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.

[1]  G. Batchelor.  Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity , 1959, Journal of Fluid Mechanics.

[2]  Raymond T. Pierrehumbert.  Lattice models of advection-diffusion. , 2000, Chaos.

[3]  N. Easwar,et al.  Turbulent suppression of spinodal decomposition , 1984 .

[4]  H. Lekkerkerker,et al.  Interfacial dynamics in demixing systems with ultralow interfacial tension , 2005 .

[5]  Chan,et al.  Light-scattering study of a turbulent critical binary mixture near the critical point. , 1987, Physical review. A, General physics.

[6]  A. Bray.  Coarsening dynamics of phase-separating systems , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  Geoffrey Ingram Taylor,et al.  Diffusion by Continuous Movements , 1922 .

[8]  Emmott,et al.  Lifshitz-Slyozov scaling for late-stage coarsening with an order-parameter-dependent mobility. , 1995, Physical review. B, Condensed matter.

[9]  A. Polyanin,et al.  Handbook of Exact Solutions for Ordinary Differential Equations , 1995 .

[10]  I. Lifshitz,et al.  The kinetics of precipitation from supersaturated solid solutions , 1961 .

[11]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  Moses,et al.  String phase in phase-separating fluids under shear flow. , 1995, Physical review letters.

[13]  Phase separation in a homogeneous shear flow: morphology, growth laws, and dynamic scaling. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[15]  C. M. Elliott,et al.  On the Cahn-Hilliard equation with degenerate mobility , 1996 .

[16]  A. Chakrabarti,et al.  Ordering of viscous liquid mixtures under a steady shear flow , 2000 .

[17]  Herbert Gajewski,et al.  On a nonlocal phase separation model , 2003 .

[18]  A. Bray.  Theory of phase-ordering kinetics , 1994, cond-mat/9501089.

[19]  Jie Shen,et al.  Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Chakrabarti,et al.  Late-stage coarsening for off-critical quenches: Scaling functions and the growth law. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  H. Aref.  Stirring by chaotic advection , 1984, Journal of Fluid Mechanics.

[22]  Raymond T. Pierrehumbert,et al.  Tracer microstructure in the large-eddy dominated regime , 1994 .

[23]  Excitable media in a chaotic flow. , 2001, Physical review letters.

[24]  Charles M. Elliott,et al.  On the Cahn-Hilliard equation , 1986 .

[25]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[26]  Furukawa.  Spinodal decomposition of two-dimensional fluid mixtures: A spectral analysis of droplet growth , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  Charles R. Doering,et al.  A bound on mixing efficiency for the advection–diffusion equation , 2004, Journal of Fluid Mechanics.