Crystalline Saffman-Taylor fingers

Journal Article · Fri Dec 01 04:00:00 EST 1995 · SIAM Journal of Applied Mathematics

DOI:https://doi.org/10.1137/S0036139993259908· OSTI ID:137058

Almgren, R ^[1]

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1. Univ. of Chicago, IL (United States). Dept. of Mathematics

The authors study the existence and structure of stead-state fingers in two-dimensional solidification, when the surface energy has a crystalline anisotropy so that the energy-minimizing Wulff shape and hence the solid-liquid interface are polygons, and in the one-sided quasi-state limit so that the diffusion field satisfies Laplace`s equation in the liquid. In a channel of finite width, this problem is the crystalline analog of the classic Saffman-Taylor smooth finger in Hele-Shaw flow. By a combination of analysis and numerical Schwarz-Christoffel mapping methods, they show that, as for solutions of the smooth problem, for each choice of Wulff shape there is a critical maximum value of the magnitude of surface tension above which no convex steady-state solutions exist. They then exhibit convergence of convex crystalline solutions to convex smooth solutions as the Wulff shape approaches a circle. They also consider the open dendrite geometry and show that there are no steady-state solutions having a finite number of sides for any crystalline surface energy. This is in striking contrast to the smooth case and an indication that the time-dependent behavior may be more complicated for crystalline surface energies.

• Sponsoring Organization: USDOE
• OSTI ID: 137058
• Journal Information: SIAM Journal of Applied Mathematics, Journal Name: SIAM Journal of Applied Mathematics Journal Issue: 6 Vol. 55; ISSN 0036-1399; ISSN SMJMAP
• Country of Publication: United States
• Language: English

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66 PHYSICS
CALCULATION METHODS
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MORPHOLOGY
SOLIDIFICATION
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