Breakdown of scaling in droplet fission at high Reynolds number

Brenner, Michael P.; Eggers, Jens; Joseph, K.M.; Nagel, Sidney R.; Shi, Xiongwei

doi:10.1063/1.869279

Cited by 123 publications

(106 citation statements)

References 34 publications

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“…While many studies have focused on the singularity occurring when the drop pinches off [37,30,7,21,41,14,36,53], less attention has been paid to the long filament that is drawn out by the falling drop [22,15,27]. Herein, we derive an exact solution for the time-dependent stretch rate and filament thickness for a cylindrical, axisymmetric viscoelastic filament stretching in a gravity-driven extensional flow, and compare our predictions to measurements of the thinning filament that forms behind a falling drop.…”

Section: Introductionmentioning

confidence: 89%

“…The fall of a fluid droplet from a faucet under the influence of gravity is an everyday occurrence in kitchen sinks, and has recently received attention due to a series of related experimental studies [37,43,54,30,7,21,25] and mathematical analyses [43,7,12,41,14,42,36,4,13,35,51,53,22] (see the review by Eggers [15]). While many studies have focused on the singularity occurring when the drop pinches off [37,30,7,21,41,14,36,53], less attention has been paid to the long filament that is drawn out by the falling drop [22,15,27].…”

Section: Introductionmentioning

confidence: 99%

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Exact solution for the extensional flow of a viscoelastic filament

et al. 2004

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We solve the free boundary problem for the dynamics of a cylindrical, axisymmetric viscoelastic filament stretching in a gravity-driven extensional flow for the Upper Convected Maxwell and Oldroyd-B constitutive models. Assuming the axial stress in the filament has a spatial dependence provides the simplest coupling of viscoelastic effects to the motion of the filament, and yields a closed system of ODEs with an exact solution for the stretch rate and filament thickness satisfied by both constitutive models. This viscoelastic solution, which is a generalization of the exact solution for Newtonian filaments, converges to the Newtonian power-law scaling as t → ∞. Based on the exact solution, we identify two regimes of dynamical behavior called the weakly-and strongly-viscoelastic limits. We compare the viscoelastic solution to measurements of the thinning filament that forms behind a falling drop for several semi-dilute (strongly-viscoelastic) polymer solutions. We find the exact solution correctly predicts the time-dependence of the filament diameter in all of the experiments. As t → ∞, observations of the filament thickness follow the Newtonian scaling 1/ √ t. The transition from viscoelastic to Newtonian scaling in the filament thickness is coupled to a stretch-to-coil transition of the polymer molecules.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Exact solution for the extensional flow of a viscoelastic filament

et al. 2004

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“…Numerous researchers have since made significant experimental and numerical contributions to the field, providing insight into the discrete behaviour of Newtonian fluids on approach to and past the "pinch" region during dripping and jetting [4][5][6][7] . In particular, the effects of viscosity [7][8][9][10][11][12][13] and surface tension 8,14 on breakup dynamics, and subsequent formation of drops have been thoroughly investigated. Simplified formulations of the Navier-Stokes equations for incompressible flow with a free surface have been produced in a number of studies in an effort to understand the physical mechanisms governing drop formation [15][16][17][18] .…”

Section: Introductionmentioning

confidence: 99%

Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration

2006

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The dynamics of drop formation and pinch-off have been investigated for a series of low viscosity elastic fluids possessing similar shear viscosities, but differing substantially in elastic properties. On initial approach to the pinch region, the viscoelastic fluids all exhibit the same global necking behavior that is observed for a Newtonian fluid of equivalent shear viscosity. For these low viscosity dilute polymer solutions, inertial and capillary forces form the dominant balance in this potential flow regime, with the viscous force being negligible. The approach to the pinch point, which corresponds to the point of rupture for a Newtonian fluid, is extremely rapid in such solutions, with the sudden increase in curvature producing very large extension rates at this location. In this region the polymer molecules are significantly extended, causing a localized increase in the elastic stresses, which grow to balance the capillary pressure. This prevents the necked fluid from breaking off, as would occur in the equivalent Newtonian fluid. Alternatively, a cylindrical filament forms in which elastic stresses and capillary pressure balance, and the radius decreases exponentially with time. A (0+1)-dimensional finitely extensible nonlinear elastic dumbbell theory incorporating inertial, capillary, and elastic stresses is able to capture the basic features of the experimental observations. Before the critical “pinch time” tp, an inertial-capillary balance leads to the expected 2∕3-power scaling of the minimum radius with time: Rmin∼(tp−t)2∕3. However, the diverging deformation rate results in large molecular deformations and rapid crossover to an elastocapillary balance for times t>tp. In this region, the filament radius decreases exponentially with time Rmin∼exp[(tp−t)∕λ1], where λ1 is the characteristic time constant of the polymer molecules. Measurements of the relaxation times of polyethylene oxide solutions of varying concentrations and molecular weights obtained from high speed imaging of the rate of change of filament radius are significantly higher than the relaxation times estimated from Rouse-Zimm theory, even though the solutions are within the dilute concentration region as determined using intrinsic viscosity measurements. The effective relaxation times exhibit the expected scaling with molecular weight but with an additional dependence on the concentration of the polymer in solution. This is consistent with the expectation that the polymer molecules are in fact highly extended during the approach to the pinch region (i.e., prior to the elastocapillary filament thinning regime) and subsequently as the filament is formed they are further extended by filament stretching at a constant rate until full extension of the polymer coil is achieved. In this highly extended state, intermolecular interactions become significant, producing relaxation times far above theoretical predictions for dilute polymer solutions under equilibrium conditions.

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“…In a particular class of systems the nonlinear character gives rise to finite-time-singularities, that is solutions which cease to be valid beyond a particular finite time span. One encounters finite-time-singularities in stellar structure, turbulent flow, and bacterial growth [2,3,4]. The phenomenon is also seen in Euler flows and in free-surface-flows [5,6].…”

Section: Introductionmentioning

confidence: 98%

Finite-time-singularity with noise and damping

Fogedby¹

2003

Braz. J. Phys.

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The combined influence of linear damping and noise on a dynamical finite-time-singularity model is considered for a single degree of freedom. The noise resolves the finite-time-singularity and replaces it by a first-passagetime distribution with a peak at the singularity and a long time tail. The damping introduces a characteristic cross-over time. In the early time regime the first-passage-time distribution shows a power law behavior with scaling exponent depending on the ratio of the non linear coupling strength to the noise strength. In the late time regime the damping prevails. The study might be of relevance in the context of hydrodynamics on a nanometer scale, in material physics, and in biophysics. I IntroductionThe influence of noise on the behavior of nonlinear dynamical system is a recurrent theme in modern statistical physics [1]. In a particular class of systems the nonlinear character gives rise to finite-time-singularities, that is solutions which cease to be valid beyond a particular finite time span. One encounters finite-time-singularities in stellar structure, turbulent flow, and bacterial growth [2,3,4]. The phenomenon is also seen in Euler flows and in free-surface-flows [5,6].In the context of hydrodynamical flow on a nanoscale [8], where microscopic degrees of freedom come into play, it is a relevant issue how noise influences the hydrodynamical behavior near a finite-time-singularity. Leaving aside the issue of the detailed reduction of the hydrodynamical equations to a nanoscale and the influence of noise on this scale to further study, we assume in the present context that a single variable or "reaction coordinate" effectively captures the interplay between the singularity and the noise. II Model without dampingIn a recent paper [13] we investigated a simple generic model system with one degree of freedom governed by a nonlinear Langevin equation driven by Gaussian white noise,The model is characterized by the coupling parameter λ, determining the amplitude of the singular term and the noise parameter ∆, determining the strength of the noise correlations. Specifically, in the case of a thermal environment at temperature T the noise strength ∆ ∝ T . In the absence of noise this model exhibits a finitetime-singularity at a time t 0 , where the variable x vanishes with a square law dependence. When noise is added the finite-time-singularity event at t 0 becomes a statistical event and is conveniently characterized by a first-passagetime distribution W (t) [9]. For vanishing noise we have W (t) = δ(t − t 0 ), restating the presence of the finite-timesingularity. In the presence of noise W (t) develops a peak about t = t 0 , vanishes at short times, and acquires a long time tail.The model in Eq.(1) has also been studied in the context of persistence distributions related to the nonequilibrium critical dynamics of the two-dimensional XY model [10] and in the context of non-Gaussian Markov processes [11]. Finally, regularized for small x, the model enters in connection with an analysis of long-...

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Breakdown of scaling in droplet fission at high Reynolds number

Cited by 123 publications

References 34 publications

Exact solution for the extensional flow of a viscoelastic filament

Exact solution for the extensional flow of a viscoelastic filament

Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration

Finite-time-singularity with noise and damping

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