## Abstract

We consider the convective flow present during the directional solidification of a dilute binary alloy having small segregation coefficient k. The constitutional undercooling is set to zero, so that morphological instabilities of the melt-crystal interface are suppressed and the interface is planar. In the linearized stability theory, the critical wave number of the convective mode a_{c} is found to behave like a_{c} ≈ k^{ 1 4} as k → 0, which is identical in behavior to the morphological mode. The weakly nonlinear theory gives rise to a Sivashinsky equation governing the spatial structure of the convection. Unlike previous long-wave evolution equations governing convective processes, this one arises in a geometrically unbounded domain with system boundaries that are neither fixed heat flux nor small Biot number; here the equivalent Biot number is of unit order.

Original language | English (US) |
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Pages (from-to) | 231-238 |

Number of pages | 8 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 39 |

Issue number | 2-3 |

DOIs | |

State | Published - Oct 2 1989 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics