Atomization and Sprays, 27(12):1041–1061 (2017)
NONLINEAR SPATIAL INSTABILITY OF A SLENDER VISCOUS JET Li-Jun Yang,1,∗ Tao Hu,1 Pi-Min Chen,2 & Han-Yu Ye1 1 2
Beijing University of Aeronautics and Astronautics, Beijing, 100191, China AVIC Aviation Powerplant Research Institute, Zhuzhou, 412002, China
*Address all correspondence to: Li-Jun Yang, School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China; Tel./Fax: +86 010 8233 9571, E-mail:
[email protected] Original Manuscript Submitted: 5/23/2017; Final Draft Received: 1/24/2018 A perturbation analysis combined with one-dimensional equations is carried out to study the nonlinear spatial instability of a slender viscous jet. The solutions and wave profiles of the second order to third order have been presented. The result indicates that, as the perturbation expression proceeds to higher orders, the main swellings become narrow and the secondary swellings are flattened, resulting in the formation of a level liquid ligament. In addition, there exist two different nonlinear regions, named herein as the strong nonlinear region and the weak nonlinear region. The division of the two regions can be explained as a result of the interactions between the higher order harmonics transferred from lower orders and the inherent higher order disturbances. In addition, as Weber number decreases or Reynolds number increases, the growth rate of the jet increases significantly; the nonlinear amplitudes increase in the strong nonlinear region but remain constant in the weak nonlinear region, resulting in a shorter breakup length and a nearly identical waveform. The critical frequency, below which the jet is in the strong nonlinear region and above which it is in the weak nonlinear region, is not affected by Weber number but decreases noticeably as the Reynolds number reduces to less than 10. The theoretical waveforms are in agreement with previous experiments and simulations.
KEY WORDS: spatial instability, one-dimensional equations, slender viscous jet, perturbation analysis
1. INTRODUCTION The instability of a liquid jet is of great scientific significance in broad practical applications such as fuel injection and atomization in combustion engines, agricultural sprays, and spray coating processes (Eggers, 1997; Eggers and Villermaux, 2008; Basaran et al., 2013). On the one hand, a good understanding of the physical mechanism of jet instability and its subsequent breakup are essential for the design and operation of the practical systems involved (Lunardelli et al., 2015). On the other hand, jet behavior is found to show abundant features in different situations, such as jets with different cross-sectional shapes (Amini et al., 2014; Gorman et al., 2014), under different physical fields (Ruo et al., 2012; Xie and Yang, 2017), or fluids that have different rheological properties (Chang et al., 2013; Tirel et al., 2017). For these reasons, numerous researchers have studied jet instability extensively in various aspects, as reviewed by Eggers (1997), Sirignano and Mehring (2000), Eggers and Villermaux (2008), and Tharakan et al. (2013). The linear instability of a liquid jet was initiated by Rayleigh (1878), who provided the first analytical description of the temporal instability of an inviscid jet moving in a vacuum. He concluded that only axisymmetric disturbances with wavelengths larger than the undisturbed jet circumference can grow in time. Weber (1931) and Chandrasekhar (1961) investigated the temporal instability of viscous jets. It was found that viscosity has a stabilizing effect by decreasing the growth rate. Tomotika (1935) studied the stability of a viscous jet in another viscous fluid; he stated that, when the ratio of viscosity of the two fluids takes finite values, the maximum growth rate always occurs at a certain definite value of the wavelength of the varicosity. Sterling and Sleicher (1975) considered the velocity
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NOMENCLATURE P Pg r R0 R1 R2 Re t U U0 V w
pressure of the jet, Pa pressure of the ambient gas, Pa displacement in the radial direction, m initial radius of the jet, m curvature radius in the transverse cross section of the jet, m curvature radius in the transverse streamwise direction of the jet, m Reynolds number time that the jet takes to develop, s velocity in the z direction of the jet, m/s initial velocity of the jet, m/s velocity in the r direction of the jet, m/s dimensionless axial velocity of the slender jet
We z
Weber number displacement in the axial direction, m
Greek Symbols δ slenderness ratio η dimensionless free surface displacement when disturbances set in η0 dimensionless initial disturbance amplitude µ viscosity of the jet, Pa·s ρ density of the jet, kg/m3 σ surface tension between the jet and the ambient gas, N/m ψ free surface displacement when disturbances set in, m ω dimensionless disturbance frequency
profile relaxation in jets to modify the theory of Weber (1931). It was shown that the velocity relaxation enhances instability. In many practical situations, oscillation grows with increasing distance from the nozzle along the jet, so spatial instability is more appropriate in describing liquid jets subjected to a continuously oscillating source of a given frequency at the atomizer exit. Spatial instability was first studied by Keller et al. (1973). They found that Rayleigh’s results are relevant only in the case of high Weber numbers. For low Weber numbers, they found a new mode of faster growing disturbances with much larger wavelengths than those of the Rayleigh mode. This mode was related to an absolute instability by Leib and Goldstein (1986a,b), who then determined the critical Weber number below which the viscous jet is absolutely unstable as a function of Reynolds number. This study was then extended by Lin and Lian (1989), who carried out research on the instability of a viscous liquid jet in a stationary inviscid gas medium. It was shown that gas density has the effect of raising the critical Weber number. In the theoretical works cited above, linear instability analysis has been employed. However, some experimental studies have shown that the main drops are interspersed with smaller satellite drops, contradicting the uniform breakup model of linear theory (Donnelly and Glaberson, 1966; Rutland and Jameson, 1971; Chaudhary and Maxworthy, 1980a). To overcome the limitation that linear theory could not account for satellite drops, a weakly nonlinear theory that considers higher order terms in the perturbation series was established. Yuen (1968) first studied a weakly nonlinear temporal instability for an inviscid liquid jet with a third-order perturbation approach. He predicted the existence of satellite drops for all wavenumbers that Goedde and Yuen (1970) discussed in their experimental results. He also indicated that the cutoff wavenumber is shifted to larger values. Nayfeh (1970) pointed out that Yuen’s treatment of the stability boundary is far from rigorous and predicted an even lower cutoff wavenumber. Subsequently, Lafrance (1975), following an analysis similar to Yuen’s, presented a third-order theory that predicted no shift in the cutoff at all. This apparent discrepancy can be attributed to an algebraic error in which Lafrance assumed a third-order solution does not satisfy the boundary conditions to the same order. Away from the cutoff wavenumber, the growth rate of higher order harmonics has been measured by Taub (1976). Excellent agreement was found between the measurement of the growth rate and the calculations based on the nonlinear analysis of Yuen (1970). Chaudhary and Redekopp (1980) further developed the solution for an infinite jet with an initial velocity disturbance consisting of a fundamental and one harmonic component, to complement
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experiments by Chaudhary and Maxworthy (1980a,b), in which jets were forced with a fundamental frequency plus its harmonics in order to suppress satellite drops. Busker et al. (1989) performed a nonlinear spatial instability analysis in an inviscid liquid jet and compared their theoretical model to experimental data. It was shown that the spatial model describes the physical reality better than models based on temporal instability. However, even the weakly nonlinear theory is not able to describe the shape of the fluid anywhere near the breakup. Near the breakup point, the separation of a drop corresponds to a singularity where expansions in the radius and velocity are bound to break down. Therefore, a complete treatment of the essential nonlinearity is necessary. Because the three-dimensional Navier-Stokes equations are extremely complicated for both analytical and numerical studies, it is interesting to generate simplified equations that still capture the essential nonlinear physics of the problem. Lee (1974), Pimbley (1976), and Pimbley and Lee (1977) studied the nonlinear dynamics of inviscid jets based on one-dimensional equations. It was shown that the behavior of main and satellite drop was empirically controlled by applied oscillations at the nozzle. Lee’s study was extended by Green (1976), who considered equations for the flow of viscous jets. His approach was inherently different from Lee’s and it developed into a so-called Cosserat theory. Bogy (1978, 1979) used this Cosserat theory to study the instability and drop formation of viscous jets. A systematic one-dimensional model was derived and analyzed for the stability of a viscous jet by other authors (Garcia and Castellanos, 1994; Bechtel et al., 1995). They obtained one-dimensional equations for arbitrary high orders using a perturbative approach. It was found that the equations derived therein are consistent with Rayleigh’s theory and can be used as the basis for constructing a solution to the stability for a viscous jet. Chesnokov (2000) analyzed the second-order temporal development of capillary waves in a viscous jet through one-dimensional equations obtained by Bechtel et al. (1995). It was shown that the size of satellite drops decreases with the Reynolds number at a constant wavenumber. Independently, Eggers and Dupont (1994) gave essentially the same leading-order equations and compared their numerical results to the previous experiments of Chaudhary and Maxworthy (1980a) and Peregrine et al. (1990). Their computed predictions coincide well with the experiments. Eggers (1993, 1995, 2005) further developed the approach and proposed a self-similar solution that could describe the ultimate breakup process of a liquid jet. Brenner et al. (1996) proved that the Eggers solution is the least unstable with respect to perturbations; they numerically constructed two infinite families of self-similar solutions for capillary pinching of a liquid jet. Recently, some researchers have investigated the beads-on-string structure formed during the jet breakup process based on the model developed by Eggers (Li and Fontelos, 2003; Oliveira and McKinley, 2005; Ardekani et al., 2010; Li et al., 2017). Still, few analytical approaches have described the nonlinear spatial evolution of a liquid jet satisfactorily. Discrepancies between theoretical results and experiments were attributed mainly to: (i) the neglect of the effects of the higher-order terms (Yuen, 1968); (ii) the neglect of viscosity of the liquid (Busker et al., 1989); and (iii) the use of temporal analysis to describe spatial jet evolution (Chaudhary and Maxworthy, 1980b). Noting that the numerical simulation is difficult to utterly reflect the physical mechanics and essence, an analytical perturbation analysis is combined with one-dimensional equations to study the nonlinear spatial instability of a slender viscous jet in the present study. The general solutions of high-order disturbances are obtained, and the first three terms are solved. A third-order waveform is drawn, and the effect of the high-order disturbance on the waveform is studied. Furthermore, the effects of vibration frequency, Weber number, and Reynolds number on the instability of the jet are discussed. Finally, comparisons are made between the waveform predicted by present theory and the experimental observations.
2. GOVERNING EQUATIONS As shown in Fig. 1, an axisymmetric, incompressible, laminar slender viscous jet with density ρ, viscosity µ, and surface tension σ is considered. The gravity force and hydrodynamic effect of the ambient air are neglected. The cylindrical polar coordinate system is chosen so that the z-axis is parallel to the direction of the basic flow, with the origin located at the initial middle of the liquid jet. For the basic flow, the jet has a constant radius R0 and a uniform velocity U0 . When disturbances set in, the jet has a new free surface ψ and the velocity in the z and r directions become U and V , respectively. The dimensional governing equations are giving as follows (where subscript r denotes ∂/∂r, etc.):
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V
r
U
R0
z
FIG. 1: Schematic description of a disturbed slender jet
The mass conservation equation is Uz + V r +
V =0 r
(1)
The equations of axial and radial momentum are given by
ρ (Ut + U Uz + V Ur ) = −Pz + µ Urr + Uzz
ρ (Vt + U Vz + V Vr ) = −Pr + µ Vrr + Vzz
Ur + r
Vr V + − 2 r r
(2)
(3)
The kinematic boundary condition requires V = ψt + U ψz
at r = ψ
(4)
The liquid shear stress is continuous across the free surface, and the normal stress is discontinuous with a jump given by the surface tension. This yields the tangential and normal dynamic boundary conditions h i 2 (Ur + Vz ) 1 − (ψz ) + 2ψz (Vr − Uz ) = 0 at r = ψ (5) P−
2µ
2
1 + (ψz )
h
i 1 1 2 Vr + (ψz ) Uz − (Ur + Vz ) ψz = Pg + σ + R1 R2
at
r=ψ
(6)
where Pg is the pressure of the ambient gas, assumed to be constant. In the cylindrical coordinate system, the principal radii of curvature in the transverse cross-section and streamwise direction are calculated as follows: 1 1 = 1/2 R1 ψ [1 + ψ2z ]
(7)
1 ψzz =− 3/2 R2 [1 + ψ2z ]
(8)
For the slender flow, the radial scale R0 is small compared to its transverse scale Z0 . The governing equations would then involve the slenderness ratio δ = R0 /Z0 . For proper simplification of the analysis and universality of the results, all the flow parameters and variables are nondimensionalized. The nondimensional Weber number and Reynolds number are defined as We = ρU02 R0 /σ and Re = ρU0 Z0 /µ. The length, velocity, time, and pressure are scaled with R0 , U0 , Z0 /U0 , and ρU02 respectively. To simplify the governing equations, we use a method similar to that of Bechtel et al. (1995). The nondimensional jet surface η, axial velocity u, radial velocity v, and pressure P are sought in the following double expansions: X η= δ2m η(2m) (z, t) (9) m>0
u=
X
δ(2n+2m) r2n un,2m (z, t)
(10)
n,m>0
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X
v = δr
δ(2n+2m) r2n v n,2m (z, t)
(11)
n,m>0
p − pg =
X
δ(2n+2m) r2n pn,2m (z, t)
(12)
n,m>0
Substituting these expansions into the governing equations of the jet (1)–(6), we obtain equations of the leading order slender 1D model and the first asymptotic corrections in δ as follows: The leading order slender 1D model (0) u0,0 (0) z η ηt + u0,0 η(0) + =0 (13) z 2 " # (0) (0) 1 ηz 3 2ηz 0,0 0,0 0,0 0,0 (14) u + u + u0,0 z zz t + u uz = Re η(0) We η(0) 2 The first asymptotic corrections in δ
i3 h (2) (0) u0,0 + u0.2 1 (2) z η z η 0,2 (0) (0) ηt + u0,0 η(2) + u0,0 z + u ηz + zzz η 16 2 i2 h 5 0,0 h (0) i2 (0) 3 0,0 h (0) i2 (0) =0 ηz + uz ηzz + 3η(0) η(0) η + uzz η z 8 8 ( ( ) ) (0) (0) (2) 0,0 (2) 3 u0,2 u0,0 0,2 z ηz + uz ηz z ηz η 0,0 0,2 0,2 0,0 0,2 ut + u uz + u uz = 2 + uzz − 2 Re η(0) η(0) ( ) (2) (0) ηz 1 2ηz η(2) + + L u0,0 , η(0) 2 − 3 We η(0) η(0)
where L(f, g) is a partial differential operator and its expression is given in Eq. (A.1). In the present study, we define the free surface and the axial velocity of the slender jet as follows: η = η(0) + δ2 η(2) + O δ4 w = u0,0 + δ2 u0,2 + O δ4
(15)
(16)
(17) (18)
2
Multiplying Eqs. (15) and (16) by δ , summing the resulting relations with Eqs. (13) and (14) respectively, and neglecting the effects of the terms of order δ4 and high orders, one obtains the one-dimensional governing equations of a slender jet: 1 5 3 wz η 2 3 2 2 2 +δ wzzz η + wzz η ηz + wz η ηzz + 3ηηz = 0 ηt + wηz + (19) 2 16 8 8 1 ηz 3 2ηz wz + δ2 L (w, η) (20) + wzz + wt + wwz = Re η We η2
In the present study, we shall seek the solutions to Eqs. (19) and (20) with an initial condition by using a perturbation expansion technique with the initial disturbance amplitude η0 as the perturbation parameter. η and w can be expanded as follows: ∞ X i η= η · ηi0 = 1 + 1 ηη0 + 2 ηη20 + · · · (21) i=0
w=
∞ X
i
w · ηi0 = 1 + 1 wη0 + 2 wη20 + · · ·
i=0
where 0 η and 0 w are set to be 1 as they correspond to the basic flow.
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An initial condition is given as follows to indicate the initial disturbance: η = 1 + η0 cos (ωt)
at
z=0
(23)
Substituting Eqs. (21) and (22) into Eqs. (19) and (20), collecting the terms with ηn0 , one can obtain the governing equations of the nth order: n
1
ηt + n ηz +
n
δ2 wz + n wzzz = n f 2 16
1
w, 1 η, · · ·
3 n 1 n wzz + ηz − n wt − n wz Re We 3 n 1 n 2 +δ wzzzz + ηzzz = n g 8Re We
n−1
w, n−1 η
(24)
(25) 1
w, 1 η, · · ·
n−1
w, n−1 η
f and 1 g are equal to zero because the first order is associated with the linear instability. The expressions for the nonlinear terms 2 f , 2 g, 3 f , and 3 g are given in Eqs. (A.2)–(A.5). 3. MATHEMATICAL SOLUTIONS 3.1 First-Order Solutions The first-order equations are linear in x and t; therefore, the solutions are sought in the expressions of normal mode 1 ˆ, 1,1 w η, 1 w = 1,1 η ˆ exp (ik1 z − iωt) + c.c. (26)
where k1 and ω are the first-order wavenumber and oscillation frequency of the disturbance, respectively, c.c. represents the complex conjugate of all the terms ahead of itself aiming to eliminate the imaginary parts and make the solutions suitable for the practice, andˆdenotes the disturbance amplitudes. In the spatial instability situation, the oscillation frequency ω is given by the initial condition (23) and treated as a real number. The wavenumber k1 = k1r + k1i is a complex number. The opposite number of k1 ’s imaginary part, −k1i , represents the spatial growth rate. Substituting Eq. (26) into the initial condition (23) and the first-order governing equations, the first-order solutions can be solved to give 1 1,1 ˆ= η (27) 2 According to Cramer’s rule in linear algebra, the nontrivial solutions of the first-order governing equations exist only if D (k1 , ω) = 0 (28) Where the dispersion function D (k, ω) is defined as follows: D (k, ω) = C (k, ω) (k − ω) − A (k) B (k)
(29)
k δ2 − k3 (30) 2 16 k 1 − δ2 k 2 (31) B (k) = We 6 ikA (k) − (k − ω) (32) C (k, ω) = Re Once the dispersion relation is satisfied, we substitute Eq. (26) into the first-order governing equations, which yields: k1 − ω 1,1 1,1 ˆ w ˆ=− η (33) A (k1 ) A (k) =
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The dispersion relation (28) is an approximation for Rayleigh exact solution, which is compactly written by Bechtel et al. (1995) as follows: k2 1 − k2 ωR k 2 2k 4 2 F (k) + [2F (k) − 1] + 2 [F (k) − F (l)] − =0 (34) ωR 2 Re 2We Re where l2 = k 2 + ωR Re; F (x) = xI0 (x)/I1 (x); I0 and I1 are the modified Bessel functions of the first kind of order 0 and 1, respectively. Representing F (k) as a series in power of k F (k) = 2 + Then, Eq. (34) can be written as follows: F ω2R
k2 + O k4 4
k2 1 − k2 (k) 3k 2 + ωR − = O k6 2 Re 2We
(35)
(36)
For temporal instability, the wavenumber k is treated as a real number; the frequency ω = ωr + iωi is treated as a complex number and its imaginary part ωi represents the temporal growth rate. Through a preliminary numerical calculation, it is found that ωr = k (37) On the other hand, A (k) F (k) = 1 + O k4 k
(38)
Multiplying Eq. (36) by 2A(k)/k, subtracting Eq. (29) from the resulting relation, and using Eqs. (37) and (38), one obtains the relation of ωR and ωi 6k (ωi − ωR ) ωi + ωR + A (k) = O k 6 (39) Re 3.2 Second-Order Solutions According to the forms of the first-order solutions expressed in Eq. (26), the nonhomogeneous terms consisting of the products of the first-order disturbances involved in the second-order governing equations can be expressed in the following forms: 2 (40) f 1 w, 1 η = 2,2,1 fˆ exp (2ik1 z − 2iωt) + 2,0,1 fˆ exp i k1 − k¯1 z + c.c. 2
g
1
w, 1 η = 2,2,1 gˆ exp (2ik1 z − 2iωt) + 2,0,1 gˆ exp i k1 − k¯1 z + c.c.
(41)
The expressions for 2,2,1 fˆ, 2,0,1 fˆ, 2,2,1 gˆ and 2,0,1 gˆ are given in Eqs. (B.1)–(B.4). Because the second-order governing equations are inhomogeneous differential equations, both the particular and general solutions to the corresponding homogeneous equations should be included in the solutions. The second-order solutions are assumed to be 2 ˆ, 2,2,1 w ˆ , 2,0,1 w η, 2 w = 2,2,1 η ˆ exp [2i (k1 z − ωt)] + 2,0,1 η ˆ exp i k1 − k¯1 z (42) ˆ, 2,2 w ˆ, 2,0 w + 2,2 η ˆ exp [i (k2 z − 2ωt)] + 2,0 η ˆ exp (ik2′ z) + c.c. where the terms with superscripts 2,2,1 and 2,0,1 are the particular solutions generated from the first-order disturbances, representing the energy transfer from the first order to the second order. The terms with superscripts 2,2 and 2,0 are the general solutions of the homogeneous equations and represent the second-order inherent disturbances. They have the second-order inherent wavenumbers k2 and k2′ , respectively.
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Substituting Eqs. (42) and (27) into the initial condition (23) yields 2,2
ˆ = −2,2,1 η ˆ, η
2,0
ˆ = −2,0,1 η ˆ η
(43)
SubstitutingEq. (42) into the second-order governing equations, collecting the terms with coefficient exp [2i (k1 z − ωt)] and exp i k1 − k¯1 z , one can obtain the disturbances with superscripts 2,2,1 and 2,0,1, respectively. Disturbances with superscript 2,2,1 i h i C (2k1 , 2ω) · 2,2,1 fˆ − A (2k1 ) · 2,2,1 gˆ 2,2,1 ˆ=− η (44) D (2k1 , 2ω) i h i (2k1 − 2ω) · 2,2,1 gˆ − B (2k1 ) · 2,2,1 fˆ 2,2,1 w ˆ=− (45) D (2k1 , 2ω) Disturbances with superscript 2,0,1 i h i C k1 − k¯1 , 0 · 2,0,1 fˆ − A k1 − k¯1 · 2,0,1 gˆ 2,0,1 ˆ=− η D k1 − k¯1 , 0 h i i k1 − k¯1 · 2,0,1 gˆ − B k1 − k¯1 · 2,0,1 fˆ 2,0,1 w ˆ=− D k1 − k¯1 , 0
(46)
(47)
It is significant to note that the governing equations with superscripts 2,2 and 2,0 do not contain inhomogeneous products, so they are solved using the same method as first-order analysis. The second-order dispersion relations and the disturbances with superscripts 2,2 and 2,0 are solved as follows: Terms with superscript 2,2 D (k2 , 2ω) = 0 (48) 2,2
w ˆ=−
k2 − 2ω 2,2 ˆ η A (k2 )
(49)
Terms with superscript 2,0 D (k2′ , 0) = 0 which is solved as k2′ = k0 .
2,0
(50)
w ˆ=0
(51)
3.3 Third-Order Solutions Similar to the analysis for the second-order instability, the third-order solution can be assumed to be 3
ˆ exp [i (k1 + k2 ) z − 3iωt] ˆ exp (3ik1 z − 3iωt) + 3,3,2 η η = 3,3,1 η 3,1,1 ˆ exp i k2 − k¯1 z − iωt ˆ exp [i (k1 + 2ik1i ) z − iωt] + 3,1,2 η η + +
3,1,3
ˆ exp (ik1 z − iωt) + η
3,3
ˆ exp (ik3 z − 3iωt) + η
3,1
ˆ exp (ik3′ z η
(52)
− iωt) + c.c.
The terms with superscripts 3,3,1 to 3,1,3 are the particular solutions generated from the first- and secondorder disturbances, representing the energy transfer from the first and second order to the third order. The terms with superscripts 3,3 and 3,1 are the general solutions of the homogeneous equations and represent the third-order inherent disturbances. They have the third-order inherent wavenumbers k3 and k3′ , respectively. Substituting Eqs. (52), (27), and (43) into the initial condition (23) yields 3,3 ˆ ˆ + 3,1,3 η ˆ = − 3,1,1 η ˆ + 3,1,2 η ˆ , 3,1 η ˆ = − 3,3,1 η ˆ + 3,3,2 η (53) η Atomization and Sprays
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The third-order solutions are solved to be, as follows: h i i C (3k1 , 3ω) · 3,3,1 fˆ − A (3k1 ) · 3,3,1 gˆ 3,3,1 ˆ=− η D (3k1 , 3ω) i h i C (k1 + k2 , 3ω) · 3,3,2 fˆ − A (k1 + k2 ) · 3,3,2 gˆ 3,3,2 ˆ=− η D (k1 + k2 , 3ω) h i i C 2k1 − k¯1 , ω · 3,1,1 fˆ − A 2k1 − k¯1 · 3,1,1 gˆ 3,1,1 ˆ=− η D 2k1 − k¯1 , ω h i i C k2 − k¯1 , ω · 3,1,2 fˆ − A k2 − k¯1 · 3,1,2 gˆ 3,1,2 ˆ=− η D k2 − k¯1 , ω i h i C (k1 , ω) · 3,1,3 fˆ − A (k1 ) · 3,1,3 gˆ 3,1,3 ˆ=− η D (k1 , ω)
k3′
(54)
(55)
(56)
(57)
(58)
D (k3 , 3ω) = 0
(59)
D (k3′ , ω)
(60)
=0
k3′
is solved as = k1 ; thus, the disturbance with superscript 3,1,3 is counteracted by the terms in the disturbance with superscript 3,1 and has no need to be solved out. The expressions for 3,3,1 fˆ, 3,3,2 fˆ, 3,1,1 fˆ, 3,1,2 fˆ, 3,3,1 gˆ, 3,3,2 gˆ, 3,1,2 gˆ, and 3,1,1 gˆ are given in Eqs. (B.5)–(B.12). 4. RESULTS AND DISCUSSION 4.1 Validation of Linear Instability A comparison of the temporal growth rate predicted by the present slender model, Eq. (28), and Rayleigh model, Eq. (34), is made in Fig. 2. Figure 2(a) shows that the curves of the present predictions and the Rayleigh solutions overlap with each other at different Reynolds numbers, indicating that present slender model coincides with Rayleigh model. The relative error ∆, defined as ∆ = |ωi − ωR |/ωR , is depicted in Fig. 2(b). It can be seen that the present solutions and Rayleigh solutions are in good agreement with relative error of < 1% at different Reynolds numbers. Especially in the long-wavelength unstable region of k < 0.4, the error can even be < 0.1%. Besides, the relative error is increased generally as Reynolds number increases, as Eq. (39) indicates. 1.0 %
0.04 Rayleigh Exact 0.03
Re 10 Re 100 Re 1000
0.6
1i
Slender Model
0.02
Re 10 Re 100 Re 1000
0.01 0.00 0.0
Re = 10 Re = 100 Re = 1000
0.8
0.4 0.2
0.2
0.4 k 0.6 (a)
0.8
1.0
0.0 0.0
0.2
0.4 k 0.6
0.8
1.0
(b)
FIG. 2: Comparison between the present slender model and the Rayleigh model at We = 100
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4.2 Effects of High-Order Disturbances Figure 3 shows the second- to third-order interface waves just before the breakup of a slender jet at Weber number of 200, Reynolds number of 4000, and the frequency of 0.25. The property of the jet is chosen according to the experiment conducted by Rutland and Jameson (1971). The initial disturbance amplitude is chosen to be 0.05. It should be noted that the initial disturbance amplitude does not affect the pinch-off dynamics of the jet (Eggers, 2005), which provides a base for the comparison of the waveforms close to the breakup point when a sufficiently accurate value of the initial disturbance amplitude is not obtainable. The wave deformation of the jet is obtained without introduced different reference length for r and z, i.e., δ = 1, since the value of δ does not affect the dimensional solutions of wavenumber, growth rate, surface displacement, etc. Thus, the same reference length is not inconsistent with the slenderness assumption. It can be seen from Fig. 3 that, as perturbation expansion proceeds to third order, the main swellings are narrowing and change from a sinusoidal wave into a nearly round shape, while the secondary swellings become flatter, forming a slender and level liquid ligament. Table 1 gives the lengths of liquid ligaments 1 and 2, c1n , c2n , and their standard deviation ξ1n , ξ2n for each order. The length of the ligament is calculated by measuring the distance of the troughs at its ends. The standard deviation, reflecting the uniformity of the liquid ligament, is obtained by discretizing the continuous waveform with the same interval 0.1. It is shown that liquid ligament 2 is slightly longer (c2n > c1n ) and less uniform (ξ2n > ξ1n ) than liquid ligament 1, due to the nonlinearity growing along the z-axis. In addition, taking the high-order disturbances into account will get a generally longer and leveler liquid ligament. This phenomenon can be attributed to the fact that the disturbance of the nth order could eliminate the peaks and troughs of the interface wave of the (n − 1)th order in the ligament, which generates an energy feedback from higher harmonics to the fundamental and a damping effect on the detachment of the pinch point. It can be also seen that the lengths of ligaments of higher orders can conform better to the experimental result of Rutland and Jameson (1971), in which the dimensional length of ligaments 1 and 2 are measured to be 16 and 18 mm, respectively, and the jet radius is 1.151 mm. Moreover, at the early stage of the growth, the jet shape remains almost sinusoidal for the majority of its length (which is not observable on the scale shown in Fig. 3), since the high-order disturbances are very weak but the fundamental wave is quite pronounced at the beginning of the evolution. As the interface wave travels, the nonlinear effect becomes more and more prominent; the high-order disturbances enlarge the curvature of the interface wave at the pinch point. As a result, the jet becomes more distorted and breaks up at a shorter length. 3
secondary swelling
(a) 0
-3
180
3
190
200 z
ligament1
210
220
main swelling
230
ligament2
(b) 0
-3
180
190
200 z
210
220
230
FIG. 3: Spatial evolution of a slender jet at We = 200, Re = 4000, η0 = 0.05, and ω = 0.25 for (a) second order and (b) third order
TABLE 1: The length and standard deviation of ligaments 1 and 2 n 2 3 Experiment
C1n 9.3 12.9 13.9
ξ1n 0.0827 0.0532 —
C2n 10.4 13.9 15.6
ξ2n 0.239 0.237 —
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4.3 Effects of Vibration Frequency Through a preliminary numerical calculation, it is found that the disturbances with superscripts 2,2,1 and 3,3,1 are the leading terms affecting the of the slender waveform jet; they are all generated directly from the fundamental first ˆ2 = 2,2,1 η ˆ and η ˆ3 = 3,3,1 η ˆ as the measurements of the second- to third-order amplitudes order. Thus, we define η to judge the value of the nonlinear disturbances. Figure 4(a) shows the second- to third-order amplitudes varied with frequency at a Weber number of 400 and ˆ 2 and Reynolds number of 400. Obviously, there exist a strong nonlinear region (approximately ω < 0.35), where η ˆ3 are relatively large and decrease notably with ω; and a weak nonlinear region (approximately 0.35 < ω < 1), η where they remain constant and minor. This phenomenon can be explained as a result of the harmonics. In the strong nonlinear region, the nonlinear disturbances transferred from lower orders are approximately satisfied with the dispersion relation and tightly coupled with the inherent disturbances, resulting in the transfer of considerable vibrational energy and a larger oscillation amplitude. However in the weak nonlinear region, the opposite is the case. As shown in Fig. 4(b), the reciprocal of the dispersion function Dn−1 , defined as Dn−1 = 1/D(k1 , nω), tends to be infinite in the strong nonlinear region, resulting in nk1 = kn since the inherent disturbances satisfy the relation D(kn , nω) = 0. In this sense, the effect of harmonics is supposed to be more prominent when their frequency approaches that of the inherent disturbances of higher order. As the frequency increases into the weak nonlinear region, Dn−1 decreases rapidly and leads to a deviation of the wavenumber of the nonlinear transferred disturbance from its inherent disturbance; thus, the effect of harmonics is damped. The variation tendency of Dn−1 and the corresponding nonlinear amplitude are mostly consistent. Figure 5 shows the wave deformations for different frequencies. For ω = 0.2, there exists an evident ligament linking the main drops, which will shrink into a satellite drop similar in size to the main drops. For ω = 0.5, the wave shape remains sinusoidal for the majority of the length and only the second-order disturbance is visible near the breakup point, and the length of the ligament, and consequently, the size of the satellite drop decrease as the wavelength decreases. No apparent satellite drop, however, is predicted for the frequency of 0.7, since the nonlinear effect diminishes as the frequency increases so that no intermediate swelling is formed between nodes of the fundamental frequency. After examining the three different wave profiles, one can see that higher order corrections are mainly introduced with the harmonics whose wave numbers consist of 2k1 , 3k1 ,... The result is in good agreement with the experiments of Rutland and Jameson (1970) and Lafrance (1975), in which the main drops are found to be larger than the satellites for dimensionless frequency of > 0.35. 4.4 Effects of Weber Number Figure 6(a) shows the dispersion relation at different Weber numbers. It can be seen that an increase in Weber number yields a smaller growth rate and thus reduces the linear instability of the slender jet. For this liquid jet without 10
2.0
n=2 n=3
n=2 n=3
1.5
ˆn
Dn 1 1.0
strong nonlinear
5
104
strong nonlinear
0.5
weak nonlinear
0
weak nonlinear
0.0
0.2
0.4 0.6 (a)
0.8
1.0
0.2
0.4 0.6
0.8
1.0
(b)
FIG. 4: Effect of frequency on (a) the nonlinear amplitudes and (b) the reciprocal of the dispersion function
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4 (a) 0
-4 530 4
540
550
560 z 570
580
590
600
(b) 0
-4
270
280
290
z 300
310
320
330
230
240
250
z 260
270
280
290
4 (c) 0
-4
FIG. 5: Spatial evolution of interface deformation for different frequencies at We = 400, Re = 400, and η0 = 0.01: (a) ω = 0.2, (b) ω = 0.5, and (c) ω = 0.7
0.02
We = 200 We = 400 We = 800
We = 200 We = 400 We = 800
4
ˆ2
k1i
ˆ3
3 2
2
1 0.15
(a)
0.8
2 0.20
0.25
0
0.65
0.4 0.6
1.0
4
2
0.01
0.2
We = 200 We = 400 We = 800
4
0.20
0.25
0.30
0 0.2
0.4 0.6
(b)
0.8
1.0
0.2
0.4
0.6
0.8
1.0
(c)
FIG. 6: Effect of Weber number on (a) growth rate, (b) second-order initial amplitude and (c) third-order initial amplitude at Re = 400
considering the disturbance of surrounding gas, there exists only Rayleigh instability but not Taylor instability, as Lin (2003) stated; thus, the surface tension always enhances the linear instability of the liquid jet. Besides, the cutoff frequency and dominant frequency (corresponding to the maximum growth rate) remain at 1 and 0.65, respectively. The result coincides nicely with the study of Rayleigh (1878) for temporal instability of inviscid jets, in which the cutoff and dominant wavenumbers were proved to be 1 and 0.697, respectively. The variations of nonlinear amplitudes with frequency at different Weber numbers are depicted in Figs. 6(b)– ˆ2 and η ˆ3 decrease slightly in the 6(c). It can be seen that, as the Weber number is increased, nonlinear amplitudes η strong nonlinear region but remain the same in the weak nonlinear region. The critical frequency below which the jet is in the strong nonlinear region and above which it is in the weak nonlinear region does not change with the Weber number. The effect of the Weber number on the wave deformation for ω = 0.2 is presented in Fig. 7. It is apparent that the breakup length increases correspondingly with the increase of the Weber number. This is because the increased Weber number decreases the growth rate distinctly for a fixed frequency. In addition, the amplitude of the swelling and the length and diameter of the ligament remain almost constant for different Weber numbers, causing the main and satellite drops to have the same size. This is because the Weber number has less of an effect on the nonlinear amplitudes. The conclusion agrees with the measurements of the main and satellite drops performed by Rutland and Jameson (1970) and Lafrance (1975), in which no significant dependence of drop sizes on the Weber
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4 (a) 0 -4 320
340
360
z
380
400
4 (b) 0 -4
500
520
540
z
560
580
4 (c) 0 -4
760
780
800
820
z
840
FIG. 7: Spatial evolution of interface deformation for different Weber numbers at Re = 400, ω = 0.2, and η0 = 0.01: (a) We = 200, (b) We = 400, and (c) We = 800
number were reported. It should be pointed out that only the frequency in the strong nonlinear region is considered here since Weber number has no effect on the nonlinear amplitudes in the weak nonlinear region. It can be concluded that a larger Weber number yields a longer breakup length and the same wave shape in the weak nonlinear region. 4.5 Effects of Reynolds Number The effects of Reynolds number on the instability of slender jets are shown in Figs. 8–10. Figure 8 shows the growth ˆ2 and η ˆ 3 as a function of frequency ω at different Reynolds numbers. rate –k1i and the nonlinear initial amplitudes η It can be seen from Fig. 8(a) that the maximum growth rate and dominant frequency increase substantially with the increase of Reynolds number. Thus, a larger Reynolds number enhances the linear instability of the slender jet. Figures 8(b)–8(c) indicate that a larger Reynolds number enlarges the nonlinear amplitudes in the strong nonlinear region but has no effect on the nonlinear amplitudes in the weak nonlinear region. Moreover, the critical frequency reduces noticeably as the Reynolds number is reduced to < 10, which is in agreement with the numerical analysis of Ashgriz and Mashayek (1995), who stated that the variation of drop size with Reynolds number becomes more pronounced for Re < 10. 0.02
k1i
Re = 10 Re = 100 Re = 1000
ˆ2
0.01
0.00 0.0
Re = 10 Re = 100 Re = 1000
4 3
ˆ3
0.4 0.6
(a)
0.8
1.0
3
2
2
1
1
0 0.2
Re = 10 Re = 100 Re = 1000
4
0 0.2
0.4 0.6
(b)
0.8
1.0
0.2
0.4 0.6
0.8
1.0
(c)
FIG. 8: Effect of Reynolds number on (a) growth rate, (b) second-order initial amplitude and (c) third-order initial amplitude at We = 400
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0.01
0.00 0.0
Re = 10 Re = 100 Re = 1000
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ˆ3
0.4 0.6
0.8
3
2
2
1
1
0 0.2
0 0.2
1.0
Re = 10 Re = 100 Re = 1000
4
0.4 0.6
(a)
0.8
1.0
0.2
(b)
0.4 0.6
0.8
1.0
(c)
FIG. 9: Spatial evolution of interface deformation for different Reynolds numbers at We = 400, ω = 0.2, and η0 = 0.01: (a) Re = 5, (b) Re = 10, (c) Re = 30, and (d) Re = 100 40
Re = 0.1 Re = 10 Re = 200
0.05
Linear
30
Re = 0.1 Re = 10 Re = 200
0.05
Nonlinear
lb 20
10 Re = 0.1 Re = 10 Re = 200
0.05
Numerical
0 0.0
0.2
0.4
0.6
0.8
1.0
FIG. 10: Comparison of the breakup length lb predicted by present nonlinear theory with those from the linear theory of Chandrasekhar (1961) and the numerical simulation of Ashgriz and Mashayek (1995) at η0 = 0.05
Figure 9 shows the interface waveforms of a slender jet for ω = 0.2 at different Reynolds numbers. It is apparent that a larger Reynolds number leads to a shorter breakup length and a fatter ligament and, consequently, a larger satellite drop. This can be explained by the effect of the Reynolds number on the growth rate and nonlinear amplitudes discussed previously. For Re > 100, there is no significant change in the ligament with Reynolds number. In addition, for a high-viscosity jet (Re = 5, 10), the ligament connecting the main swellings becomes a tenuous thread as observed in the experiments. This phenomenon originates from the need for an increased pressure difference between the ligament and the main swellings to overcome the dissipative and inhibiting effects of viscosity. Higher viscosity strengthens the inhibiting effects, resulting in a more slender and threadlike ligament to effect detachment of the ligament from the main swellings. Moreover, only for very small Reynolds numbers (Re = 5, 10) does the breakup occur at the middle point of the ligament and, as the Reynolds number increases, the breakup point moves to the end of the ligament. The result is in accordance with the numerical simulation of Ashgriz and Mashayek (1995). Figure 10 shows the breakup lengths for each frequency at different Reynolds numbers. The linear solutions predicted by Chandrasekhar (1961) are obtained using the relation lb = ln(η0 )/k1i . It can be seen that with the increase of the oscillation frequency, the breakup length first decreases until it reaches a minimum and then increases toward infinity. And the breakup lengths predicted by the present nonlinear theory are generally shorter than the linear results, due to the effects of nonlinear disturbances. But the discrepancy between the nonlinear and linear solution becomes very small around the dominant frequency since the fundamental disturbance with the maximum growth rate is much more dominant than the nonlinear disturbances. In addition, the numerical result of Ashgriz and
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Mashayek (1995) is larger than both the linear and nonlinear solutions at low frequencies. As the frequency increases, the numerical result gets close to the linear solution for Re = 200 and Re = 10, but it agrees better with the nonlinear solution for Re = 0.1. 4.6 Comparison to Experiments and Simulations Figure 11 compares breakup length versus initial disturbance amplitude with experiment and numerical simulation conducted by Moallemi et al. (2016), where a relation between the velocity disturbance amplitude δ′0 and initial surface amplitude η0 is given by 3 ′ 1 δ = η0 + (η0 − δ′0 ) (61) 2 0 We Equation (61) is applied to transform δ′0 used by Moallemi et al. (2016) into η0 adopted in the present calculation. Clearly, a logarithmic relation is predicted that is in agreement with the experiment and simulation. Note that it is also verified by Figs. 4(a) and 5(c), which reveal that the nonlinear effect is very weak when ω = 0.697. A comparison between the wave deformations close to the breakup point, predicted by the present nonlinear theory, and the experimental results of Goedde and Yuen (1970) is made in Fig. 12. In the experiment, the liquid jet issued vertically downward into the atmosphere from a round nozzle, and an oscillating voltage between a metal plate concentric with the jet and the jet itself was applied to impose an axisymmetric disturbance on the liquid jet. Pictures of fast-moving liquid jets were taken with a 70 mm rotating drum camera. Photograph (1,1), Photograph (1,3) and Photograph (1,6) of Fig. 3 in their paper were chosen for comparison. As the paper points out, the time interval t0 between each photograph is 1.25 × 10−3 s. In the theory, the parameters are chosen to be identical to the experiment and waveforms just before breakup are presented. The resulting waveform is aligned with the picture of the experimental jets to enhance contrast. It can be seen readily from Fig. 12 that the present theoretical wave deformations coincide nicely with the experimental observations during the early stage of disturbance evolution. However, there exists apparent deviation of the theoretical wave profile from the experiment in the main swelling amplitude and the ligament length for Fig. 12(c). It is because that the border between the main swelling and the ligament is corresponding to a breakup point, in which the nonlinearity is so strong that the effects of the higher-order disturbances should be accounted for. Figure 13 shows the wave deformation predicted by the present theory with the experimental observation of Chaudhary and Maxworthy (1980a) and the numerical simulation of Eggers and Dupont (1994). In the experiment, pressurized fluid is injected at a high speed to form a liquid jet from a round nozzle, where periodic perturbations are applied using a piezoelectric transducer with one face in contact with the fluid. A synchronized light-emittingdiode (LED) stroboscope is used for instantaneous observation of the jet. Eggers and Dupont (1994) modeled the experiment by solving a one-dimensional equation of motion for the velocity and the radius. They developed a fully 140 120
Experiment Numerical Present theory
lb 100 80 60 40 20 0 1E-6 1E-5 1E-4 0.001 0.01 0.1 FIG. 11: Comparison of breakup length versus initial disturbance amplitude with experiment and numerical simulation of Moallemi et al. (2016) at ω = 0.697, Re = 408, and We = 14.8
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FIG. 12: Comparison of our theoretical interface waves (solid line) and experimental observations of Goedde and Yuen (1970). For water jet: k = 0.43, σ = 5.9 × 10−2 N/m, ν = 10−6 m2 /s, R0 = 0.86 mm, and U0 = 3.7 m/s, (a) 6 × t0 before evident pinch occurs; (b) 4 × t0 before evident pinch occurs; and (c) t0 before evident pinch occurs
FIG. 13: Comparison between (a) present theoretical interface wave, (b) the experimental result of Chaudhary and Maxworthy (1980a), and (c) the simulation of Eggers and Dupont (1994) at k = 0.4312 and We = 239
implicit second-order scheme to integrate the equation near the breakup point. It is clear that present nonlinear theory can predict apparent level ligaments and pinch points at the ends of the ligaments, showing satisfactory coincidence with the experiment and the simulation. However, this perturbation analysis cannot precisely describe the evolution of the jet at the breakup point as the simulation does, since the breakup point corresponds to a singularity of the equations of motion, where the velocity and gradients of the local radius break down. 5. CONCLUSIONS A nonlinear spatial instability of a slender viscous jet has been investigated by combining a perturbation analysis with the one-dimensional equations. The mathematical solution of nth-order interface disturbances has been derived and the second- to third-order wave deformations have been plotted. As the perturbation expansion proceeds to third orders, the main swellings are narrowing and the secondary swellings become flatter, forming a level liquid ligament as observed in the experiments. This phenomenon is attributed to the fact that the high-order disturbances eliminate the crests and troughs of the lower order interface waves. Besides, there exist two independent nonlinear regions, i.e., the strong nonlinear region and the weak nonlinear region. The nonlinear amplitudes are large and decrease remarkably with the increased frequency in the strong nonlinear region but remain minimal and invariable in the weak nonlinear region, the discrepancy of which could be due to the resonance between the nonlinear transferred disturbances and the inherent disturbances. In addition, as the Weber number decreases or Reynolds number increases, the growth rate of the slender jet increases significantly. The nonlinear amplitudes increase in the strong nonlinear region but remain
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constant in the weak nonlinear region, resulting in a shorter breakup length but nearly the same wave deformation. The critical frequency, which divides the strong and weak nonlinear region, is not affected by Weber number but decreases obviously as the Reynolds number reduces to < 10. The higher order wave deformations show reasonably better agreement with the experiments and the numerical simulation. ACKNOWLEDGMENTS This work was supported by the China’s National Natural Science Funds for Distinguished Young Scholar (Grant No. 11525207) and China’s National Natural Science Foundation (Grants Nos. 11272036 and 11672025). The authors are grateful to the reviewers for their valuable comments that helped to greatly improve the paper. The authors also thank Luo Xie for stimulating discussions. REFERENCES Amini, G., Lv, Y., Dolatabadi, A., and Ihme, M., Instability of elliptic liquid jets: temporal linear stability theory and experimental analysis, Phys. Fluids, vol. 26, no. 11, p. 114105, 2014. Ardekani, A.M., Sharma, V., and Mckinley, G.H., Dynamics of bead formation, filament thinning and breakup in weakly viscoelastic jets, J. Fluid Mech., vol. 665, no. 12, pp. 46–56, 2010. Ashgriz, N. and Mashayek, F., Temporal analysis of capillary jet breakup, J. Fluid Mech., vol. 291, no. 5, pp. 163–190, 1995. Basaran, O.A., Gao, H., and Bhat, P.P., Nonstandard inkjets, Annu. Rev. Fluid Mech., vol. 45, pp. 85–113, 2013. Bechtel, S.E., Carlson, C.D., and Forest, M.G., Recovery of the Rayleigh capillary instability from slender 1-D inviscid and viscous models, Phys. Fluids, vol. 7, no. 12, pp. 2956–2971, 1995. Bogy, D.B., Drop formation in a circular liquid jet, Annu. Rev. Fluid Mech., vol. 11, no. 1, pp. 207–228, 1979. Bogy, D.B., Use of one-dimensional cosserat theory to study instability in a viscous liquid jet, Phys. Fluids, vol. 21, no. 2, pp. 190– 197, 1978. Brenner, M.P., Lister, J.R., and Stone, H.A., Pinching threads, singularities and the number 0.0304..., Phys. Fluids, vol. 8, no. 11, pp. 2827–2836, 1996. Busker, D.P., Lamers, A.T., and Nieuwenhuizen, J.K., The non-linear break-up of an inviscid liquid jet using the spatial-instability method, Chem. Eng. Sci., vol. 44, no. 2, pp. 377–386, 1989. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford, UK: Clarendon Press (Oxford University Press), 1961. Chang, Q., Zhang, M., Bai, F., Wu, J., Xia, Z., Jiao, K., and Du, Q., Instability analysis of a power law liquid jet, J. Non-Newtonian Fluid Mech., vol. 198, pp. 10–17, 2013. Chaudhary, K.C. and Maxworthy, T., The nonlinear capillary instability of a liquid jet part 2: Experiments on jet behaviour before droplet formation, J. Fluid Mech., vol. 96, no. 2, pp. 275–286, 1980a. Chaudhary, K.C. and Maxworthy, T., The nonlinear capillary instability of a liquid jet part 3: Experiments on satellite drop formation and control, J. Fluid Mech., vol. 96, no. 2, pp. 287–297, 1980b. Chaudhary, K.C. and Redekopp, L.G., The nonlinear capillary instability of a liquid jet part 1: Theory, J. Fluid Mech., vol. 96, no. 2, pp. 257–274, 1980. Chesnokov, Y.G., Nonlinear development of capillary waves in a viscous liquid jet, Tech. Phys., vol. 45, no. 8, pp. 987–994, 2000. Donnelly, R.J. and Glaberson, W., Experiments on the capillary instability of a liquid jet, Proc. R. Soc. A, vol. 290, no. 1423, pp. 547–556, 1966. Eggers, J. and Dupont, T.F., Drop formation in a one-dimensional approximation of the Navier–Stokes equation, J. Fluid Mech., vol. 262, no. 10, pp. 205–221, 1994. Eggers, J. and Villermaux, E., Physics of liquid jets, Rep. Prog. Phys., vol. 71, no. 3, p. 036601, 2008. Eggers, J., Drop formation—An overview, ZAMM-Z. Angew. Math. Mech., vol. 85, no. 6, pp. 400–410, 2005. Eggers, J., Nonlinear dynamics and breakup of free-surface flows, Rev. Mod. Phys., vol. 69, no. 3, pp. 865–930, 1997. Eggers, J., Theory of drop formation, Phys. Fluids, vol. 7, no. 5, pp. 941–953, 1995.
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Yang et al.
Eggers, J., Universal pinching of 3D axisymmetric free-surface flow, Phys. Rev. Lett., vol. 71, no. 21, pp. 3458–3460, 1993. Garcia, F.J. and Castellanos, A., One-dimensional models for slender axisymmetric viscous liquid bridges, Phys. Fluids, vol. 6, no. 11, pp. 2676–2689, 1994. Goedde, E.F. and Yuen, M.C., Experiments on liquid jet instability, J. Fluid Mech., vol. 40, no. 3, pp. 495–511, 1970. Gorman, J.M., Sparrow, E.M., and Abraham, J., Slot jet impingement heat transfer in the presence of jet-axis switching, Int. J. Heat Mass Transfer, vol. 78, pp. 50–57, 2014. Green, A.E., On the non-linear behaviour of fluid jets, Int. J. Eng. Sci., vol. 14, no. 1, pp. 49–63, 1976. Keller, J.B., Rubinow, S.I., and Tu, Y.O., Spatial instability of a jet, Phys. Fluids, vol. 16, no. 12, pp. 2052–2055, 1973. Lafrance, P., Nonlinear breakup of a laminar liquid jet, Phys. Fluids, vol. 18, no. 4, pp. 428–432, 1975. Lee, H.C., Drop formation in a liquid jet, IBM J. Res. Dev., vol. 18, no. 4, pp. 364–369, 1974. Leib, S.J. and Goldstein, M.E., The generation of capillary instabilities on a liquid jet, J. Fluid Mech., vol. 168, no. 7, pp. 479–500, 1986a. Leib, S.J. and Goldstein, M.E., Convective and absolute instability of a viscous liquid jet, Phys. Fluids, vol. 29, no. 4, pp. 952–954, 1986b. Li, F., Yin, X.Y., and Yin, X.Z., Oscillation of satellite droplets in an Oldroyd-B viscoelastic liquid jet, Phys. Rev. Fluids, vol. 2, no. 1, p. 013602, 2017. Li, J. and Fontelos, M.A., Drop dynamics on the beads-on-string structure for viscoelastic jets: A numerical study, Phys. Fluids, vol. 15, no. 4, pp. 922–937, 2003. Lin, S.P. and Lian, Z.W., Absolute instability of a liquid jet in a gas, Phys. Fluids A, vol. 1, no. 3, pp. 490–493, 1989. Lin, S.P., Breakup of Liquid Sheets and Jets, New York, NY: Cambridge University Press, 2003. Lunardelli, A., Wentz, J.E., Abraham, J.P., and Plourde, B.D., Shielding nozzle design and analysis for atomization-based cutting fluid systems in micromachining, J. Micro Nano-Manuf., vol. 3, no. 2, p. 024501, 2015. Moallemi, N., Li, R., and Mehravaran, K., Breakup of capillary jets with different disturbances, Phys. Fluids, vol. 28, no. 1, p. 012101, 2016. Nayfeh, A.H., Nonlinear stability of a liquid jet, Phys. Fluids, vol. 13, no. 4, pp. 841–847, 1970. Oliveira, M.S. and McKinley, G.H., Iterated stretching and multiple beads-on-a-string phenomena in dilute solutions of highly extensible flexible polymers, Phys. Fluids, vol. 17, no. 7, p. 071704, 2005. Peregrine, D.H., Shoker, G., and Symon, A., The bifurcation of liquid Bridges, J. Fluid Mech., vol. 212, no. 3, pp. 25–39, 1990. Pimbley, W.T. and Lee, H.C., Satellite droplet formation in a liquid jet, IBM J. Res. Dev., vol. 21, no. 1, pp. 21–30, 1977. Pimbley, W.T., Drop formation from a liquid jet: A linear one-dimensional analysis considered as a boundary value problem, IBM J. Res. Dev., vol. 20, no. 2, pp. 148–156, 1976. Rayleigh, L., On the instability of jets, P. Lond. Math Soc., vol. 10, no. 1, pp. 4–13, 1878. Ruo, A., Chen, K., Chang, M., and Chen, F., Instability of a charged non-Newtonian liquid jet, Phys. Rev. E, vol. 85, no. 1, p. 016306, 2012. Rutland, D.F. and Jameson, G.J., A non-linear effect in the capillary instability of liquid jets, J. Fluid Mech., vol. 46, no. 2, pp. 267–271, 1971. Rutland, D.F. and Jameson, G.J., Theoretical prediction of the sizes of drops formed in the breakup of capillary jets, Chem. Eng. Sci., vol. 25, no. 11, pp. 1689–1698, 1970. Sirignano, W.A. and Mehring, C., Review of theory of distortion and disintegration of liquid streams, Prog. Energy Combust., vol. 26, no. 4, pp. 609–655, 2000. Sterling, A.M. and Sleicher, C.A., The instability of capillary jets, J. Fluid Mech., vol. 68, no. 3, pp. 477–495, 1975. Taub, H.H., Investigation of nonlinear waves on liquid jets, Phys. Fluids, vol. 19, no. 8, pp. 1124–1129, 1976. Tharakan, T.J., Mukhopadhyay, A., Datta, A., and Jog, M.A., Trends in comprehensive modeling of spray formation, Int. J. Spray Combust., vol. 5, no. 2, pp. 123–180, 2013. Tirel, C., Renoult, M.C., Dumouchel, C., Lisiecki, D., Crumeyrolle, O., and Mutabazi, I., Multi-scale analysis of a viscoelastic liquid jet, J. Non-Newtonian Fluid Mech., vol. 245, pp. 1–10, 2017.
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Tomotika, S., On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid, Proc. R. Soc. A-Math. Phys. Eng. Sci., vol. 150, no. 870, pp. 322–337, 1935. Weber, C., Zum Zerfall eines Fl¨ussigkeitsstrahles, ZAMM-Z. Angew. Math. Mech., vol. 11, no. 2, pp. 136–154, 1931. Xie, L. and Yang, L., Axisymmetric and non-axisymmetric instability of a charged viscoelastic jet under an axial magnetic field, J. Non-Newtonian Fluid Mech., vol. 248, pp. 92–98, 2017. Yuen, M., Non-linear capillary instability of a liquid jet, J. Fluid Mech., vol. 33, no. 1, pp. 151–163, 1968.
APPENDIX A. PARTIAL DIFFERENTIAL OPERATORS IN GOVERNING EQUATIONS 3 3 1 2 2 fzzzz g + fzzz ggz + fzz 3gz + 2ggzz + fz (3gz gzz + ggzzz ) L (f, g) = Re 8 2 " # 3 gz gzz 1 (gz ) 1 gzzz + + [3fz (gz gt − ggzt ) − + We g 2g 2 4 − fzt ggz + 3f fz gz2 − ggzz − ggz f fzz + fz2
1 wz 1 η δ2 1 w, 1 η = −1 w1 ηz − 3 wzzz 1 η + 101 wzz 1 ηz + 61 wz 1 ηzz − 2 16 1 1 1 1 2 η ηz 1 1 1 1 1 6 wz ηz 1 2 g 1 w, 1 η = + w wz − − δ2 −31 wz 1 ηzt − 1 wzt 1 ηz − 31 wz 1 ηzz ηz ηzz + We Re We 4 31 3 11 1 1 1 1 1 1 1 1 wzzzz η + wzzz ηz + 2 wzz ηzz + wz ηzzz − wzz ηz + Re 4 2 2
f
1
δ2 1 1 1 2 wz η + 2 wz 1 η − 3 wzzz 2 η + 2 wzzz 1 η w, 1 η, 2 w, 2 η = −1 w2 ηz − 2 w1 ηz − 2 16 + 10 1 wzz 2 ηz + 2 wzz 1 ηz + 6 1 wz 2 ηzz + 2 wz 1 ηzz + 31 wzzz 1 η2 + 201 wzz 1 η1 ηz + 61 wz 21 η1 ηzz + 31 η2z 6 1 wz 2 ηz + 2 wz 1 ηz − 1 wz 1 ηz 1 η 2 1 η2 ηz + 2 η1 ηz − 31 η2 · 1 ηz 3 1 2 2 1 + g = w wz + w wz − Re We 3 1 3 1 − δ2 2 1 wzzzz 2 η + 2 wzzzz 1 η + 1 wzzzz 1 η2 + wzzz 2 ηz + 2 wzzz 1 ηz + 1 wzzz 1 ηz 1 η Re 8 2 1 1 2 2 1 1 2 1 1 + 2 wzz ηzz + wzz ηzz + wzz 3 ηz + 2 ηzz η + 1 wz 2 ηzzz + 2 wz 1 ηzzz + 1 wz 31 ηz 1 ηzz + 1 η1 ηzzz 1 3 1 1 η 1 1 2 3 wz 1 ηz 1 ηt − 1 η1 ηzt ηz ηzz + 2 ηz 1 ηzz − 1 ηz 1 ηzz 1 η − z + + We 2 4 1 2 2 1 1 2 2 1 1 − wz ηzt + wz ηzt − wzt ηz + wzt ηz + wzt 1 ηz 1 η + 3 1 wz 1 η2z − 1 η1 ηzz − 1 w1 ηzz − 1 wz 2 ηzz + 2 wz 1 ηzz − 1 wzz 2 ηz + 2 wzz 1 ηz + 1 ηz 1 w1 wzz + 1 η1 wzz + 1 wz2 3
f
(A.1)
(A.2)
(A.3)
1
(A.4)
(A.5)
APPENDIX B. NONLINEAR TERMS IN SECOND-AND-THIRD-ORDER EQUATIONS 2,2,1
Volume 27, Issue 12, 2017
fˆ = −ik1
3 19δ2 2 − k 2 16 1
1,1
ˆ w ˆ · 1,1 η
(B.1)
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Yang et al.
3 2 5 ¯ 3 ¯2 1,1 k1 2 ˆ ¯ ˆ w ˆ · 1,−1 η + δ k1 − k1 + k1 k1 − k1 f = −i −k1 + 2 16 8 8 2 2 2 6 ik1 2,2,1 2 2 57k1 ˆ+ ˆ + ik1 1,1 w gˆ = −k1 − ˆ · 1,1 η ˆ +δ + i (k1 − ω) 1,1 w 2 + δ2 k12 1,1 η Re 4Re We 1,1 i 3 k13 3k12 k¯1 6 ¯ 2 2 3 2 2,0,1 ¯ ¯ ¯ ¯ ¯ ˆ k1 + δ − + 2k1 k1 − k1 + −k1 k1 + 3k1 − 2k1 ω w ˆ · 1,−1 η gˆ = −k1 Re Re 4 2 4 ik1 ˆ + ik1 1,1 w ˆ · 1,−1 η + 2 − δ2 k1 k¯1 1,1 η ˆ · 1,−1 w ˆ We 2 5 47δ2 2 1,1 19δ2 2 2,2,1 53δ2 2 1,1 3,3,1 ˆ 2,2,1 1,1 1,1 ˆ+ 2− ˆ− ˆ f = −ik1 w ˆ· w ˆ· η η − k k k w ˆ· η 2 16 1 4 1 16 1 1,1 δ2 k1 k1 2 2 3,3,2 ˆ ˆ w ˆ · 2,2 η − 3k1 + 10k1 k2 + 6k2 f = −i k2 + 2 16 δ2 k2 k2 ˆ − 3k22 + 10k1 k2 + 6k12 2,2 w ˆ · 1,1 η − i k1 + 2 16 1,1 ik1 δ2 3,1,1 ˆ 3 2 2 ˆ ˆ + 2,0,1 η f = 2k1i − w ˆ · 2,0,1 η − −3ik1 + 20k1 k1i + 24ik1 k1i 2 16 1,−1 δ2 ik¯1 3 2 2 ¯ ¯ ¯ ˆ w ˆ · 2,2,1 η 3ik1 − 20ik1 k1 + 24ik1 k1 − + −2ik1 + 2 16 δ2 3 2 ˆ · 2,0,1 w ˆ ˆ + 2,0,1 w −24k1i + 40ik1 k1i + 12k12 k1i 1,1 η − ik1 − k1i + 16 δ2 ˆ + ik¯1 − ik1 − ˆ · 1,−1 η −24ik13 + 40ik12 k¯1 − 12ik1 k¯12 2,2,1 w 16 δ2 ˆ ˆ · 1,−1 η − 40k12 k1i − 12ik1 k¯12 − 18ik13 + 36ik12 k¯1 1,1 w ˆ · 1,1 η 16 2 δ2 ˆ ˆ · 1,1 η 3ik¯13 − 20ik1 k¯12 + 30ik¯1 k12 1,−1 w − 16 δ2 k¯1 ¯2 k¯1 3,1,2 ˆ ˆ ˆ · 2,2 η + 3k1 − 10k¯1 k2 + 6k22 1,−1 w f = −i k2 − 2 16 δ2 k2 k2 ˆ ˆ · 1,−1 η + −3k22 + 10k¯1 k2 − 6k¯12 2,2 w − i −k¯1 + 2 16 12 231 2 7i 3,3,1 ˆ gˆ = k12 ˆ · 2,2,1 η − δ2 k1 + (k1 − ω) 1,1 w Re 4Re 2 78 2 5i 12 ˆ − δ2 k1 + (k1 − ω) 2,2,1 w ˆ · 1,1 η + k12 Re Re 2 2 6 255 2 i ˆ − k12 + δ2 k1 + (k1 − ω) 1,1 w ˆ · 1,1 η Re 8Re 4 3 3δ2 k12 ik1 1,1 ˆ− 3+ ˆ · 2,2,1 η ˆ η 6 + 6δ2 k12 1,1 η + We 2 2 5δ i 3 1,1 2 1,1 ˆ + 3ik1 1,1 w w ˆ · η ˆ · 2,2,1 w ˆ k − 4 1 2,0,1
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
Atomization and Sprays
Nonlinear Spatial Instability of a Slender Viscous Jet
1061
1,1 ik1 3k1 k13 3k 2 k2 6k1 k2 ˆ − δ2 + 1 + 2k11 k22 + k23 + −7k2 ω + 3k22 + k1 k2 w ˆ · 2,2 η Re Re 4 2 4 2,2 ik2 3k2 k23 3k1 k22 6k1 k2 ˆ −5k1 ω + 3k12 + k1 k2 w ˆ · 1,1 η − δ2 + + 2k12 k2 + k13 + + Re Re 4 2 4 " # ˆ · 2,2 η ˆ 1,1 2 + δ2 k1 k2 1,1 η 2,2 + i (k1 + k2 ) + w ˆ· w ˆ We 1,−1 i 3 k¯13 3k¯12 k2 6k2 2 2 2 3 3,1,2 ¯ ¯ ¯ ˆ 5k2 ω − 3k2 + k1 k2 w ˆ · 2,2 η +δ − + 2k1 k2 − k2 + gˆ = −k1 Re Re 4 2 4 3 ¯ 2,2 6k1 3 k2 3k¯1 k22 i ˆ w ˆ · 1,−1 η − k2 + δ2 − + 2k¯12 k1 − k¯13 + −k¯1 ω + 3k¯12 − k¯1 k2 Re Re 4 2 4 " # 2 − δ2 k¯1 k2 1,−1 η ˆ · 2,2 η ˆ 1,−1 ¯ + i k2 − k1 + w ˆ · 2,2 w ˆ We 3 k14 1 12ik1 k1i 2 3 2 3,1,1 − δ2 + 3ik13 k1i − 8k12 k1i − 8ik1 k1i + k1 k1i ω − 6ik1 k1i gˆ = Re Re 4 2 12k k¯ 1,1 3 k¯14 1 1 ˆ + 2,0,1 η ˆ − − k12 k1i w ˆ · 2,0,1 η + δ2 − 3k1 k¯13 + 8k12 k¯12 − 8k13 k¯1 Re Re 4 12ik1 k1i i 1,−1 ˆ+ w ˆ · 2,2,1 η 5k1 k¯1 ω − 6k12 k¯1 + k1 k¯12 + 2 Re 1 3 4 3 2 4k1i − 12ik1 k1i − 8k12 k1i + 2ik13 k1i + 3k1 k1i ω + 3k12 k1i − δ2 Re 2 12k k¯ 1,1 2,0,1 3 1 1 2 2 ˆ· w ˆ + 2,0,1 w η − 2ik1 k1i ˆ − +δ 4k14 − 12k13 k¯1 Re Re 2,2,1 i ˆ −k1 k¯1 ω + 3k1 k¯12 − 2k12 k¯1 w ˆ · 1,−1 η + 8k12 k¯12 − 2k1 k¯13 + 2 3 19 4 21 3 ¯ 1 12ik1 k1i 2 2 ¯2 3 ¯ +δ k − k1 k1 + 5k1 k1 − k1 k1 + −5ik1 k¯1 ω − Re Re 4 1 2 4 1,1 6k1 k¯1 1,1 1,−1 2 2¯ 2 3 ¯ ˆ· ˆ+ w ˆ· η η − 7ik1 ω + 5ik1 k1 + 3ik1 k1 + 4ik1 Re 4 1,−1 2 1 3 ¯3 3 k1 2¯ 3¯ 2 ¯2 2 ¯ ˆ −ik1 k1 ω + ik1 k w ˆ 1,1 η − k1 k1 + 5k1 k1 − 4k1 k1 + −δ Re 8 2 4 1 2 ˆ · 2,0,1 η ˆ + 2,0,1 η ˆ + 2k12 k1i 1,1 η 4k1i − 2ik1 + δ2 4ik1 k1i − We 2 ¯ 1,−1 2,2,1 2 ¯ ¯ ˆ· ˆ − 3i k1 − 2k1 + δ −ik13 − ik1 k¯12 + 4ik1 − 2ik1 −1 + δ k1 k1 η η 2 δ2 h 5 2¯ 1,−1 1,1 ˆ ˆ· ˆ ˆ · 1,−1 wˆ · 1,1 η − η η 4ik13 − 5ik12 k¯ + ik1 k¯12 1,1 w + ik1 k1 2 4 i 2 ˆ + 2ik1 − ik¯1 1,−1 w + 3ik1 k¯12 − 2ik12 k¯1 1,1 w ˆ 1,−1 η ˆ · 2,2,1 w ˆ ˆ + 2,0,1 w ˆ + (ik1 − 2k1i ) 1,1 w ˆ · 2,0,1 w
3,3,2
gˆ =
where
1,−1
Volume 27, Issue 12, 2017
ˆ = (1,1 η ˆ), η
1,−1
w ˆ = (1,1 w) ˆ
(B.10)
(B.11)
(B.12)
