Non-linear capillary instability of a liquid jet

Yuen, M. C.

doi:10.1017/s0022112068002429

Cited by 247 publications

(118 citation statements)

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“…Linear theory predicts well the disturbance growth rate measured by Donnelly and Glaberson (1966) and Goedde and Yuen (1970); however, linear analyses cannot explain the formation of satellite droplets (i.e., small droplets that accompany the primary train of larger droplets) that are observed in both liquid-liquid and liquid-gas systems. Yuen (1968) was the first to analyze the formation of non-uniform-size droplets. He developed a third-order perturbation solution for a cylindrical, inviscid, liquid jet in gas, and showed that non-uniform-size droplets form as results of nonlinear effects which were neglected in Rayleigh's analysis.…”

Section: Theoretical Studiesmentioning

confidence: 99%

Laboratory Experiments to Simulate Co2 Ocean Disposal

Masutani¹

1999

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Section: Theoretical Studiesmentioning

confidence: 99%

Laboratory Experiments to Simulate Co2 Ocean Disposal

Masutani¹

1999

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“…the wavelength with maximum growth rate, is often used to get a good approximation for the main drop size in liquid jet breakup. The effect of nonlinearities was first studied by Yuen [1]. In his weakly nonlinear analysis of an inviscid jet in a vacuum he described the jet interface shape up to order three at small initial deformation amplitude.…”

Section: Introductionmentioning

confidence: 99%

“…Thereafter we solve the equations derived in the sequence of the order and present our method of approximation of one part of the viscous contribution containing products of Bessel functions with different arguments. Results on surface shapes are then presented and the effect of both the liquid viscosity and the level of approximation discussed by comparison to the inviscid solution of Yuen [1]. The paper ends with the conclusions.…”

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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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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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“…Solving the second-order equations with the restriction to inviscid jet liquid yields the results of Yuen [3]. Taking the divergence of the vectorial second-order momentum equation for the viscous case, the following Poisson equation for the second-order pressure p 2 is obtained:…”

mentioning

confidence: 95%

“…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.…”

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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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Non-linear capillary instability of a liquid jet

Cited by 247 publications

References 1 publication

Laboratory Experiments to Simulate Co2 Ocean Disposal

Laboratory Experiments to Simulate Co2 Ocean Disposal

Weakly nonlinear instability of a viscous liquid jet

Nonlinear instability of a viscous liquid jet

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