Flow-parametric regulation of shear-driven phase separation in two and three dimensions

TL;DR

This study uses high-resolution simulations of the Cahn-Hilliard equation with chaotic shear flow to explore how phase separation and domain alignment depend on dimensionality, flow parameters, and anisotropy, revealing regimes of coarsening arrest and hyperdiffusivity.

Contribution

It provides a detailed comparison of phase separation in 2D and 3D under shear flow, identifying conditions for coarsening arrest and the influence of flow anisotropy and correlation time.

Findings

Identification of a hyperdiffusive regime where concentration variance decays.

Strong dependence of concentration distribution on dimensionality in the hyperdiffusive regime.

Flow correlation time influences domain alignment direction, with late-time alignment always in the maximum flow direction.

Abstract

The Cahn-Hilliard equation with an externally-prescribed chaotic shear flow is studied in two and three dimensions. The main goal is to compare and contrast the phase separation in two and three dimensions, using high-resolution numerical simulation as the basis for the study. The model flow is parametrized by its amplitudes (thereby admitting the possibility of anisotropy), lengthscales, and multiple time scales, and the outcome of the phase separation is investigated as a function of these parameters as well as the dimensionality. In this way, a parameter regime is identified wherein the phase separation and the associated coarsening phenomenon are not only arrested but in fact the concentration variance decays, thereby opening up the possibility of describing the dynamics of the concentration field using the theories of advection diffusion. This parameter regime corresponds to long flow correlation times, large flow amplitudes and small diffusivities. The onset of this hyperdiffusive regime is interpreted by introducing Batchelor lengthscales. A key result is that in the hyperdiffusive regime, the distribution of concentration (in particular, the frequency of extreme values of concentration) depends strongly on the dimensionality. Anisotropic scenarios are also investigated: for scenarios wherein the variance saturates (corresponding to coarsening arrest), the direction in which the domains align depends on the flow correlation time. Thus, for correlation times comparable to the inverse of the mean shear rate, the domains align in the direction of maximum flow amplitude, while for short correlation times, the domains initially align in the opposite direction. However, at very late times (after the passage of thousands of correlation times), the fate of the domains is the same regardless of correlation time, namely alignment in the direction of maximum flow amplitude.