Coarsening dynamics of the convective Cahn-Hilliard equation

Stephen J. Watson*, Felix Otto, Boris Y. Rubinstein, Stephen H. Davis

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

78 Scopus citations

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 ℒM for coarsening exhibits the temporal power law scaling t1/2; provided ℒM is appropriately small with respect to the Peclet length scale ℒP. 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 ℒM emerges. It predicts both the scaling constant c of the t1/2 regime, i.e. ℒM = ct1/2, as well as the crossover to logarithmically slow coarsening as ℒM crosses ℒP. Our analytical coarsening law stands in good qualitative agreement with large-scale numerical simulations that have been performed on cCH.

Original languageEnglish (US)
Pages (from-to)127-148
Number of pages22
JournalPhysica D: Nonlinear Phenomena
Volume178
Issue number3-4
DOIs
StatePublished - Apr 15 2003

Keywords

  • Coarsening dynamical system
  • Driven phase ordering
  • Scaling laws

ASJC Scopus subject areas

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

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