Coarsening dynamics of the convective Cahn-Hilliard equation

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.