Linear stability analysis of a three-dimensional viscoelastic liquid jet surrounded by a swirling air stream

Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

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Linear stability analysis of a three-dimensional viscoelastic liquid jet surrounded by a swirling air stream Li-Jun Yang ⇑, Ming-Xi Tong, Qing-Fei Fu School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

a r t i c l e

i n f o

Article history: Received 6 July 2012 Received in revised form 7 September 2012 Accepted 10 October 2012 Available online 9 November 2012 Keywords: Linear analysis Instability Viscoelastic liquid jet Non-axisymmetric disturbances Air swirl

a b s t r a c t A theoretical model is established to investigate the instability of a viscoelastic liquid jet with axisymmetric and non-axisymmetric disturbances, which is moving in a swirling air stream. The dispersion relation is derived by a temporal linear stability analysis. Results show that the three-dimensional viscoelastic liquid jet is more unstable than its Newtonian counterpart when considering the air swirl. The effects of air swirl strength, jet velocity, surface tension, liquid viscosity and gas density on the instability of viscoelastic jet surrounded by a swirling gas are analogous with the example of the Newtonian jet. Note that air swirl is also a stabilizing factor on the instability of the viscoelastic jet. The axisymmetric mode can prevail over the non-axisymmetric when the swirl strength is strong enough, while the nonaxisymmetric mode is dominant in a liquid jet with a high liquid Weber number and a low liquid Reynolds number. It is also found that the maximum unstable growth rate of a viscoelastic liquid jet increases as liquid elasticity increases or the time constant ratio decreases. Furthermore, when air swirl velocity is introduced, it is the gas-to-liquid relative axial velocity that governs jet instability in axisymmetric disturbances, while the absolute gas axial velocity is another influencing factor in the non-axisymmetric mode. Finally, the competition between the gas rotating and axial velocities on jet instability is examined. In the case of a relatively smaller resultant gas velocity, the effect of the gas axial velocity can prevail over that of the air swirl when the gas velocity ratio is larger in the non-axisymmetric mode, whereas it can always exceed the effect of the air swirl in the axisymmetric mode. For a larger resultant gas velocity, the effect of the gas axial velocity is predominant only at a large gas velocity ratio in the axisymmetric mode. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction The liquid jet is widely used in various practical applications [1–3], such as in diesel engines, gas turbines, liquid fuel rocket engines and spray drying. Considerable research is focused on better methods for atomizing unstable liquid jets. The study of unstable liquid jets dates back to Rayleigh’s pioneering work [4,5]. He conducted a temporal linear stability analysis for an inviscid cylindrical liquid jet moving in vacuum. The results showed that only axisymmetric disturbances can grow and dominate the breakup of a jet, which unfortunately did not agree with the experiments [6–8], which indicated that in special situations non-axisymmetric disturbances can prevail over axisymmetric disturbances. Weber [9] implemented the study of a viscous liquid jet surrounded by an inviscid gas medium, which gave a detailed description of the instability of a Newtonian liquid jet. But Weber’s results were limited to axisymmetric disturbances. There have been many studies investigating the mechanisms of jet ⇑ Corresponding author. Tel./fax: +86 010 82339571. E-mail address: [email protected] (L.-J. Yang). 0377-0257/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnnfm.2012.10.011

instability since Weber’s work [10–18]. Although the effects of flow conditions, fluid properties and nozzle geometry on jet instability in the axisymmetric mode can be understood through these studies, these theories do not explain the observation of growing non-axisymmetric disturbances. In previous studies examining jet instability [12,19–21], it is accepted that the axisymmetric mode is always more dangerous than non-axisymmetric. However, a more general investigation of the non-axisymmetric mode may offer a satisfactory explanation for the results of the experiments previously mentioned [6–8]. To determine the physical mechanism of a non-axisymmetric mode, some researchers established different theoretical models to predict its growing process [22–29]. Liu and Liu [22,23] carried out a linear stability analysis for viscoelastic liquid jets when the ambient gas medium was assumed to be inviscid. For the parametric values considered in their paper, they found that the unstable growth rate of non-axisymmetric disturbances can exceed that of axisymmetric disturbances at high liquid Weber number. But their results were limited to the case of small wave number, and the effect of liquid viscosity on the transition between these two disturbances was neglected. Although Lin and Webb [24], Avital [25],

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Ibrahim [26], Yang [27] failed to discover the phenomenon observed by the experiments [6–8], Li [28] successfully proved that the non-axisymmetric mode becomes a dominant mode when the liquid Weber number is high. Ruo et al. [29] gave the boundaries between these two modes (the axisymmetric and non-axisymmetric modes), and executed the experimental conditions which can cause the predominance of non-axisymmetric disturbances clearly. It should be noted that all the models established in the above studies describe jet instability in the absence of air swirl. Liao et al. [30] examined the effect of air swirl on the instability of a threedimensional Newtonian liquid jet. Their results showed that the Newtonian jet becomes more stable by increasing the air swirl, whether the trend is the same for the viscoelastic jet is still unknown. Thus, there is a need to investigate the instability of viscoelastic liquid jet moving in a swirling air stream. In addition, it is interesting to explore the elastic – as well as other fluid parametric – effects on jet instability when air swirl is introduced. Note that the transition between the axisymmetric and non-axisymmetric modes is also discussed in this paper. Finally, the competition between the gas axial and rotating velocities on jet instability is researched. It is worth mentioning that the linear stability analysis can predict the initial growth of disturbances with small amplitudes, but it is unable to explore jet deformation due to the underlying assumption of infinitesimal perturbation magnitude. The viscoelastic jet generates breakup retardation and forms a structure of drops connected by thin ligaments with increasing deformation. Eventually, the jet breakup slows down and generates very large breakup lengths because of the strong nonlinear behavior existing in viscoelastic fluids [31]. In this sense, a nonlinear analysis may help to fully explain the breakup behavior of viscoelastic liquid jet, which will be studied in the future. Note that although the present work is confined to the linear scope, it can also give some insight into the unstable behavior of a viscoelastic jet at the initial stage of disturbance growth. Thus, it is hoped that the present study will contribute to further investigation of the instability of viscoelastic liquid jets. A three-dimensional viscoelastic liquid jet, which is subjected to a swirling gas medium, is considered in the present study. The dispersion relation is obtained using a linear stability analysis. Finally, the effects of flow conditions on the instability of liquid jet with and without air swirl are investigated by solving the dispersion relationship in the temporal mode.

s þ k1

  Ds Dc_ ; ¼ g0 c_ þ k2 Dt Dt

ð3Þ

where

c_ ¼ rv þ ðrv ÞT ; x_ ¼ rv  ðrv ÞT ; Ds @ s

1 _ ssx _ Þ; ¼ þ ðv  rÞs þ ðx 2 Dt @t Dc_ @ c_ 1 _ Þ; _  c_  c_  x ¼ þ ðv  rÞc_ þ ðx 2 Dt @t

ð4Þ ð5Þ ð6Þ ð7Þ

_ is the vorticity tensor, D/Dt is where c_ is the rate of strain tensor, x the co-rotational derivative, g0 is the zero shear viscosity, k1 is the stress relaxation time, k2 is the deformation retardation time. Similarly, the governing equations of incompressible inviscid gas are:

$  v^ ¼ 0;

ð8Þ

qg

ð9Þ



 @ ^ $ v ^ ¼ rpg ; þv @t

^ is the gas velocity vector, which can where qg is the gas density, v ^ x Þ, pg is the gas pressure. be expressed as ðv^ r ; v^ h ; v Because the jet surface is subjected to small disturbances after the liquid jet issues from the nozzle, the equation describing the surface of the liquid jet can be expressed as follows:

R ¼ a þ e expðixt þ inh þ ikxÞ;

ð10Þ

where a is the undisturbed jet radius, e is the initial amplitudes of disturbances, x is the amplification factor for a disturbance of wavelength k, n represents the disturbance mode, and k is the axial wave number of disturbances which can be expressed as 2p/k. It is worth mentioning that the disturbances are two-dimensional, and the deformation on the jet surface is axisymmetric when n = 0. The so-called axisymmetric mode (n = 0), is shown in Fig. 1a. When n = 1, as is shown in Fig. 1b, the disturbances are three-dimensional, and the jet surface becomes asymmetric in this situation. This mode is called the ‘non-axisymmetric mode’. The cross section and inter-

2. The relationship of derived dispersion Consider an incompressible infinitely long viscoelastic liquid jet moving in a coaxial swirling air stream. In terms of cylindrical coordinates (r, h, x), the r-axis is normal to the centerline of liquid jet, h-axis is in the azimuthal direction, and x-axis is along the moving direction of the liquid jet flow. By neglecting the gravity, the governing equations of liquid jet, which are the conservation laws of mass and momentum, can be expressed as follows:

$  v ¼ 0; 

@ þv $ q @t



v ¼ $p þ $  s;

ð1Þ ð2Þ

where q is the liquid density, t is the time, v is the liquid velocity vector, which can be expressed as (vr, vh, vx), p is the liquid pressure, and s is the stress tensor of liquid. The rheological equation of state relating the stress tensor to the velocity field is the Oldroyd B-constitutive equation [32–34], which is written in the objective reference frames as given below. It describes the fluid viscoelastic behavior:

Fig. 1. Schematic of a liquid jet with (a) axisymmetric and (b) non-axisymmetric disturbances.

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face of the liquid jet become more complicated in cases n > 1, which are not discussed here. To carry out a linear stability analysis for the liquid jet, the velocities, pressures, and stresses can be assumed to be equal to the initial steady state, plus the unsteady disturbances. Here, initial disturbances are assumed to be dependent only on the radial position r.

v ¼ ð0; 0; Ul Þ þ v 0 ¼ ð0; 0; U l Þ þ v 00 expðixt þ inh þ ikxÞ;

ð11Þ

where

v

0 0

¼ ðV; W; UÞ;

ð12Þ

  A v^ ¼ 0; ; U0 þ v^0 r   A ¼ 0; ; U 0 þ v^0 0 expðixt þ inh þ ikxÞ; r

¼ P þ p00 expðixt þ inh þ ikxÞ; pg ¼ Pg þ p0g ¼ Pg þ p0g0 expðixt þ inh þ ikxÞ;

s ¼ 0 þ s0 ¼ 0 þ s00 expðixt þ inh þ ikxÞ:

ð13Þ

  @v h 1 @v r v h ¼ 0; þ  r @h @r r

ð25Þ

r ¼ a;

! 1 1 @2 @2 A2 @v r p  pg ¼ c 2 þ 2 2 þ 2 R þ qg 3 R þ 2gðxÞ ; a a @h @x a @r

ð26Þ

r ¼ a; ð27Þ

where c is the coefficient of surface tension, A is the vortex strength. Substituting Eqs. (11), (15), (17) into Eqs. (1) and (2), the following linearized governing equations are obtained:



@ @ þ Ul  q @t @x

ð28Þ



v 0 ¼ $p0 þ $  s0 :

ð29Þ

Taking divergence of Eq. (29) and making use of Eqs. (4), (11), (21), (23) and (28) gives

ð14Þ

$2 p0 ¼ 0:

ð15Þ

p00

ð16Þ

ð30Þ

 @ @ R; r ¼ a; þ Ul @t @x   @ A @ @ R; r ¼ a: þ þ U0 v^ r ¼ @t r @h @x

ð17Þ



ð18Þ ð19Þ

Furthermore, the small disturbances have no effect on the gas phase which is far from the liquid jet. Thus,

v^ r ¼ 0; v^ h ¼

A ; r

v^ x ¼ U0 ;

r ! þ1:

ð20Þ

ð21Þ

where the effective viscosity g(x) can be expressed as:

1 þ ðix þ ikU l Þk2 : 1 þ ðix þ ikU l Þk1

ð22Þ

A1 ¼

ðix þ ikU l Þa s1 e; In ðkaÞ s2 "

ð34Þ

ð35Þ

2

3

kla I0n ðkaÞðix þ ikU l Þ k a2 I0n ðkaÞIn1 ðlaÞðix þ ikU l Þ e e 0 2nIn ðkaÞIn ðlaÞ 2nlIn ðlaÞIn ðkaÞIn ðlaÞ # 3 2 k aI0n ðkaÞðix þ ikU l Þ s1 k aIn1 ðlaÞðix þ ikU l Þ e þ e þ 0 2 s2 2nlIn ðlaÞIn ðlaÞ l I0n ðlaÞIn ðkaÞ

A2 ¼ 

2

laðix þ ikU l Þ k ðix þ ikU l Þ e e; 2 2nIn ðlaÞ l I0n ðlaÞ

ð36Þ

" 2 # 3 1 kla I0n ðkaÞðix þ ikU l Þ k a2 I0n ðkaÞIn1 ðlaÞðix þ ikU l Þ s1 e e A3 ¼ 0 2 nIn ðkaÞIn ðlaÞ s2 nlIn ðlaÞIn ðkaÞIn ðlaÞ 2



k aIn1 ðlaÞðix þ ikU l Þ laðix þ ikU l Þ e e; 0 2nIn ðlaÞ 2nlIn ðlaÞIn ðlaÞ ð37Þ 2

ð23Þ

ðix þ ikU l Þik ðix þ ikU l Þik aI0n ðkaÞ s1 e e; 0 0 s2 lIn ðlaÞ lIn ðlaÞIn ðkaÞ

ð38Þ

where

0

x_ ¼ 0 þ x_ _ 00 expðixt þ inh þ ikxÞ: ¼0þx

ð33Þ

where l2 = k2 + q(ix + ikUl)/g(x). Constants A1, A2, A3 and A4 are determined by utilizing Eqs. (18), (25), (26) and (28):

A4 ¼ 

¼ 0 þ c_ 00 expðixt þ inh þ ikxÞ;

ð32Þ

in W ¼ iA2 In1 ðlrÞ  iA3 Inþ1 ðlrÞ þ A1 In ðkrÞ; r U ¼ A4 In ðlrÞ þ ikA1 In ðkrÞ;

Here the undisturbed rate-of-strain tensor and the undisturbed vorticity tensor are both zero because the basic flow is plug flow, thus:

c_ ¼ 0 þ c_ 0

ð31Þ

where In(kr) is the nth-order Bessel function of the first kind. After substituting Eq. (31) into Eq. (29), the initial disturbed jet velocities for the three directions can be obtained:

þ

Note that Eq. (3) is linearized as below:

s0 ¼ gðxÞ  c_ 0 ;

¼ qðix þ ikU l ÞA1 In ðkrÞ;

V ¼ A2 In1 ðlrÞ þ A3 Inþ1 ðlrÞ þ kA1 I0n ðkrÞ;

The boundary conditions must satisfy the kinematic and dynamic conditions at the jet surface. When the interface of liquid jet is a material surface, the kinematic boundary conditions require the following:

gðxÞ ¼ g0

srh ¼ gðxÞ

r ¼ a;

Note that Eq. (30) has a specific solution as follows:

p ¼ P þ p0

vr ¼

srx

$  v 0 ¼ 0;

where

c ; UÞ; b v^0 0 ¼ ð Vb ; W

  @v x @v r ¼ gðxÞ þ ¼ 0; @r @x

ð24Þ

The tangential forces must be vanishing, and the difference between normal forces must be balanced by surface tension at liquid interface; the dynamic condition can be expressed as follows:

" # 2 i 1 k In1 ðlaÞ h 0 ðlaÞ2 I00n ðlaÞ  laIn ðlaÞ s1 ¼ n  1þ 2 0 In ðlaÞ l In ðlaÞ 2

þ

2 k n  2 l I0n ðlaÞ

 0 laIn1 ðlaÞ  In1 ðlaÞ ;

ð39Þ

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L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

s2 ¼ 2n2  þ

 0  3 kaIn ðkaÞ n2 k aI0n ðkaÞIn1 ðlaÞ 1  2 In ðkaÞ l In ðkaÞI0n ðlaÞ

3 nk aI0n ðkaÞ  2 l I0n ðlaÞIn ðkaÞ 0 kaIn ðkaÞ

In ðkaÞ

where

" # i 1 K 2 In1 ðLÞ h 2 00 S1 ¼ n  ðLÞ In ðLÞ  LI0n ðLÞ 1þ 2 0 In ðLÞ L In ðLÞ 2

 0 laIn1 ðlaÞ  In1 ðlaÞ

" 2 k In1 ðlaÞ 2

l I0n ðlaÞ

#

1

00 2 I n ðlaÞ

ðlaÞ

In ðlaÞ

 la

I0n ðlaÞ In ðlaÞ

þ



þ n2 :

ð40Þ

Substituting Eqs. (13) and (16) into the governing equations of ambient gas Eqs. (8) and (9), the linearized forms are expressed as follows:

$  v^ 0 ¼ 0;

ð41Þ

qg ix þ

ð42Þ



 inA b ¼ ikp0 ; U þ ikU 0 g0 r2 " #  0 c dp inA 2A W b qg ix þ 2 þ ikU 0 V  2 ¼  g0 ; r r dr   inA c ¼  in p0 : qg ix þ 2 þ ikU 0 W r r g0

ð43Þ ð44Þ

The initial disturbed pressure of gas can be obtained from the above functions by using the kinematic boundary conditions for gas phase Eqs. (19) and (20):

p0g0 ¼

qg k



x

An  kU 0 a2



x

 An K n ðkrÞ  kU e; 0 r2 K 0n ðkaÞ

ð45Þ

where Kn(kr) is the nth-order Bessel function of the second kind. Finally, the dispersion equation relating the amplification factor x to the wave number k is obtained when substituting Eqs. (10), (31), (32) and (45) into the dynamic boundary condition Eq. (27). After considerable manipulation to reduce the function to a concise form, the following form can be obtained:

 2 qg s An K n ðkaÞ ix þ i 2 þ ikU 0 a s2 k K 0n ðkaÞ 2g 1 þ ðix þ ikU l Þk2 þ 0 ðix þ ikU l Þ a 1 þ ðix þ ikU l Þk1 ( 2  0  2 0 k a2 I00n ðkaÞ s1 k aI0n1 ðlaÞ kaIn ðkaÞ s1 laI ðlaÞ  In ðlaÞ  þ 1  n  0 In ðkaÞ s2 In ðkaÞ s2 In ðlaÞ lIn ðlaÞ " 0 #) 3 2 0 kaIn ðkaÞ s1 k aIn1 ðlaÞIn ðkaÞ s1 k In1 ðlaÞ 1  20  20  In ðkaÞ s2 l In ðlaÞIn ðkaÞ s2 l In ðlaÞ   1 n2 A2 2 ð46Þ ¼ c 2  2  k  qg 3 : a a a

qaðix þ ikU l Þ2 1 

Note that Eq. (46) can be reduced to proper forms that agree with previous work [5,9,12,22,23,28,30] by setting specific parameters to zero. In this sense, the dispersion equation displayed above is a general one. 3. Results and discussion It is more convenient to use the dimensionless form of the dispersion relation to analyze the instability of liquid jet, which can be written as below:

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 S1 q pffiffiffiffiffiffiffiffiffi 2 K n ðKÞ  X þ K Wel  X  n Wes  KU Wel S2 K K 0n ðKÞ  pffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffi Z þ i X þ K Wel kEl  pffiffiffiffiffiffiffiffiffi  2Z iX þ iK Wel Z þ i X þ K Wel El (   K 2 I00n ðKÞ S1 K 2 I0n1 ðLÞ KI0n ðKÞ S1 LI0 ðLÞ  In ðLÞ  þ 1  n  0 In ðKÞ S2 In ðKÞ S2 In ðLÞ LIn ðLÞ " #) 0 3 0 2 KI ðKÞ S1 K In1 ðLÞIn ðKÞ S1 K In1 ðLÞ  20  20  n 1 In ðKÞ S2 L In ðLÞIn ðKÞ S2 L In ðLÞ

 Wes ; ð47Þ ¼  1  n2  K 2 þ q

K2n  L2 I0n ðLÞ

S2 ¼ 2n2 

 LI0n1 ðLÞ  In1 ðLÞ ;

ð48Þ

 0  KIn ðKÞ n2 K 3 I0n ðKÞIn1 ðLÞ 1  In ðKÞ L2 In ðKÞI0n ðLÞ

nK 3 I0n ðKÞ  0 LIn1 ðLÞ 2 0 L In ðLÞIn ðKÞ

  In1 ðLÞ

" #  KI0n ðKÞ K 2 In1 ðLÞ I00 ðLÞ I0 ðLÞ þ L n þ n2 ;  1 L2 n 2 0 In ðKÞ In ðLÞ In ðLÞ L In ðLÞ

ð49Þ

where K = ka is the dimensionless wave number, L = la. Wel ¼ qU 2l a=c is the liquid Weber number, which expresses the ratio of inertial forces to surface tension forces. Wes = qA2/(c a) is the gas swirl Weber number, which represents the ratio of centrifugal forces to surface tension forces. Z = g0/( qca)0.5 is the Ohnesorge number, which denotes the ratio of viscous forces to surface tension forces. The elasticity number El = k1g0/(qa2) is used to describe the relationship between viscous and elastic effects in the liquid jet. q ¼ qg =q is the ratio of gas-to-liquid density, k ¼ k2 =k1 is the ratio of deformation retardation time to stress relaxation time, U ¼ U 0 =U l is the ratio of gas-to-liquid axial velocity. X = x( qa3/c)0.5 is the dimensionless amplification factor. A temporal linear stability analysis is carried out to examine the effects of flow parameters and fluid properties on jet instability. The parameter set studied in the present paper consists of the liquid Weber number Wel, gas swirl Weber number Wes, elasticity number El, Ohnesorge number Z, time constant ratio  k, gas-to li , gas-to-liquid axial velocity ratio U and the raquid density ratio q tio of gas rotating velocity to gas axial velocity K r ¼ ðA=aÞ=U 0 ¼ pffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffi Wes = U Wel . The degree of jet instability is represented by the positive imaginary part of the dimensionless amplification factor Xi, called the ‘unstable growth rate’. It can be written as follows:

Xi ¼ i  ðX  Xr Þ:

ð50Þ

A liquid jet with a larger maximum unstable growth rate Xi,max can behave with greater instability and break up more easily. Here, the maximum unstable growth rate is the growth rate which corresponds to the peak of the growth rate curve. It is noted that the dimensionless amplification factor X is derived by means of Muller [35]. The results for two kinds of unstable disturbances, which are the axisymmetric and non-axisymmetric modes, are solved respectively. In order to investigate the real property of viscoelastic fluids, ‘Boger fluids’ were selected for this study; the shear viscosity is constant, and the elastic effects are easily distinguished from viscous effects in Boger fluids. Under small shear rates, the rheological equation of Boger fluids can be characterized by the Oldroyd-B model, which makes them applicable for the analysis [34,36]. A specific variety of Boger fluids (PIB Boger fluid 4000 ppm), is used as an example in the parametric study here. The physical properties of this fluid, measured by Carroll and Joo [37], are listed in Table 1. In addition, the jet radius a is fixed at 103 m, while the jet velocity Ul ranges from 1 to 10 m/s in the present study. Table 1 Physical properties of PIB Boger fluid 4000 ppm.

q (kg m3)

g0 (Pa s)

k1 (s)

k2 (s)

c (N m1)

850

0.102

3.11  103

1.464  103

0.0146

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L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

A Newtonian liquid jet can behave with greater stability than a viscoelastic jet when considering only axisymmetric disturbances in a previous study [12]. In this sense, the maximum unstable growth rate Xi,max of viscoelastic jet is larger than its Newtonian counterpart. The competition between the unstable growth rates of these two fluids is investigated for the axisymmetric case plotted in Fig. 2a, and the non-axisymmetric case plotted in Fig. 2b respectively. The parameters are fixed at Wel ¼ 2500; Z ¼  ¼ 0:001; U ¼ 0 with El ¼ 0:3732;  0:9156; q k ¼ 0:471 for a viscoelastic liquid jet, and El ¼ 0;  k ¼ 0 for a Newtonian liquid jet. It is obvious that the trends are analogous for the axisymmetric and non-axisymmetric cases, whether the gas stream swirls or not (a viscoelastic liquid jet is always more unstable than a Newtonian jet). The result is not surprising because the viscoelastic liquid jet does not have the elastic constraint for deformation when compared to a rigid Newtonian jet. It is further noted that the unstable growth rates of non-axisymmetric disturbances may not grow when the wave number is zero in the case of air swirl, with the parametric values considered here. It is also found that the cutoff wave numbers of these two different fluids are almost the same for a uniform set of parameters. Note that the cutoff wave number is the wave number at which the unstable growth rate remains at zero. Those disturbances between the lower cutoff number and upper cutoff number are unstable. Similar results were reported by Brenn et al., Liu and Liu [12,22,23], who considered the viscoelastic liquid jet with axisymmetric and non-axisymmetric disturbances moving in an air stream without swirl.

a

0.35

Newtonian jet, We s=0

3.1. Effect of gas swirl Weber number It can also be seen from Fig. 2 that the effect of air swirl on jet instability is notable. Therefore, it is necessary to examine its effect on the unstable growth rates of Newtonian and viscoelastic liquid jets, which are plotted in Fig. 3a and b respectively. The parameters ¼ held constant for these plots are Wel ¼ 2500; Z ¼ 0:9156; q 0:001; U ¼ 0, with El ¼ 0;  k ¼ 0 for a Newtonian liquid jet in Fig. 3a, and El ¼ 0:3732;  k ¼ 0:471 for a viscoelastic liquid jet in Fig. 3b. It is evident that the unstable growth rates decrease as the swirl strength increases for all the cases. The unstable growth rate of non-axisymmetric disturbances is reduced more rapidly than that of axisymmetric disturbances, which leads to the domination of axisymmetric disturbances when swirl strength is notable during the jet breakup process. The resulting enhancement of jet stability by air swirl has also been obtained by previous researchers [30,38], which can be explained by the competition between the air static pressure and the movement of liquid surface. Air static pressure increases in the positive radial orientation of rotating air, due to the presence of the gas pressure gradient. Then the disturbed liquid surface will be pushed back to the undisturbed position by the static pressure in the air medium, which eventually causes the liquid jet to stabilize. Note that the dominant wave number is the wave number at which the liquid jet has maximum unstable growth rate; it decreases as the gas swirl Weber number increases for axisymmetric disturbances, while it increases as the gas swirl Weber number increases for non-axisymmetric disturbances.

a

0.5

Newtonian jet, We s=30

0.30

Viscoelastic jet, We s=0

0.4

Viscoelastic jet, We s=30

0.25

0.3

We s=0, n=1 We s=10, n=1 We s=30, n=1

We s=1000, n=0 We s=1500, n=0

We s=50, n=1

Ωi

Ωi

0.20

We s=0, n=0 We s=50, n=0 We s=500, n=0

0.15

0.2

0.10 0.1 0.05 0.00 0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.0

3.0

0.5

1.0

2.0

2.5

3.0

K

K

b

1.5

0.5

Viscoelastic jet, We s=0

0.4

Viscoelastic jet, We s=30

b

We s=0, n=0 We s=50, n=0 We s=500, n=0 We s=1000, n=0 We s=1500, n=0

0.5

Newtonian jet, We s=0 0.4

Newtonian jet, We s=30

We s=0, n=1 We s=10, n=1 We s=30, n=1 We s=50, n=1

0.3

Ωi

Ωi

0.3

0.2

0.2

0.1

0.1

0.0 0.0

0.5

1.0

1.5

2.0

2.5

K Fig. 2. The unstable growth rates Xi of Newtonian (El = 0,  k ¼ 0) and viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets with (a) axisymmetric and (b) non-axisymmetric  ¼ 0:001; U ¼ 0 disturbances versus the wave number K at Wel = 2500, Z = 0.9156, q for Wes = 0 (solid lines) and Wes = 30 (dash lines).

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

K Fig. 3. Effect of the gas swirl Weber number Wes on the unstable growth rate Xi  ¼ 0:001; U ¼ 0 for (a) the versus the wave number K at Wel = 2500, Z = 0.9156, q Newtonian (El = 0,  k ¼ 0) and (b) viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets with axisymmetric (solid lines) and non-axisymmetric (dash lines) disturbances.

6

L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

3.2. Effect of liquid Weber number According to classical linear stability analysis [12,19–21], the axisymmetric mode is always the dominant mode, which is obviously in disagreement with the appearance of the non-axisymmetric mode observed in the experiments [6–8]. The main reason that these researchers failed to predict the predominance of non-axisymmetric disturbances may be the low liquid Weber number, or the high liquid Reynolds number, chosen in their studies [29]. Note that Liao et al. [30] neglected the viscous effects on the competition between the axisymmetric and non-axisymmetric modes. In fact, different combinations of the liquid Weber number Wel and Ohnesorge number Z can cause the transition between these two disturbance modes. The relationship between these two dimensionless numbers and the liquid Reynolds number Re can be written as below:

Re ¼

pffiffiffiffiffiffiffiffiffi Wel : Z

ð51Þ

The effect of jet velocity on the jet instability is plotted in Fig. 4.  ¼ 0:001; U ¼ The parameters are chosen as Z ¼ 0:9156; q 0; Wes ¼ 0 with El ¼ 0;  k ¼ 0 for a Newtonian liquid jet, and El ¼ 0:3732;  k ¼ 0:471 for a viscoelastic liquid jet respectively. For the liquid jet with a low liquid Reynolds number which is the order of 10, the maximum unstable growth rate Xi,max, increases notably as the jet velocity increases. The physical mechanism for the trend displayed above can be revealed as follows: It is the aerodynamic interaction between the liquid and gas medium that basically causes the liquid jet to be more unstable, which can be enhanced by increasing the jet velocity. It can be seen from Fig. 4 that the aerodynamic effect is a significant destabilizing factor on jet instability. It is also found that non-axisymmetric disturbances cannot grow in the Rayleigh regime (Wel < 1000). To further increase the liquid Weber number, the maximum unstable growth rate of non-axisymmetric mode generates and exceeds its axisymmetric counterpart in the Taylor regime (Wel > 1000). This result provides a good explanation for the phenomenon observed by the experiments [6–8]. For a comprehensive study, the case when the liquid Reynolds number is above 104 is also considered. The fixed parameters in Fig. 5 are the same as these in Fig. 4, except for Z = 0.001. It can be seen from Fig. 5 that axisymmetric disturbances are always predominant for both fluids. Thus, it can be concluded from Figs. 4 and 5 that the non-axisymmetric mode can prevail over the axisym-

1.2

metric mode in the Taylor regime when the liquid Reynolds number is relatively lower. Note that the interfacial shear force is stronger at a lower liquid Reynolds number; it distributes asymmetrically on the interface of liquid and gas, generating a bending moment to twist the liquid jet due to the geometric asymmetry of non-axisymmetric mode. Then the non-axisymmetric mode is maintained strongly by the bending moment and dominates the jet breakup in this situation. Furthermore, it is found that under the same conditions, the curves standing for the maximum unstable growth rates of Newtonian and viscoelastic liquid jets respectively are almost coincident, though the viscoelastic liquid jet is still more unstable than its Newtonian counterpart. When the strong air swirl is introduced, as discussed in Section 3.1, the trends displayed above will be changed. Fig. 6 shows the effect of the air swirl on the transition between the axisymmetric and non-axisymmetric modes. The parameters held constant for  ¼ 0:001; U ¼ 0; Wes ¼ 200 with these plots are Z ¼ 0:9156; q El ¼ 0;  k ¼ 0 for a Newtonian liquid jet in Fig. 6a, and El ¼ 0:3732;  k ¼ 0:471 for a viscoelastic liquid jet in Fig. 6b. Subjected to air swirl with strong swirl strength, the jet is dominated by axisymmetric disturbances, since the air swirl damps the nonaxisymmetric mode more severely. Note that, with the parameters selected here, non-axisymmetric disturbances cannot develop when the liquid Weber number is below 2500. Furthermore, the unstable growth rates and unstable ranges for these two modes are increased notably by the liquid Weber number. In this sense, the Newtonian and viscoelastic liquid jets will break up more easily by increasing the jet velocity. In order to examine the effect of surface tension on the unstable growth rates for axisymmetric and non-axisymmetric disturbances, we redefine the dimensionless amplification factor as follows:

X0 ¼

xa Ul

which causes it to omit the coefficient of surface tension c. When the liquid Reynolds number is fixed at 83.33, the other parameters pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi  ¼ 0:001; U ¼ 0 with are chosen as Z ¼ Wel =Re ¼ Wel =83:33; q El ¼ 0;  k ¼ 0 for a Newtonian liquid jet in Fig. 7a, and El ¼ 0:3732;  k ¼ 0:471 for a viscoelastic liquid jet in Fig. 7b. It is obvious that the maximum unstable growth rate X0i;max for axisymmetric disturbances first decreases, then increases as the surface tension decreases, while that for non-axisymmetric disturbances always increases as the surface tension decreases. It implies that, for the axi-

4

Newtonian jet, n= 0 Newtonian jet, n=1 Viscoelastic jet, n= 0 Viscoelastic jet, n=1

1.0

ð52Þ

;

Newtonian jet, n= 0 Newtonian jet, n=1 Viscoelastic jet, n= 0 Viscoelastic jet, n=1

3

Ωi,max

Ωi,max

0.8 0.6

2

0.4 1 0.2 0.0

0 0

1000

2000

3000

4000

5000

We l Fig. 4. Effect of the liquid Weber number Wel on the maximum unstable growth  ¼ 0:001; U ¼ 0, Wes = 0 for the Newtonian (El = 0,  rate Xi,max at Z = 0.9156, q k ¼ 0) and viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets with axisymmetric (solid lines) and non-axisymmetric (dash lines) disturbances.

0

1000

2000

3000

4000

5000

We l Fig. 5. Effect of the liquid Weber number Wel on the maximum unstable growth  ¼ 0:001; U ¼ 0, Wes = 0 for the Newtonian (El = 0,  rate Xi,max at Z = 0.001, q k ¼ 0) and viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets with axisymmetric (solid lines) and non-axisymmetric (dash lines) disturbances.

7

L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

a

0.8

a 0.020

We l=3500, n= 0

n= 0 , We s= 0 n=1, We s= 0

We l=4500, n= 0 We l=5000, n= 0

0.6

0.4

We l=5000, n=1

2

2

2

0.012

0.008

0.2

0.004

0.0 0

1

2

3

4

5

0

6

1000

2000

3000

4000

5000

4000

5000

We l

K

b

2

2

n=1, We s=A We l /a Ul

Ω 'i,max

Ωi

We l=4500, n=1

2

n=0, We s=A We l /a Ul

0.016

We l=3500, n=1

1.0

b

We l=3500, n= 0 We l=4500, n= 0 We =5000, n= 0

0.8

l

0.020

n=0, We s= 0 n=1, We s= 0

We l=3500, n=1

2

2

2

2

2

2

n=0, We s=A We l /a Ul

0.016

n=1, We s=A We l /a Ul

We l=4500, n=1

0.6

Ω 'i,max

Ωi

We l=5000, n=1 0.4

0.012

0.008 0.2

0.004

0.0 0

1

2

3

4

5

6

0

1000

2000

3.3. Effect of the elasticity number The ultimate difference between Newtonian and viscoelastic fluids may be the notable elastic effect generated in the latter. In terms of mathematics, the rheological equation Eq. (3) in Section 2, includes two elastic parameters, k1 and k2, which makes it different from that of a Newtonian jet. Fig. 8 illustrates the effective liquid viscosity g(x), versus different values of K and Xi,max, with other parameters fixed at Wel ¼ 2500; Z ¼ 0:9156; El ¼ 0:3732;   ¼ 0:001; U ¼ 0; Wes ¼ 0. Due to the presence of the k ¼ 0:471; q two elastic parameters displayed above, the effective liquid viscosity of a viscoelastic fluid is not a constant value when the wave number and maximum unstable growth rate vary, which may make the unstable behavior of a viscoelastic fluid differ from that of a Newtonian fluid. As a result, it is necessary to seek the elastic effects on the jet instability. The deformation retardation time, k2, will be examined in Section 3.5, while the stress relaxation time k1 is discussed as below.

0.102

-1

symmetric mode, surface tension is a destabilizing factor when the liquid Weber number is relatively lower, which can magnify the disturbances on the interface between the liquid and gas. However, the source of jet instability is the aerodynamic interaction that exists at the surface of a liquid jet with a higher liquid Weber number. Because the non-axisymmetric mode cannot grow at low liquid Weber number, the enhanced aerodynamic interaction is a dominant factor and makes the jet always become more unstable in this situation.

Fig. 7. Effect of the surface tension on the maximum unstable growth rate X0i;max at pffiffiffiffiffiffiffiffiffi  ¼ 0:001; U ¼ 0 for (a) the Newtonian (El = 0,  Z ¼ Wel =83:33, q k ¼ 0) and (b) viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric (solid

lines) and non-axisymmetric (dash lines) disturbances with Wes ¼ A2 Wel = a2 U 2l and without (Wes = 0) the air swirl.

η (ω) / N • m

Fig. 6. Effect of the liquid Weber number Wel on the unstable growth rate Xi versus  ¼ 0:001; U ¼ 0, Wes = 200 for (a) the Newtothe wave number K at Z = 0.9156, q nian (El = 0,  k ¼ 0) and (b) viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets with axisymmetric (open dots) and non-axisymmetric (solid dots) disturbances.

3000

We l

K

0.100

0.098

0.096

0.094

0.0

0.5

1.0

K

1.5

2.0

2.5

0.32 0.25 0 6 0.19 2 0.12 8 ax 0.06 Ω i, m 4 0.00 0

Fig. 8. The effective liquid viscosity g(x) versus the wave number K and maximum unstable growth rate Xi,max at Wel = 2500, Z = 0.9156, El = 0.3732,  k ¼ 0:471,  ¼ 0:001; U ¼ 0, Wes = 0. q

The corresponding results for the cases with (Wes = 30) and without (Wes = 0) air swirl are shown in Fig. 9. The other parame ¼ 0:001; ters are fixed at Wel ¼ 2500; Z ¼ 0:9156;  k ¼ 0:471; q U ¼ 0. It can be seen that the jet instability can be enhanced by

8

L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

0.55 0.50

We s= 0, n= 0

Ω i,max

0.45

We s= 0, n=1 We s=30, n= 0

0.40

We s=30, n=1

In addition, the response of the unstable growth rate to the rise of elasticity in the axisymmetric mode is more notable than the response in the non-axisymmetric mode. Although not plotted, it should be noted that the unstable range of a viscoelastic liquid jet is unchangeable when the elasticity varies; this implies that the lower and upper cutoff wave numbers are irrelevant to the value of liquid elasticity.

0.35

3.4. Effect of the Ohnesorge number 0.30 0.25 0.20 0.0

0.2

0.4

0.6

0.8

1.0

El Fig. 9. Effect of the elasticity number El on the maximum unstable growth rate  ¼ 0:001; U ¼ 0 for the viscoelastic Xi,max at Wel = 2500, Z = 0.9156, k ¼ 0:471, q liquid jet subject to axisymmetric (solid lines) and non-axisymmetric (dash lines) disturbances with (Wes = 30) and without (Wes = 0) the air swirl.

increasing the liquid elasticity, due to the stress relaxation. According to Eq. (3), the increasing stress relaxation time k1 decreases the stress tensor of viscoelastic liquid jet, upon which the energy dissipation is dependent. Thus, the viscoelastic jet with a more durable stress relaxation time becomes more unstable, and can break up by ultimately dissipating less energy.

a

1.4

Newtonian jet, n= 0 Newtonian jet, n=1 Viscoelastic jet, n= 0 Viscoelastic jet, n=1

1.2

Section 3.2 investigates the competition between the axisymmetric and non-axisymmetric modes, which can be governed by viscous effects in the Taylor regime. The non-axisymmetric mode can dominate the process of jet breakup at a small liquid Reynolds number and a high liquid Weber number. Another significant question – how the Newtonian and viscoelastic liquid jets respond to the variation of liquid viscosity – is discussed in this section. The cases Wes = 0 and Wes = 30 are plotted in Fig. 10a and b, respec¼ tively. The other parameters are fixed at Wel ¼ 2500; q 0:001; U ¼ 0 with El ¼ 0;  k ¼ 0 for a Newtonian liquid jet, and El ¼ 0:3732;  k ¼ 0:471 for a viscoelastic liquid jet. It can be clearly seen from Fig. 10 that the maximum unstable growth rates move to smaller values as the Ohnesorge number increases in all the situations. It should be noted that increasing the liquid viscosity leads to an increase in stress tensor, which indicates that liquid jet with a high viscosity is difficult to break up into droplets since it will dissipate more energy than a low viscosity jet. Furthermore, liquid viscosity more notably damps the unstable growth rate for the axisymmetric mode. Although not plotted, it is ⎯λ=0.1, n=0 ⎯λ=0.3, n=0 ⎯λ=0.5, n=0 ⎯λ=0.7, n=0 ⎯λ=0.9, n=0

a 0.6 0.5 0.4

0.8

Ωi

Ωi,max

1.0

⎯λ=0.1, n=1 ⎯λ=0.3, n=1 ⎯λ=0.5, n=1 ⎯λ=0.7, n=1 ⎯λ=0.9, n=1

0.3

0.6 0.2 0.4 0.1 0.2 0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.0

Z

1.0

1.5

2.0

2.5

3.0

K

b 1.4

Newtonian jet, n= 0 Newtonian jet, n=1 Viscoelastic jet, n= 0 Viscoelastic jet, n=1

1.2

b

1.0

⎯λ=0.1, n=0 ⎯λ =0.3, n=0 ⎯λ =0.5, n=0 ⎯λ =0.7, n=0 ⎯λ =0.9, n=0

0.35

0.28

0.8

⎯λ =0.1, n=1 ⎯λ =0.3, n=1 ⎯λ =0.5, n=1 ⎯λ =0.7, n=1 ⎯λ =0.9, n=1

0.21

Ωi

Ωi,max

0.5

0.6

0.14 0.4 0.07

0.2 0.0

0.2

0.4

0.6

0.8

1.0

Z

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

K Fig. 10. Effect of the Ohnesorge number Z on the maximum unstable growth rate  ¼ 0:001; U ¼ 0 for the Newtonian (El = 0,  Xi,max at Wel = 2500, q k ¼ 0) and viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric (solid lines) and non-axisymmetric (dash lines) disturbances (a) without (Wes = 0) and (b) with (Wes = 30) the air swirl.

Fig. 11. Effect of the time constant ratio  k on the unstable growth rate Xi versus the  ¼ 0:001; U ¼ 0 for the wave number K at Wel = 2500, Z = 0.9156, El = 0.3732, q viscoelastic liquid jet subject to axisymmetric (solid lines) and non-axisymmetric (dash lines) disturbances (a) without (Wes = 0) and (b) with (Wes = 30) the air swirl.

9

L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

also noted that the unstable range cannot be changed by the variation of liquid viscosity. Similar results can be found in previous studies [23,28,30].

25

Stable range for We s=30, n=1 Unstable range for the other cases

20

3.5. Effect of the time constant ratio

We s=0, n=0

15

3.6. Effect of the gas-to-liquid density ratio The effect of gas-to-liquid density ratio on jet instability is examined in Figs. 12 and 13. The parameters held constant for these plots are Wel = 2500, Z = 0.9156, U ¼ 0 with El = 0,  k ¼ 0 for

1.4

Ωi,max :

Newtonian, n=0 Λ : Newtonian, n=1 Viscoelastic, n=0 Viscoelastic, n=1

1.2

Newtonian, n=0 Newtonian, n=1 Viscoelastic, n=0 Viscoelastic, n=1

5

0

Stable range for all the cases 0.0010

0.0012

0.0014

a Newtonian liquid jet, and El = 0.3732,  k ¼ 0:471 for a viscoelastic liquid jet. The case without air swirl (Wes = 0), is shown in Fig. 12a, while the case considering air swirl (Wes = 30), is shown in Fig. 12b. The results show that the maximum unstable growth rates are increased notably when the gas-to-liquid density ratio increases for axisymmetric disturbances and non-axisymmetric disturbances. However, the dominant wavelength K decreases as the density ra-

9

a 0.1218

8

0.1212

Newtonian, ⎯ρ = 0.0001 Newtonian, ⎯ρ = 0.001 Newtonian, ⎯ρ = 0.01 Newtonian, ⎯ρ = 0.1 Viscoelastic, ⎯ρ = 0.0001 Viscoelastic, ⎯ρ = 0.001 Viscoelastic, ⎯ρ = 0.01 Viscoelastic, ⎯ρ = 0.1

6

0.8

5

0.1206

4 0.4

0.1200 3

0.0010

0.0012

0.0014

0.0016

0.0018 0.1194 0.38

⎯ρ 1.0

Ωi,max :

Newtonian, n=0 Λ : Newtonian, n=1 Viscoelastic, n=0 Viscoelastic, n=1

Newtonian, n=0 Newtonian, n=1 Viscoelastic, n=0 Viscoelastic, n=1

0.39

0.40

0.41

0.42

9 8

b 0.13

7

0.12

4 3

0.0010

0.0012

0.0014

0.0016

0.0018

0.11

Ωi

5

0.4

0.44

Newtonian, ⎯ρ =0.0001 Newtonian, ⎯ρ =0.001 Newtonian, ⎯ρ =0.01 Viscoelastic, ⎯ρ =0.0001 Viscoelastic, ⎯ρ =0.001 Viscoelastic, ⎯ρ =0.01

6 0.6

0.43

K

Λ

Ωi,max

0.8

0.2

0.0018

 on the cutoff wavelength K0 at Fig. 13. Effect of the gas-to-liquid density ratio q Wel = 2500, Z = 0.9156, U ¼ 0 for the Newtonian (El = 0,  k ¼ 0) and viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric and non-axisymmetric disturbances with (Wes = 30) and without (Wes = 0) the air swirl.

0.6

b

0.0016

⎯ρ

Λ

Ωi,max

We s=30, n=1

7

1.0

0.2

We s=30, n=0

Unstable range for all the cases

10

Ωi

a

We s=0, n=1

Λ0

The time constant ratio is defined as the ratio of deformation retardation time k2 to stress relaxation time k1. Its effect on the jet instability is explored in Fig. 11, with the fixed parameters cho ¼ 0:001; U ¼ 0. sen as Wel ¼ 2500; El ¼ 0:3732; Z ¼ 0:9156; q Note that the cases Wes = 0 and Wes = 30 are plotted in Fig. 11a and b respectively. It can be seen from Fig. 11 that a viscoelastic jet can behave with greater stability as the time constant ratio increases, while the unstable range remains the same. Since the stress tensor of a viscoelastic liquid jet is increased by increasing the deformation retardation time k2, according to Eq. (3), then the liquid jet will become more stable and will dissipate more energy to break up in this situation. Similar to the liquid elasticity and viscosity, the effect of the time constant ratio on the unstable growth rate of axisymmetric disturbances is more remarkable.

0.10

0.09

2

⎯ρ

0.08

0.3

0.6

0.9

K  on the maximum unstable growth Fig. 12. Effect of the gas-to-liquid density ratio q rate Xi,max and dominant wavelength K at Wel = 2500, Z = 0.9156, U ¼ 0 for the Newtonian (El = 0,  k0) and viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric (solid lines) and non-axisymmetric (dash lines) disturbances (a) without (Wes = 0) and (b) with (Wes = 30) the air swirl.

 on the unstable growth rate Xi Fig. 14. Effect of the gas-to-liquid density ratio q versus the wave number K at Wel = 0.001, Z = 0.9156, U ¼ 0 for the Newtonian (El = 0,  k ¼ 0) and viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric disturbances (a) without (Wes = 0) and (b) with (Wes = 30) the air swirl.

L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

tio increases. Here, the dominant wavelength is the wavelength corresponding to the dominant wave number. In addition, the unstable ranges in Fig. 13 are expanded by raising the gas-to-liquid density ratio. Note that the cutoff wavelength K0 is the wavelength corresponding to the cutoff wave number. In terms of physics, the aerodynamic force of gas medium is positively related to the gas density. It can arouse the aerodynamic interaction existing at the interface of liquid jet to become stronger by increasing the gas density, which will make the jet more unstable. This conclusion is in agreement with the results obtained by Liao et al. [23,30]. It is necessary to note that gas density can be improved by increasing the gas pressure. Thus, in this sense, a high gas pressure may be beneficial for jet instability. Furthermore, as air swirl is introduced, non-axisymmetric disturbances can prevail over axisymmetric disturbances again when the gas-to-liquid density ratio is large enough. Among others, the effect of the gas-to-liquid density on jet instability may exhibit a different trend in the Rayleigh regime [30]. According to Li and Shen [39], the instability of an axisymmetric Newtonian liquid jet can be enhanced by decreasing the gas-to-liquid ratio, when the liquid Weber number is smaller than the critical Weber number (below which the jet is absolutely unstable). Through the effect of the density ratio plotted in Fig. 14, the same trend can be seen for a viscoelastic liquid jet in the axisymmetric mode. The parameters are fixed at Wel ¼ 0:001; Z ¼ 0:9156; U ¼ 0 with Wes = 0 in Fig. 14a and Wes = 30 in

a

0.5

Ωi,max:

n=0 n=1 n=0 n=1

Λ:

10

n=0 n=1

Unstable range for all 8 the cases

Unstable range for n =0 Stable range for n =1

Unstable range for all thecase s

6

4 Stable range for all the cases

2 0.0

0.5

1.0 ⎯U

1.5

2.0

Fig. 16. Effect of the gas-to-liquid velocity ratio U on the cutoff wavelength K0 at  ¼ 0:001, Wes = 0 for the Newtonian (El = 0,  Wel = 2500, Z = 0.9156, q k ¼ 0) and viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric and nonaxisymmetric disturbances.

a

25

0.4

Fig. 14b respectively. Note that the elastic parameters are El ¼ 0;  k ¼ 0 for a Newtonian liquid jet, and El ¼ 0:3732;  k ¼ 0:471 for a viscoelastic liquid jet. Since the non-axisymmetric mode cannot grow in this regime, further discussion is unnecessary.

Λ0

10

⎯U=0, 2, n=0 ⎯U=0.1, 1.9, n=0 ⎯U=0.3, 1.7, n=0 ⎯U=0, n=1 ⎯U=0.01, n=1 ⎯U=0.03, n=1

0.8

0.6

20

⎯U=0.05, n=1 ⎯U=1.3, n=1 ⎯U=1.5, n=1 ⎯U=1.7, n=1 ⎯U=1.9, n=1 ⎯U=2, n=1

Ωi

15

Λ

Ωi,max

0.3 0.4

0.2 0.2

10 0.1

0.0 0.0

0.5

1.0

1.5

0.0 0.0

5 2.0

0.5

1.0

b

25

0.5

Ωi,max:

n=0 n=1 n=0 n=1

0.4

Λ:

20

0.6

Ωi

Λ

Ωi,max

2.5

⎯U=0, 2, n=0 ⎯U=0.1, 1.9, n=0 ⎯U=0.3, 1.7, n=0 ⎯U=0, n=1 ⎯U=0.01, n=1 ⎯U=0.03, n=1

0.8

0.3

15

2.0

3.0

K

⎯U

b

1.5

⎯U=0.05, n=1 ⎯U=1.3, n=1 ⎯U=1.5, n=1 ⎯U=1.7, n=1 ⎯U=1.9, n=1 ⎯U=2, n=1

0.4

0.2

10

0.2

5

0.0

0.1

0.0 0.0

0.5

1.0

1.5

2.0

⎯U Fig. 15. Effect of the gas-to-liquid axial velocity ratio U on the maximum unstable growth rate Xi,max and dominant wavelength K at Wel = 2500, Z = 0.9156,  ¼ 0:001, Wes = 0 for (a) the Newtonian (El = 0,  q k ¼ 0) and (b) viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric and non-axisymmetric disturbances.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

K Fig. 17. Effect of the gas-to-liquid axial velocity ratio U on the unstable growth rate  ¼ 0:001, Wes = 30 for (a) Xi versus the wave number K at Wel = 2500, Z = 0.9156, q the Newtonian (El = 0,  k ¼ 0) and (b) viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric (solid lines) and non-axisymmetric (dash lines) disturbances.

11

L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

3.7. Effect of the gas-to-liquid axial velocity ratio In practical air-blasted atomization, the process of jet breakup can be improved by using the coaxial gas stream moving around the liquid jet. Hence, it is essential to investigate the effect of gas axial velocity on jet instability. Figs. 15 and 16 illustrate the case without air swirl (Wes = 0); Fig. 17 shows the case with air swirl (Wes = 30). In addition, the Newtonian case is shown in Figs. 15a and 17a, while the viscoelastic case is exhibited in Figs. 15b and 17b. The other parameters held constant in Figs. 15–17 are  ¼ 0:001 with El = 0,  Wel = 2500, Z = 0.9156, q k ¼ 0 for a Newtonian liquid jet, and El = 0.3732,  k ¼ 0:471 for a viscoelastic liquid jet. It is obvious that the maximum unstable growth rate decreases as the gas-to-liquid axial velocity ratio increases, when the gas axial velocity is smaller than its liquid counterpart (U <1), while it increases when the gas axial velocity (U >1), increases further. The reason for this is explained below. The gas-to-liquid axial relative velocity governs the instability of liquid jet. The jet becomes more stable by increasing the gas axial velocity continuously until it equals the jet velocity, which actually decreases the gas-to-liquid relative axial velocity DU ¼ jU  1j in the process. To further increase the gas axial velocity to make the gas-to-liquid relative axial velocity increase, the jet instability will also be enhanced. Thus under the same conditions, the unstable growth rate curves with the same gas-to-liquid relative axial velocity are coincident, such as the curves U ¼ 0:1 and U ¼ 1:9 for the axisymmetric mode in Fig. 17, which have an identical relative axial velocity of

a 0.45

Kr=0, n=1

Kr=0.5, n=0

Kr=0.5, n=1

Kr=1, n=0

Kr=1, n=1

Kr=1.5, n=0

Kr=1.5, n=1

Kr=2, n=0

Kr=2, n=1

Kr=2.5, n=0

Kr=2.5, n=1

Kr=5, n=0

Kr=5, n=1

3.8. Effect of the ratio of gas rotating velocity to gas axial velocity The effects of gas velocity ratio Kr on the unstable growth rate are plotted in Figs. 18 and 19. The parameters are fixed at Wel ¼  ¼ 0:001 with El ¼ 0;  2500; Z ¼ 0:9156; q k ¼ 0 for a Newtonian

a

0.14

Kr=0, n=0 Kr=0.5, n=0

0.12

Kr=1, n=0 Kr=1.5, n=0

0.10

Kr=2, n=0 0.08

Kr=2.5, n=0

Ωi

Ωi

0.30

Kr=0, n=0

DU = 0.9. It is worth mentioning that the jet subject to non-axisymmetric disturbances becomes stable when the gas axial velocity approaches the liquid velocity. In fact, the difference between the fluctuating pressures of liquid and gas phases (which is determined by the gas-to-liquid relative axial velocity), has a great impact on the jet instability [30]. The liquid jet becomes stable when the aerodynamic interaction between the liquid and gas medium is weak, which can be achieved by reducing the fluctuating pressure difference, or decreasing the gas-to-liquid relative axial velocity practically. It should be noted that the breakup process of liquid jet subjected to non-axisymmetric disturbances is influenced not only by the gas-to-liquid relative axial velocity, but also by the absolute gas axial velocity when air swirl is introduced. It is apparent from Fig. 17a and b, that in the non-axisymmetric mode, the maximum unstable growth rate for the curve U ¼ 0 is much smaller than that for the curve U ¼ 2, whereas these two curves have the same gasto-liquid relative axial velocity of DU = 2. Furthermore, it can be clearly seen from Figs. 15 and 16 that the dominant wavelength first increases, then decreases as the gas-toliquid axial velocity ratio increases, while the unstable range presents a polar trend.

Kr=5, n=0

0.06

0.15

0.04 0.02 0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.0

K

0.6

1.2

1.8

K

Ωi

0.30

Kr=0, n=0

Kr=0, n=1

Kr=0.5, n=0

Kr=0.5, n=1

Kr=1, n=0

Kr=1, n=1

Kr=1.5, n=0

Kr=1.5, n=1

Kr=2, n=0

Kr=2, n=1

Kr=2.5, n=0

Kr=2.5, n=1

Kr=5, n=0

Kr=5, n=1

b

0.14

Kr=0, n=0 Kr=0.5, n=0

0.12

Kr=1, n=0 Kr=1.5, n=0

0.10

Kr=2, n=0 0.08

Kr=2.5, n=0

Ωi

b 0.45

Kr=5, n=0

0.06

0.15

0.04 0.02

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

K

0.00 0.0

0.6

1.2

1.8

K Fig. 18. Effect of the ratio of gas rotating velocity to gas axial velocity Kr on the unstable growth rate Xi versus the wave number K at Wel = 2500, Z = 0.9156,  ¼ 0:001, Uall = 0.1095 for (a) the Newtonian (El = 0,  q k ¼ 0) and (b) viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric (solid lines) and nonaxisymmetric (dash lines) disturbances.

Fig. 19. Effect of the ratio of gas rotating velocity to gas axial velocity Kr on the unstable growth rate Xi versus the wave number K at Wel = 2500, Z = 0.9156,  ¼ 0:001, Uall = 0.6325 for (a) the Newtonian (El = 0,  q k ¼ 0) and (b) viscoelastic (El = 0.3732,  k ¼ 0:471) liquid jets subject to axisymmetric disturbances.

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L.-J. Yang et al. / Journal of Non-Newtonian Fluid Mechanics 191 (2013) 1–13

liquid jet in Figs. 18a and 19a, while with El ¼ 0:3732;  k ¼ 0:471 for a viscoelastic liquid jet in Figs. 18b and 19b. In addition, the fixed dimensionless gas resultant velocity Uall is 0.1095 in Fig. 18, and 0.6325 in Fig. 19 respectively. Here, the dimensionless gas qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi resultant velocity Uall can be expressed as

Wes =Wel þ U 2 . Note

that the gas axial velocity decreases as the gas velocity ratio Kr increases, which makes the jet more unstable; while the strength of air swirl increases as the gas velocity ratio Kr increases, which makes the jet more stable, with the parameters chosen here. The gas axial velocity is smaller than the liquid jet velocity in Figs. 18 and 19. It can be seen from Fig. 18a and b that the maximum unstable growth rate for non-axisymmetric disturbances first decreases, then increases as the gas velocity ratio Kr increases, while it always increases for axisymmetric disturbances as the gas velocity ratio Kr increases. This indicates that when the gas resultant velocity is relatively smaller, the effect of gas axial velocity on jet instability can prevail over that of the air swirl when the gas velocity ratio Kr is great enough for the non-axisymmetric mode, whereas it can always be a dominant factor for the axisymmetric mode. However, the effect of gas axial velocity on jet instability with axisymmetric disturbances can prevail over that of the air swirl only at large gas velocity ratio Kr when the gas resultant velocity is larger, which can be clearly seen form Fig. 19a and b. It should also be noted that liquid jet subject to non-axisymmetric disturbances is stable in the case of Uall = 0.6325, which is not shown in Fig. 19. Although it is not plotted in the present paper, to further increase the resultant gas velocity to a greater value will arouse the maximum unstable growth rate of liquid jet always to decrease, as the gas velocity ratio Kr increases, due to the stronger swirl strength and decreasing gas axial velocity which, in this case is larger than – or close to – the liquid velocity. 4. Conclusion A linear stability analysis has been conducted on a three-dimensional viscoelastic liquid jet, moving in a swirling gas stream. The derived dispersion relation was solved by Muller’s method. The effects of some flow parameters and fluid properties (including the liquid Weber number Wel, gas swirl Weber number Wes, elasticity number El, Ohnesorge number Z, time constant ratio  k, gas-to li , gas-to liquid axial velocity ratio U and the raquid density ratio q tio of gas rotating velocity to gas axial velocity Kr), on jet instability were investigated. Compared to a Newtonian liquid jet in swirling gas, the air swirl strength, jet velocity, surface tension, liquid viscosity and gas-to-liquid density ratio were predicted to exhibit analogous effects for a viscoelastic jet. It should be noted that air swirl was also predicted to stabilize viscoelastic liquid jet. As a result, some conclusions are specified as below. The effects of some rheological parameters, when air swirl is introduced to jet instability, were examined. A high liquid elasticity, or a small time constant ratio, was predicted to make the jet more unstable, while the unstable range was predicted to remain the same, as these two elastic parameters vary. Furthermore, the reactions of unstable growth rates of axisymmetric disturbances to the variations of elasticity number and time constant ratio were predicted to be more sensitive than those for non-axisymmetric disturbances. The distinction between the effects of gas-to-liquid axial velocity ratio on the two disturbance modes, when considering the air swirl, was studied. It is the gas-to-liquid relative axial velocity that predicted to govern jet instability for the axisymmetric mode. However, the breakup process of liquid jet surrounded by a swirling gas stream was predicted to be influenced not only by the gasto-liquid relative axial velocity, but also by the absolute gas axial velocity in the case of non-axisymmetric disturbances.

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