Dynamics of three-dimensional thin film rupture

Witelski, Thomas P.; Bernoff, Andrew J.

doi:10.1016/s0167-2789(00)00165-2

Cited by 90 publications

(109 citation statements)

References 43 publications

Supporting

Mentioning

103

Contrasting

Unclassified

Order By: Relevance

“…In terms of self-similar solutions, both spreading and blow up self-similar solutions are possible. Rupture self-similar solutions have also been observed and studied [53,51]. If m = n + 2, there exist spreading self-similar solutions if 0 < n < 3, and none if n 3 [1].…”

Section: Introductionmentioning

confidence: 95%

The thin film equation with backwards second order diffusion

Novick-Cohen¹,

Shishkov²

2011

Interfaces Free Bound.

View full text Add to dashboard Cite

We focus on the thin film equation with lower order "backwards" diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a "porous media cutoff" of van der Waals forces. We treat in detail the equationwhere ν = ±1, n > 0, M > m, and A 0. Global existence of weak nonnegative solutions is proven when m − n > −2 and A > 0 or ν = −1, and when −2 < m − n < 2, A = 0, ν = 1. From the weak solutions, we get strong entropy solutions under the additional constraint that m−n > −3/2 if ν = 1. A local energy estimate is obtained when 2 n < 3 under some additional restrictions. Finite speed of propagation is proven when m > n/2, for the case of "strong slippage", 0 < n < 2, when ν = 1 based on local entropy estimates, and for the case of "weak slippage", 2 n < 3, when ν = ±1 based on local entropy and energy estimates.

show abstract

Section: Introductionmentioning

confidence: 95%

The thin film equation with backwards second order diffusion

Novick-Cohen¹,

Shishkov²

2011

Interfaces Free Bound.

View full text Add to dashboard Cite

show abstract

“…Analysis of finite-time blow-up and rupture in many higher order PDEs starts along similar lines [35,11,76,77,79,34,22,25].…”

Section: Self-similar Finite-time Blow-up In a Higher-order Pdementioning

confidence: 99%

“…Evans et al [35] give a matched asymptotic construction of the self-similar solutions for a more general version of (4.3) in the limit of a large number of extrema. Their computational approach is comparable to [73,75,76] and is described in terms of a one-parameter shooting search as a function of the far-field parameter c 0 . While these solutions are very reminiscent of the families of discrete localized multi-bump solutions of the Swift-Hohenberg equation, it is not clear how to embed the self-similar solutions in continuous families as in "homoclinic snaking" [26,7].…”

Section: Similarity Variablesmentioning

confidence: 99%

“…In the pinch-off of a thin liquid filament, as in a lava lamp or the break-up of a fluid jet, the profile of the pinching neck assumes a universal shape as detachment is approached [18,32]. Other examples include rupture of a solid filament due to surface diffusion [11], rupture of thin films [79,69,70,75,76], and the ignition of a combustible substance [6].…”

Section: Introductionmentioning

confidence: 99%

“…For infinite-time spreading dynamics, it may be appropriate to assume that the solution is advancing into an undisturbed medium [74]. For finite-time singular behavior, boundary conditions are posed in a matching region that is near to the blow-up in primitive variables but lies in the far-field with respect to the self-similar variables [75,76].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

Self Cite

View full text Add to dashboard Cite

Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

show abstract

Population Balance Modeling – Experimental Study with Bubble Columns

Hlawitschka

Seiberth

Gao

et al. 2013

Chemie Ingenieur Technik

View full text Add to dashboard Cite

Ein gerührter Blasenkontaktor im Gegenstrom wird mithilfe der Computational Fluid Dynamics (CFD) in Kombination mit der Populationsbilanzmodellierung (PBM) untersucht. Zur Simulation der Hydrodynamik wird auf das Euler-Euler-Modell zurückgegriffen, welches jedoch die Widerstandskräfte zwischen den Phasen modelliert. Die Widerstandskraft wird daher in einem Einzelblasenprüfstand bestimmt, bei dem die Blase durch einen Gegenstrom in Schwebe gehalten wird. Hierbei konnte der experimentell bestimmte Widerstandskoeffizient für sphärische Blasen mittels Literaturkorrelationen abgebildet werden. Die Einzelblasenapparatur ermöglicht zudem weitere Untersuchungen zum Stofftransport an ruhenden Blasen. Die durchgeführten CFD-PBM-Simulationen des gerührten Blasenkontaktors zeigen eine gute Übereinstimmung mit der experimentell ermittelten Blasengrößenverteilung.A stirred bubble column contactor is investigated using of computational fluid dynamics (CFD) and population balance equations. For the simulation, the Euler-Euler model is applied which accounts for the interactions between the phases by experimentally derived correlations. Therefore, the resistance coefficient of singe bubbles is measured using a single bubble test rig keeping the bubble in a plane using a counter-current flow. The resulting drag coefficient for spherical bubbles could be described by literature correlations. The system allows further measurements of the mass transfer at a stagnant bubble. Computational fluid dynamics population balance simulations of the stirred bubble column contactor show good agreement between the simulated bubble size and experimental data. EinleitungBlasensäulen werden in der chemischen, petrochemischen, biochemischen und Metallindustrie [1] eingesetzt. Dort dienen diese u. a. zur Oxidation, Chlorierung, Alkylierung, Polymerisation und Hydrierung, Erstellung von synthetischen Kraftstoffen sowie zur Fermentation und Abwasserbehandlung [2 -4]. Trotz der weiten Verbreitung von Blasensäulen stellen sich deren Dimensionierung und Maßstabsvergrößerung durch die komplexen Interaktionen des Mehrphasensystems als problematisch dar [5]. Aufgrund der recht einfachen Bauweise existiert eine Vielzahl an unterschiedlichen Bauformen, wobei generell zwischen der einfachen Blasensäule, der Blasensäulenkaskade sowie dem Schlaufenreaktor unterschieden wird. Die auftretenden Blasenregime (Abb. 1), homogener Blasenaufstieg, heterogener Blasenaufstieg sowie Schwallströmung, hängen von der Blasengrößenverteilung und der relativen Aufstiegsgeschwindigkeit ab [6]. Das homogene Blasenregime ist durch eine homogene, kleine Blasengrößenverteilung gekennzeichnet und tritt bei niedrigen Gasgeschwindigkeiten von weniger als 5 cm s -1 auf [4,6,7]. Aufgrund der relativ gleichmäßigen Verteilung der Blasen sowie der homogenen, kleinen Blasengröße sind die Blasenkoaleszenz sowie der Blasenzerfall gering [6]. Die Blasengröße wird somit haupt-Abbildung 2. Koaleszenzrate des Modells von Hibiki und Ishii [49] unter Verwendung der Modellkonstanten C rc = 0,03 und C c...

show abstract

Dynamics of three-dimensional thin film rupture

Cited by 90 publications

References 43 publications

The thin film equation with backwards second order diffusion

The thin film equation with backwards second order diffusion

Stability and dynamics of self-similarity in evolution equations

Population Balance Modeling – Experimental Study with Bubble Columns

Contact Info

Product

Resources

About