Complex variable methods in Hele–Shaw moving boundary problems

Howison, Sam

doi:10.1017/s0956792500000802

Cited by 185 publications

(223 citation statements)

References 27 publications

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“…1, 31 Explicit solutions can be sometimes found by studying the singularity structure of (3) via a time dependent conformal map from an auxiliary ζ-plane to the physical z-plane. This gives an initial value problem in terms of the unknown time dependent parameters of the conformal map.…”

Section: Mathematical Modelmentioning

confidence: 99%

“…Hence, the Schwarz function of the interface is defined as S = z = f (1/ζ,t), and (3) can be analytically continued away from the interface to hold over Ω(t). This idea has been used to derive explicit solutions, e.g., the evolution of fluid blobs due to hydrodynamic singularities 31 or even external fields 32 and is also used here to understand the evolution of bubbles.…”

Section: Mathematical Modelmentioning

confidence: 99%

See 1 more Smart Citation

On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

2015

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Unsteady propagating bubbles in an unbounded Hele-Shaw cell are considered numerically in the case of zero surface tension. The instability of elliptical bubbles and their evolution toward a stable circular boundary, with speed twice that of the fluid speed at infinity, is studied numerically and by stability analysis. Numerical simulations of bubbles demonstrate that the important role played by singularities of the Schwarz function of the bubble boundary in determining the evolution of the bubble. When the singularity lies close to the initial bubble, two types of topological change are observed: (i) bubble splitting into multiple bubbles and (ii) a finite fluid blob pinching off inside the bubble region. C 2015 AIP Publishing LLC.

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Section: Mathematical Modelmentioning

confidence: 99%

Section: Mathematical Modelmentioning

confidence: 99%

On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

2015

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“…Hele-Shaw [47]) describes the pressure of two immiscible viscous fluids trapped between two parallel glass plates and has attracted considerable attention in the literature, both on the analytical side [23,27,28,29,33,38,51,54,63] and on the numerical side [2,6,7,8,12,25,26,49,50,65]. This list of references also encompasses work related to the one-sided Hele-Shaw model, which arises as a limit when the viscosity of one of the fluids approaches zero.…”

Section: Introductionmentioning

confidence: 99%

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Mayer

2000

Eur. J. Appl. Math

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Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow to treat such free boundary problems in both IR 2 and IR 3 . The advantage of such an approach is the re-usability of much of the setup for all of the different problems.As an example of the method we will compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity. IntroductionIn this paper we will study geometric evolution problems for surfaces driven by curvature. These moving boundary problems arise from models in physics and the material sciences, and they describe phase changes, or more generally, phenomena in fluid flow.The Hele-Shaw model (named after H.S. ) describes the pressure of two immiscible viscous fluids trapped between two parallel glass plates and has attracted considerable attention in the literature, both on the analytical side [23,27,28,29,33,38,51,54,63] and on the numerical side [2,6,7,8,12,25,26,49,50,65]. This list of references also encompasses work related to the one-sided Hele-Shaw model, which arises as a limit when the viscosity of one of the fluids approaches zero. Furthermore, the Hele-Shaw model has classically been considered on a bounded domain, but many of the references above also address the problem on an unbounded domain. We shall only consider the problem on all of IR n ; for the precise formulation see (4.4).The Mullins-Sekerka model was first proposed by (and much later named after) Mullins and Sekerka to study solidification of materials of negligible specific heat [58]. For a while this model was also called the Hele-Shaw model, leading to a somewhat confused literature. Pego [61], and then Alikakos, Bates, and Chen [3], and also Stoth [68], established this model as a singular limit of the Cahn-Hilliard equation [17], a fourth order partial differential equation modelling nucleation and coarsening phenomena in a melted binary alloy. The Mullins-Sekerka model is considered to be a good model to describe the stage of Ostwald ripening in phase transitions, which is the stage after the initial nucleation has 2 U. F. Mayer essentially been completed and where some particles grow at the cost of others in an effort to decrease interfacial energy (surface area). The literature focuses mainly on the modelling and analytical aspects of the problem [11,16,18,19,21,32,34,46,56,66,71,72] though there are numerical simulations for the two-dimensional case [10,73]. We shall consider the two-phase problem on all of IR n , see (4.3).The mean curvature flow is probably the most studied geometric moving boundary problem, it is in some way also the simplest: the normal velocity is equal to the mean curvature of the interface. This sha...

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“…Reviews by Saffman [38], Bensimon, Kadanoff, Liang, Shraiman, & Tang [.5], and Homsy [16] summarize the state of affairs as of the mid-eighties, while_some of the the more recent developments are reviewed by Pelce [31], Kessler, Koplik, & Levine [22], nowison [21], and Tanveer [42] from a range of different perspectives.…”

Section: Introductionmentioning

confidence: 99%