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 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 LM crosses L-P. Our analytical coarsening law stands in good qualitative agreement with large-scale numerical simulations that have been performed on cCH. (C) 2003 Elsevier Science.