A Well-Posed Numerical Method to Track Isolated Conformal Map Singularities in Hele-Shaw Flow

Baker, Greg; Siegel, Michael; Tanveer, S.

doi:10.1006/jcph.1995.1170

Cited by 19 publications

(11 citation statements)

References 33 publications

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“…This fact explains early measurements of the singularity exponent by Dai et al [5] that yield values much closer to Ϫ1 than Ϫ 4 3 . Our simulations also show that even the secondary singularities come from a close neighborhood of the singularities of the initial condition, and not from zeros located at infinity as previously suggested [1]. Finally, the present method seems to open a new way to further characterize the spatial shapes observed during the growth of the interface, simply because the latter is determined by the asymptotic position of the singularities in the complex plane, for which the TWM yields explicit equations.…”

Section: Numerical Resultssupporting

confidence: 56%

Traveling Wavelets for the Simulation of a Hele–Shaw Flow with Surface Tension

Elezgaray¹

1999

Applied and Computational Harmonic Analysis

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Abstract-The singularity dynamics of two immiscible fluids in a Hele-Shaw cell is studied using the traveling wavelet method. The singularities of the conformal mapping of the flow are accurately approximated by the solutions of a system of ordinary differential equations for the position of couples of zeros and poles. This model also describes secondary tip-splitting instabilities and remains valid for values of the surface tension and time intervals where previous perturbative calculations do not apply.

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Section: Numerical Resultssupporting

confidence: 56%

Traveling Wavelets for the Simulation of a Hele–Shaw Flow with Surface Tension

Elezgaray¹

1999

Applied and Computational Harmonic Analysis

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“…The reason is that the leading-order equation zt = q1zc + q2 is well-posed in > 1. (Tanveer 1993 provides analytical evidence for the well-posedness of this equation, whereas computational evidence is presented in Baker et al 1995. ) In contrast, the leading-order equation of the problem formulated in < 1 is ill-posed (Howison 1986b;Tanveer 1993).…”

Section: Governing Equationsmentioning

confidence: 99%

Singular effects of surface tension in evolving Hele-Shaw flows

Siegel

Tanveer

Dai³

1996

J. Fluid Mech.

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In this paper, we present evidence to show that a smoothly evolving zero-surface tension solution of the Hele-Shaw equations can be singularly perturbed by the presence of arbitrarily small non-zero surface tension in order-one time. These effects are explained by the impact of ‘daughter singularities’ on the physical interface, whose formation was suggested in a prior paper (Tanveer 1993). For the case of finger motion in a channel, it is seen that the daughter singularity effect is strong enough to produce the transition from a finger of arbitrary width to one with the selected steady-state width in O(1) time.

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“…However, conformal mapping methods apply most naturally to singly connected domains, and can have difficulties with efficiently including the effect of surface tension. As a numerical method, the most sophisticated version of conformal mapping seems that due to Baker et al [12], who solve a well-posed evolution problem for zero surface tension by analytically continuing initial data and equations of motion into the complex plane, and explicitly tracking the solution's poles and other singularities.…”

Section: Historical Perspectivementioning

confidence: 99%

Boundary Integral Methods for Multicomponent Fluids and Multiphase Materials

Hou

Lowengrub

Shelley³

2001

Journal of Computational Physics

191

135

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We present a brief review of the application of boundary integral methods in two dimensions to multicomponent fluid flows and multiphase problems in materials science. We focus on the recent development and outcomes of methods which accurately and efficiently include surface tension. In fluid flows, we examine the effects of surface tension on the Kelvin-Helmholtz and Rayleigh-Taylor instabilities in inviscid fluids, the generation of capillary waves on the free surface, and problems in Hele-Shaw flows involving pattern formation through the Saffman-Taylor instability, pattern selection, and singularity formation. In materials science, we discuss microstructure evolution in diffusional phase transformations, and the effects of the competition between surface and elastic energies on microstructure morphology. A common link between these different physical phenomena is the utility of an analysis of the appropriate equations of motion at small spatial scales to develop accurate and efficient time-stepping methods.

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A Well-Posed Numerical Method to Track Isolated Conformal Map Singularities in Hele-Shaw Flow

Cited by 19 publications

References 33 publications

Traveling Wavelets for the Simulation of a Hele–Shaw Flow with Surface Tension

Traveling Wavelets for the Simulation of a Hele–Shaw Flow with Surface Tension

Singular effects of surface tension in evolving Hele-Shaw flows

Boundary Integral Methods for Multicomponent Fluids and Multiphase Materials

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