Coarsening dynamics of the convective Cahn-Hilliard equation

Watson, S.J., Otto, F., Rubinstein, B.Y. and Davis, S.H. (2003) Coarsening dynamics of the convective Cahn-Hilliard equation. Physica D: Nonlinear Phenomena, 178(3-4), pp. 127-148. (doi: 10.1016/S0167-2789(03)00048-4)

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 (Image ). Theoretical predictions on Image include: • The characteristic length Image for coarsening exhibits the temporal power law scaling t1/2; provided Image is appropriately small with respect to the Peclet length scale Image. • 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 Image and cCH confirm each of these predictions. A linear stability analysis of Image 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 Image emerges. It predicts both the scaling constant c of the t1/2 regime, i.e. Image , as well as the crossover to logarithmically slow coarsening as Image crosses Image . Our analytical coarsening law stands in good qualitative agreement with large-scale numerical simulations that have been performed on cCH.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Watson, Dr Stephen
• Authors: Watson, S.J., Otto, F., Rubinstein, B.Y., and Davis, S.H.
• Subjects: Q Science > QC Physics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Physica D: Nonlinear Phenomena
• ISSN: 0167-2789
• ISSN (Online): 1872-8022
• Published Online: 19 March 2003