are about 0.007,0.00005 and 0.0007; the leftmost frequency is associated with the forcing.

The interfacial gas concentration and mass absorption rates presented in Fig. 15 indicate that the mass transfer rate increases as the amplitude of the forcing is increased. For the values of the parameters used in Fig. 15,the

Fig. 12. Gas concentration at the jet's inner interface (left) and mass absorption rate at the jet's inner interface (right). (Sievert's solubility law. Top: cˆ4:4;aˆ0:01;Sˆ0:25. Bottom: cˆ4:4;aˆ0:01;Sˆ0:50.)

largest mass absorption rates were about 0.008 and 0.006,respectively,i.e.,200 and 150 percent larger,respectively,than those in the absence of forcing.

Although not shown here,the oscillations in the volume enclosed by the annular liquid jet exhibit a similar behaviour to those of the mass absorption rates. Moreover,these oscillations exhibit a phase lag with respect to those of the pressure coecient and mass absorption rates. This lag is associated with the characteristic times of di€usion and forcing; note that the forcings em-ployed here have smaller time scales than those of di€usion and that,because of the assumption of equilibrium at the interfaces,the interfacial gas concen-tration is instantaneously adjusted to the instantaneous pressure of the gases enclosed by the annular liquid jet. However,this does not occur with the in-terfacial concentration gradients on account of the di€usion and convection of the liquid.

Forcˆ5 andaˆ0:01,it has been observed that the solution exhibits two

frequencies for Sˆ0:25,and quasiperiodic motion for S ˆ0:50 and S ˆ1.

Therefore,as the forcing frequency is increased,the periodic motion corre-sponding to the Hopf bifurcation ®rst generates a second frequency and then

Fig. 13. Pressure coecient (top left),jet's mean radius at the convergence point (top right),phase diagram for the jet's mean radius at the convergence point (bottom left) and power spectrum of the jet's mean radius at the convergence point (bottom right). (Sievert's solubility law; cˆ4:5;aˆ0:01;Sˆ0:25.)

becomes quasiperiodic. The area of the phase diagram of the quasiperiodic motion and the amplitude of the oscillations in the pressure coecient decrease as the forcing frequency is increased; for still larger forcing frequencies,the motion becomes again periodic and has a frequency smaller than that of the forcing.

A qualitative comparison between the results presented here and those of Gross [22] who considered a system of two ordinary di€erential equations in-dicates that,upon increasing the forcing amplitude,the motion becomes quasiperiodic but does not exhibit frequency locking. Moreover,the frequency contents of the nonlinear dynamics of annular liquid jets with mass transfer depends on the forcing frequency. The di€erences between the dynamics ob-served here and those of Gross [22] may be due to the integrodi€erential coupling of the model equations for annular liquid jets and the larger number of ordinary di€erential equations that result upon space discretization.

Comparisons between the results presented here and those above for Henry's solubility law indicate that the nonlinear dynamics of annular liquid jets with mass transfer is dynamically richer for Sievert's than for Henry's solubility law.

Fig. 14. Pressure coecient (top left),jet's mean radius at the convergence point (top right),phase diagram for the jet's mean radius at the convergence point (bottom left) and power spectrum of the jet's mean radius at the convergence point (bottom right). (Sievert's solubility law; cˆ4:5;aˆ0:01;Sˆ0:50.)

This behaviour was expected since the interfacial gas concentration depends linearly on the pressure and the square of the pressure for Henry's and Sievert's solubility laws,i.e.,Sievert's law introduces further nonlinearities.

The results presented in previous ®gures corresponded to periodically

per-turbed Hopf bifurcations forcPcH. We now consider the e€ects of periodic

forcing forc<cH,i.e.,for bifurcation parameters smaller than the critical one.

Figs. 16±18 correspond tocˆ4:2 and illustrate the e€ects of bothaandSon

the nonlinear dynamics of annular liquid jets with periodic forcing. Fig. 16

indicates that,foraˆ0:01 andS ˆ0:25,the pressure coecient,the interfacial

gas concentration and the mass absorption rate are periodic functions of time with a period equal to that of the excitation; the axial velocity component and the jet's thickness at the convergence point and the volume of the gases en-closed by the annular liquid jet are also periodic functions. The phase diagram presented in Fig. 16 is a single curve,while the power spectrum shows a single peak at the forcing frequency.

Fig. 17 shows that,for aˆ0:05 and Sˆ0:25,the pressure coecient is

not periodic but quasiperiodic as illustrated in both the phase diagram and

Fig. 15. Gas concentration at the jet's inner interface (left) and mass absorption rate at the jet's inner interface (right). (Sievert's solubility law. Top: cˆ4:5;aˆ0:01;Sˆ0:25. Bottom: cˆ4:5;aˆ0:01;Sˆ0:50.)