Coarsening dynamics of the convective Cahn-Hilliard equation

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. © 2003 Elsevier Science B.V. All rights reserved. PACS: 05.45.−a; 68.35.Ja; 81.10.Aj