Three-dimensional effects in directional solidification in Hele-Shaw cells: Nonlinear evolution and pattern selection

V. S. Ajaev, S. H. Davis

Research output: Contribution to journalArticlepeer-review

Abstract

Directional solidification of a dilute binary alloy in a Hele-Shaw cell is modeled by a long-wave nonlinear evolution equation with zero flux and contact-angle conditions at the walls. The basic steady-state solution and its linear stability criteria are found analytically, and the nonlinear system is solved numerically. Concave-down (toward the solid) interfaces under physically realistic conditions are found to be more unstable than the planar front. Weakly nonlinear analysis indicates that subcritical bifurcation is promoted, the domain of modulational instability is expanded and transition to three-dimensional patterns is delayed due to the contact-angle condition. In the strongly nonlinear regime fully three-dimensional steady-state solutions are found whose characteristic amplitude is larger than that for the two-dimensional problem. In the subcritical regime secondary bifurcation to stable solutions is promoted.

Original languageEnglish (US)
Pages (from-to)1275-1284
Number of pages10
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume61
Issue number2
DOIs
StatePublished - 2000

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

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