## Abstract

We characterize the coarsening dynamics associated with a convective Cahn-Hilliard equation (cCH) in one space dimension. First, we derive a sharp-interface theory through a matched asymptotic analysis. Two types of phase boundaries (kink and anti-kink) arise, due to the presence of convection, and their motions are governed to leading order by a nearest-neighbors interaction coarsening dynamical system (CDS). Theoretical predictions on CDS include: The characteristic length ℒ_{M} for coarsening exhibits the temporal power law scaling t^{1/2}; provided ℒ_{M} is appropriately small with respect to the Peclet length scale ℒ_{P}. Binary coalescence of phase boundaries is impossible. Ternary coalescence only occurs through the kink-ternary interaction; two kinks meet an anti-kink resulting in a kink. Direct numerical simulations performed on both CDS and cCH confirm each of these predictions. A linear stability analysis of CDS identifies a pinching mechanism as the dominant instability, which in turn leads to kink-ternaries. We propose a self-similar period-doubling pinch ansatz as a model for the coarsening process, from which an analytical coarsening law for the characteristic length scale ℒ_{M} emerges. It predicts both the scaling constant c of the t^{1/2} regime, i.e. ℒ_{M} = ct^{1/2}, as well as the crossover to logarithmically slow coarsening as ℒ_{M} crosses ℒ_{P}. Our analytical coarsening law stands in good qualitative agreement with large-scale numerical simulations that have been performed on cCH.

Original language | English (US) |
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Pages (from-to) | 127-148 |

Number of pages | 22 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 178 |

Issue number | 3-4 |

DOIs | |

State | Published - Apr 15 2003 |

Externally published | Yes |

### Funding

SJW was supported by the Max-Planck-Institute for Mathematics in the Sciences (MIS), Leipzig in the initial stages of this project and later by the NSF N.I.R.T. grant #DMR-0102794. SHD was supported by the NSF N.I.R.T. grant #DMR-0102794. The authors would like to thank Prof. A.A. Golovin for kindly providing numerical simulations of the cCH equation to validate the kink-ternary prediction of our theory.

## Keywords

- Coarsening dynamical system
- Driven phase ordering
- Scaling laws

## ASJC Scopus subject areas

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