Weakly nonlinear instability of planar viscous sheets

Yang, Lijun; Wang, Chen; Fu, Qing-fei; Du, Mingyue; Tong, Ming-Xi

doi:10.1017/jfm.2013.502

Cited by 31 publications

(23 citation statements)

References 46 publications

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“…By including the viscous stresses in the liquid, the model complements the results of Yuen [1] for the inviscid case and Yang and co-authors [2] for a plane viscous liquid sheet. In a weakly nonlinear analysis, velocity, pressure and jet surface shape are expanded in series with respect to a small deformation parameter, here the initial amplitude of the jet deformation, yielding a set of equations with different powers of the parameter.…”

Section: Discussionmentioning

confidence: 56%

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Weakly nonlinear instability of a viscous liquid jet

Renoult¹,

Brenn²,

Mutabazi³

2017

Proceedings ILASS–Europe 2017. 28th Conference on Liquid Atomization and Spray Systems

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A weakly nonlinear stability analysis of an axisymmetric viscous liquid jet is performed. The calculation is based on a small-amplitude perturbation method and restricted to second order. Contrary to the inviscid jet and the planar viscous sheet cases studied by Yuen in 1968 [1] and Yang et al. in 2013 [2], respectively, a part of the solution results from a polynomial approximation of Bessel functions. Results on interface shapes for a small wave number and initial perturbation amplitude, four different Ohnesorge numbers, taking into account the approximate part or not, are used to predict the influence of liquid viscosity on satellite drop formation and evaluate the influence of the approximation. It is observed that the liquid viscosity has a retarding effect on satellite drop formation, in agreement with previous experimental and numerical work. In addition, it is found that the approximate terms can be reasonably ignored, providing a simpler viscous weakly nonlinear model for the description of the first nonlinearity growth in liquid jets. The present work replaces the ILASS 2016 paper [3] by the authors on the same subject. KeywordsViscous liquid jet, nonlinear capillary instability, satellite drop formation. IntroductionA liquid jet breaks up forming main and satellite drops. The present work is concerned with the influence of both liquid viscosity and nonlinearities on satellite drop formation. The first linear stability analysis of the capillary instability of a liquid jet in an ambient medium was conducted by Rayleigh [4,5], more than a century ago. In this reference work, the liquid is assumed inviscid and the ambient medium is the vacuum. His analysis shows that, in order to destabilize the jet, the wavelength λ of a varicose surface disturbance must be greater than the circumference of the undeformed circular jet cross section, as observed initially by Savart and Plateau [6,7]. Two associated amplitude growth rates correspond to such an unstable perturbation wavelength, one being the opposite of the other, with the magnitude given by the well-known Rayleigh's linear dispersion relation for the inviscid jet in a vacuum. The effect of liquid viscosity was then investigated by Weber [8], about fifty years later. The generalization of the previous dispersion relation introduces a growth rate and viscosity dependent modified wave number making the dispersion relation transcendental, yet numerically solvable, which is turned into a closed-form expression in the long-wave approximation. Contrary to the inviscid case, there are two different real growth rates with opposite signs for each unstable wavelength, with distinct absolute values below the corresponding inviscid one. This difference leads to dispersion relation curves always lower than the inviscid ones, as expected by the classical damping effect inferred from liquid viscosity. Due to the linearity of the previous analysis, the interaction of disturbances with different wavelengths is not accounted for, and the drops produced by the jet br...

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Section: Discussionmentioning

confidence: 56%

“…sheet stability [2]. In a 2016 ILASS paper [3], the jet geometry was considered by the present authors, introducing the need for a polynomial approximation of one part of the viscous contribution, that could not be fully solved analytically, due to the presence of Bessel function products with different arguments.…”

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confidence: 99%

Weakly nonlinear instability of a viscous liquid jet

Renoult¹,

Brenn²,

Mutabazi³

2017

Proceedings ILASS–Europe 2017. 28th Conference on Liquid Atomization and Spray Systems

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“…Early nonlinear studies on the capillary instability of inviscid liquid jets were carried up to the third order contributions to the jet deformation and showed the nonlinear interaction between different modes. A recent study on the weakly nonlinear instability of planar Newtonian liquid sheets revealed the role of the liquid viscosity in the sheet stability behavior and showed a complicated influence [1]. Here, the instability of a liquid jet is examined as the axisymmetric counterpart of the sheet, in search for corresponding insight into the role of the liquid viscosity in the jet instability mechanism.…”

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confidence: 99%

Nonlinear instability of a viscous liquid jet

Renoult

Brenn

Mutabazi

2016

Proc Appl Math and Mech

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We investigate the weakly nonlinear temporal instability of an axisymmetric Newtonian liquid jet. Early nonlinear studies on the capillary instability of inviscid liquid jets were carried up to the third order contributions to the jet deformation and showed the nonlinear interaction between different modes. A recent study on the weakly nonlinear instability of planar Newtonian liquid sheets revealed the role of the liquid viscosity in the sheet stability behavior and showed a complicated influence [1]. Here, the instability of a liquid jet is examined as the axisymmetric counterpart of the sheet, in search for corresponding insight into the role of the liquid viscosity in the jet instability mechanism. The weakly nonlinear instability of an incompressible, Newtonian axisymmetric liquid jet is studied as an alternative to and for comparison with numerical simulations such as in [2]. The problem is formulated in cylindrical coordinates. Body forces are not accounted for, since the Froude number is large. The variables and equations of motion are non-dimensionalized with the undeformed jet radius a, the capillary time-scale (ρa 3 /σ) 1/2 and the pressure σ/a where ρ is the liquid density and σ the liquid-air surface tension. The jet surface is described as a place where r(z, t) = 1 + η(z, t), where η is the deformation against the undisturbed cylindrical shape. The initial surface disturbance is assumed to be purely sinusoidal with amplitude η 0 and wavenumber k. With this hypothesis, mass conservation leads to the expression η(z, 0) = η 0 cos kz + (1 − η 2 0 /2) 1/2 − 1 for the initial deformation [3]. As usual in weakly nonlinear analysis, the initial deformation is supposed to be small, so that η 0 ≪ 1. This allows the velocity components, the pressure field and the deformed interface shape to be represented by series expansions with respect to η 0 . The expansion for the axial flow velocity, as an example, reads u z = u z1 η 0 + u z2 η 2 0 + . . . . Introducing the expansions into the equations of motion and into the boundary conditions, we obtain equations determining the linear problem in terms of the quantities with subscript 1, and the second-order problem by quantities with subscript 2. The related equations are linear in the quantities of the respective order, but non-linear in the quantities of the next lower order.The first-order contribution to the jet surface deformation is formulated as η 1 =η 1 exp(ikz − α 1 t), where α 1 is a complex frequency andη 1 is the first-order initial amplitude fixed to 1/4 by the initial condition. Representing the first-order velocities as derivatives of a disturbance stream function ψ, and taking the curl of the first-order momentum equation, the fourth-order partial differential equationfor the disturbance stream function is obtained, where E 2 = r∂/∂r((1/r)∂/∂r) + ∂ 2 /∂z 2 . Its solution readswhere I 1 is the first-order modified Bessel function of the first kind. We have discarded the modified Bessel functions of the second kind for regularity of the solution on the...

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“…The asymptotic analytical perturbation solution for weakly nonlinear instability of the linear sinuous mode was investigated in the study of Clark & Dombrowski (1972), Jazayeri & Li (2000) and Yang et al (2013). It was found that the first harmonic of the linear sinuous mode is varicose, which explains the disintegration of sinuous waves.…”

mentioning

confidence: 99%

“…In the theoretical analysis of Squire (1953), Clark & Dombrowski (1972), Asare, Takahashi & Hoffman (1981), Jazayeri & Li (2000) and Yang et al (2013), only the linear sinuous mode was investigated, and in the experiments of Squire (1953), Hagerty & Shea (1955), Crapper, Dombrowski & Pyott (1975), Asare et al (1981) and Tammisola et al (2011), the waves were evidently characterized by sinuous characters. However, Mitra, Li & Renksizbulut (2001) offered a different opinion.…”

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confidence: 99%

Nonlinear dual-mode instability of planar liquid sheets

Wang

Yang

2015

J. Fluid Mech.

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The nonlinear temporal instability of gas-surrounded planar liquid sheets, whose linear instability contains both sinuous and varicose modes, is studied. Both the weakly nonlinear analysis using a second-order perturbation expansion and the numerical simulation using a boundary integral method have been applied. Their comparison shows that the weakly nonlinear analysis can precisely predict the shapes of sheets for most of the time of disturbance evolution and qualitatively explain the instability mechanism when sheets break up. Both the first harmonics of the linear sinuous mode and linear varicose mode are varicose; they contribute to the breakup of sheets, but the first harmonic generated by the coupling between the linear sinuous and varicose modes is sinuous; it plays an important role in modulating the wave profile. The instability with various initial phase differences between the upper and lower interfaces is examined. Except for the varicose initial disturbance, the linear sinuous mode dominates in the shapes of sheets when their amplitudes grow large. Within the second-order analysis, the major modes that can cause the breakup include the linear varicose mode, the first harmonic of the linear sinuous mode and the first harmonic of the linear varicose mode. The effects of various flow parameters have been investigated. At relatively large wavenumbers where approximate analytical and numerical results agree well when sheets break up, increasing the wavenumber reduces the wave amplitude. Reducing the initial disturbance amplitude makes the first harmonic of the linear sinuous mode the dominant mode in causing the breakup. Increasing the Weber number or gas-to-liquid density ratio significantly reduces breakup time and enhances instability.

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Weakly nonlinear instability of planar viscous sheets

Cited by 31 publications

References 46 publications

Weakly nonlinear instability of a viscous liquid jet

Weakly nonlinear instability of a viscous liquid jet

Nonlinear instability of a viscous liquid jet

Nonlinear dual-mode instability of planar liquid sheets

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