An Accurate von Neumann's Law for Three-Dimensional Foams

Hilgenfeldt, Sascha; Kraynik, Andrew M.; Koehler, Stephan A.; Stone, Howard A.

doi:10.1103/physrevlett.86.2685

Cited by 132 publications

(96 citation statements)

References 23 publications

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“…9, Bottom), diffusion causes some phases to collapse and others to grow under the large shear. In 3D, the rate of volume change matches with a recently found generalization of von Neumann's law (19,20).…”

Section: Fluid Flow With Permeabilitysupporting

confidence: 48%

The Voronoi Implicit Interface Method for computing multiphase physics

Saye

Sethian

2011

Proc. Natl. Acad. Sci. U.S.A.

100

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We introduce a numerical framework, the Voronoi Implicit Interface Method for tracking multiple interacting and evolving regions (phases) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.), intricate jump conditions, internal constraints, and boundary conditions. The method works in two and three dimensions, handles tens of thousands of interfaces and separate phases, and easily and automatically handles multiple junctions, triple points, and quadruple points in two dimensions, as well as triple lines, etc., in higher dimensions. Topological changes occur naturally, with no surgery required. The method is first-order accurate at junction points/lines, and of arbitrarily high-order accuracy away from such degeneracies. The method uses a single function to describe all phases simultaneously, represented on a fixed Eulerian mesh. We test the method's accuracy through convergence tests, and demonstrate its applications to geometric flows, accurate prediction of von Neumann's law for multiphase curvature flow, and robustness under complex fluid flow with surface tension and large shearing forces.multiple interface dynamics | level set methods | foams | minimal surfaces T here are a host of physical problems that involve interconnected moving interfaces, including dry foams, crystal grain growth, mixing of multiple materials, and multicellular structures. These interfaces are the boundaries of individual regions/cells, which we refer to as phases. The physics, chemistry, and mechanics that drive the motion of these interfaces are often complex, and include topological challenges when interfaces are destroyed and created.Producing good mathematical models that capture the motion of these interfaces, especially at degeneracies, such as triple points and triple lines where multiple interfaces meet, is challenging. Building robust numerical methods to tackle these problems is equally difficult, requiring numerical resolution of sharp corners and singularities, and recharacterization of domains when topologies change. A variety of methods have been proposed to handle these problems, including (i) front tracking methods, which explicitly track the interface, modeled as moving segments in two dimensions and moving triangles in three dimensions, (ii) volume of fluid methods, which use fixed Eulerian cells and assign a volume fraction for each phase within a cell, (iii) level set methods (1), which use an implicit formulation to represent the interface, and treat each region/phase separately, followed by a repair procedure which reattaches the evolving regions to each other (1-3), and (iv) diffusion generated motion which combine diffusion via convolution with reconstruction procedures to simulate multiphase mean curvature flow (4). Although there are advantages and disadvantages to each of these approaches, it has remained a challenge to robustly and accurately handle the wide range of possible motions of evolving, highly complex interconnected interfaces separating a large number of phases un...

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Section: Fluid Flow With Permeabilitysupporting

confidence: 48%

The Voronoi Implicit Interface Method for computing multiphase physics

Saye

Sethian

2011

Proc. Natl. Acad. Sci. U.S.A.

100

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“…These are like regular polygons but with edges replaced by circular arcs, all of radius R, that meet at 120 • as required by Plateau's laws. Isotropic bubbles have been used to model both two- [57] and three-dimensional [58][59][60][61] foams. We find…”

Section: Shapementioning

confidence: 99%

Bubble statistics and coarsening dynamics for quasi-two-dimensional foams with increasing liquid content

2013

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. (2013). Bubble statistics and coarsening dynamics for quasi-two-dimensional foams with increasing liquid content. Physical Review E, 87(4) Bubble statistics and coarsening dynamics for quasi-two-dimensional foams with increasing liquid content AbstractWe report on the statistics of bubble size, topology, and shape and on their role in the coarsening dynamics for foams consisting of bubbles compressed between two parallel plates. The design of the sample cell permits control of the liquid content, through a constant pressure condition set by the height of the foam above a liquid reservoir. We find that in the scaling regime, all bubble distributions are independent not only of time, but also of liquid content. For coarsening, the average rate decreases with liquid content due to the blocking of gas diffusion by Plateau borders inflated with liquid; we achieve a factor of 4 reduction from the dry limit. By observing the growth rate of individual bubbles, we find that von Neumann's law becomes progressively violated with increasing wetness and decreasing bubble size. We successfully model this behavior by explicitly incorporating the border-blocking effect into the von Neumann argument. Two dimensionless bubble shape parameters naturally arise, one of which is primarily responsible for the violation of von Neumann's law for foams that are not perfectly dry. We report on the statistics of bubble size, topology, and shape and on their role in the coarsening dynamics for foams consisting of bubbles compressed between two parallel plates. The design of the sample cell permits control of the liquid content, through a constant pressure condition set by the height of the foam above a liquid reservoir. We find that in the scaling regime, all bubble distributions are independent not only of time, but also of liquid content. For coarsening, the average rate decreases with liquid content due to the blocking of gas diffusion by Plateau borders inflated with liquid; we achieve a factor of 4 reduction from the dry limit. By observing the growth rate of individual bubbles, we find that von Neumann's law becomes progressively violated with increasing wetness and decreasing bubble size. We successfully model this behavior by explicitly incorporating the border-blocking effect into the von Neumann argument. Two dimensionless bubble shape parameters naturally arise, one of which is primarily responsible for the violation of von Neumann's law for foams that are not perfectly dry. Disciplines Physical Sciences and Mathematics | Physics

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“…11,17) Mean field calculations, 14,15,18) however, indicated mostly smaller N c values of approximately 13, although the most recently reported calculation 19) shows that N c is nearly 15. In any event, the author's opinion is that the critical size corresponds to the topological mean grain size.…”

Section: Characteristics Of the Grain Coarsening Process Observed In mentioning

confidence: 99%

Grain Growth Behaviors Observed by a Modified Potts-MC-type Simulation

Ito

2017

ISIJ Int.

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Details of the Potts-MC-type simulation model modified to reduce computing time were explained and effects of parameters to control anti-freezing of the structure, orientation-relation-dependent migration rate and energy of boundaries as well as orientation-relation-dependent migration rate of triple lines on the elementary structural development process were presented. Some topological characters of the structures observed by the modified method were then compared with earlier reported ones. Most of them coincided with the earlier reported results. The critical grain size has been found to coincide with the topological mean grain size. The transition probabilities of grains between topological classes in the shortest elapsed time are reported for the first time; they suggest that the steady state topological grain size distribution could exist for only a short period.

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An Accurate von Neumann's Law for Three-Dimensional Foams

Cited by 132 publications

References 23 publications

The Voronoi Implicit Interface Method for computing multiphase physics

The Voronoi Implicit Interface Method for computing multiphase physics

Bubble statistics and coarsening dynamics for quasi-two-dimensional foams with increasing liquid content

Grain Growth Behaviors Observed by a Modified Potts-MC-type Simulation

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