  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.
