TY - JOUR

T1 - Coarsening dynamics of the convective Cahn-Hilliard equation

AU - Watson, Stephen J.

AU - Otto, Felix

AU - Rubinstein, Boris Y.

AU - Davis, Stephen H.

N1 - Funding Information:
SJW was supported by the Max-Planck-Institute for Mathematics in the Sciences (MIS), Leipzig in the initial stages of this project and later by the NSF N.I.R.T. grant #DMR-0102794. SHD was supported by the NSF N.I.R.T. grant #DMR-0102794. The authors would like to thank Prof. A.A. Golovin for kindly providing numerical simulations of the cCH equation to validate the kink-ternary prediction of our theory.

PY - 2003/4/15

Y1 - 2003/4/15

N2 - 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.

AB - 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.

KW - Coarsening dynamical system

KW - Driven phase ordering

KW - Scaling laws

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U2 - 10.1016/S0167-2789(03)00048-4

DO - 10.1016/S0167-2789(03)00048-4

M3 - Article

AN - SCOPUS:0037446088

VL - 178

SP - 127

EP - 148

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -