Drop formation in a one-dimensional approximation of the Navier–Stokes equation

Eggers, Jens; Dupont, Todd

doi:10.1017/s0022112094000480

Cited by 492 publications

(542 citation statements)

References 19 publications

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“…Downstream of breakup point, the jet solution no longer has physical meaning, since the jet in that region will have broken up into droplets, which cannot be described by this approach, as is also the case in other works. [6][7][8]11 The solutions obtained can be put on the physical plane x − z in the following way:…”

Section: Methodsmentioning

confidence: 99%

“…Gravity was included in Decent et al 3 Nonlinear one-dimensional models for axisymmetric straight jets have been developed, by assuming a periodic disturbance along the infinite jet, by many authors (see Lee,4 Mansour and Lundgren, 5 Schulkes, 6 Papageorgiou and Orellana 7 ). The presence of the orifice has also been included, first by Keller et al 33 , and in recent jet simulations, which consider the jet having a finite length (see Eggers and Dupont, 8 Hilbing and Heister, 9 Cheong 10 ). An extensive review of the work on straight axisymmetric liquid jets is given by Eggers 11 and Vanden-Broeck.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear travelling waves on a spiralling liquid jet

et al. 2006

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We describe the nonlinear evolution of a travelling wave disturbance on a spiralling slender inviscid jet which emerges from a rotating orifice neglecting gravity. One-dimensional equations are derived using asymptotic methods and solved numerically. Some results are presented for this nonlinear theory which is rather different from previous linear theories, showing the influence of surface tension and rotation on the breakup of the jets into droplets. Comparison with the experimental results shows good qualitative agreement.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonlinear travelling waves on a spiralling liquid jet

et al. 2006

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“…Eggers and Dupont [8] argued that a similarity solution of this form does indeed hold for the problem at hand. In fact they argue that such a solution is in good agreement with solutions to the Navier-Stokes equation (or approximations thereof) for low viscosities.…”

Section: Breaking Dropletsmentioning

confidence: 96%

“…7 Often in attempting to reduce a theory T c to T f in the philosopher's sense we find that we need to amend or correct the reduced theory T c . 8 Most physicists, now, would accept the idea that our concept of temperature and our conception of other "exact" terms that appear in classical thermodynamics such as "entropy", need to be modified in light of the alleged reduction to statistical mechanics. Textbooks, in fact, typically speak of the theory of "statistical thermodynamics" allowing explicitly for fluctuations that one observes in thermodynamic systems.…”

Section: A More Sophisticated Attempt At Reductionmentioning

confidence: 99%

Critical phenomena and breaking drops: Infinite idealizations in physics

Batterman

2005

Studies in History and Philosophy of Science Part B: Studies in

106

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“…In the present work, we follow the analysis by Zhang, Padgett & Basaran (1996) and use the one-dimensional model introduced by Eggers & Dupont (1994) where the full curvature term is kept. This model has a long history which goes back to Saint Venant and Cosserat (see, for instance Bogy 1978;Meseguer 1983).…”

Section: Introductionmentioning

confidence: 99%

Forced dynamics of a short viscous liquid bridge

2014

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The dynamics of an axisymmetric liquid bridge of a fluid of density ρ, viscosity µ and surface tension σ held between two co-axial disks of equal radius e is studied when one disk is slowly moved with a velocity U(t). The analysis is performed using a one-dimensional model for thin bridges. We consider attached boundary conditions (the contact line is fixed to the boundary of the disk), neglect gravity and limit our analysis to short bridges such that there exists a stable equilibrium shape. This equilibrium is a Delaunay curve characterized by the two geometric parameters 0 = V 0 /(πe 3 ) and S = (πe 2 )/V 0 , where V 0 is the volume of fluid and the length of the bridge. Our objective is to analyse the departure of the dynamical solution from the static Delaunay shape as a function of 0 , S, the Ohnesorge number Oh = µ/ √ ρσ e, and the instantaneous velocity U(t) and acceleration ∂ t U (non-dimensionalized using e and σ /ρ) of the disk. Using a perturbation theory for small velocity and acceleration, we show that (i) a non-homogeneous velocity field proportional to U(t) and independent of Oh is present within the bridge; (ii) the area correction to the equilibrium shape can be written as U 2 A i + U Oh A v + ∂ t UA a where A i , A v and A a are functions of 0 and S only. The characteristics of the velocity field and the shape corrections are analysed in detail. For the case of a cylinder (S = 1), explicit expressions are derived and used to provide some insight into the break-up that the deformation would induce. The asymptotic results are validated and tested by direct numerical simulations when the velocity is constant and when it oscillates. For the constant velocity case, we demonstrate that the theory provides a very good estimate of the dynamics for a large range of parameters. However, a systematic departure is observed for very small Oh due to the persistence of free eigenmodes excited during the transient. These same eigenmodes also limit the applicability of the theory to oscillating bridges with large oscillating periods. Finally, the perturbation theory is applied to the cylindrical solution of Frankel & Weihs (J. Fluid Mech., vol. 155 (1985), pp. 289-307) obtained for constant velocity U when the contact lines are allowed to move. We show that it can be used to compute the correction associated with acceleration. Finally, the effect of gravity is discussed and shown to modify the equilibrium shape but not the main results obtained from the perturbation theory.

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Drop formation in a one-dimensional approximation of the Navier–Stokes equation

Cited by 492 publications

References 19 publications

Nonlinear travelling waves on a spiralling liquid jet

Nonlinear travelling waves on a spiralling liquid jet

Critical phenomena and breaking drops: Infinite idealizations in physics

Forced dynamics of a short viscous liquid bridge

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