Boris Y. Rubinstein

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 t1/2; 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. nDirect 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 t1/2 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.

Sylvester Waves in the Coxeter Groups

AbstractA new recursive procedure of the calculation of partition numbers function W(s, dm) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W(s, dm) and their periods τ(G) for the irreducible Coxeter groups and a list for the first twelve symmetric group n

Theory of Pendular Rings Revisited

mathcal{S}

Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume, by Hermann Schwarz

nm are presented. A least common multiplen

On Explicit Formula for Restricted Partition Function

lcm

Stability of vertical boiling thin film

n(m) of the series of the natural numbers 1,2,...,m plays a role in the period τ(n