Bubbles and filaments: Stirring a Cahn-Hilliard fluid

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 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.