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 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.