  The advective Cahn-Hilliard equation describes the competing processes of
stirring and separation in a two-phase fluid. Intuition suggests that bubbles
will form on a certain scale, and previous studies of Cahn-Hilliard dynamics
seem to suggest the presence of one dominant length scale. However, the
Cahn-Hilliard phase-separation mechanism contains a hyperdiffusion term and we
show that, by stirring the mixture at a sufficiently large amplitude, we excite
the diffusion and overwhelm the segregation to create a homogeneous liquid. At
intermediate amplitudes we see regions of bubbles coexisting with regions of
hyperdiffusive filaments. Thus, the problem possesses two dominant length
scales, associated with the bubbles and filaments. For simplicity, we use use a
chaotic flow that mimics turbulent stirring at large Prandtl number. We compare
our results with the case of variable mobility, in which growth of bubble size
is dominated by interfacial rather than bulk effects, and find qualitatively
similar results.
