Liquid mixture convection during phase separation in a temperature gradient

Abstract

We simulate the phase separation of a low-viscosity binary mixture, assuming that the fluid system is confined between two walls that are cooled down to different temperatures below the critical point of the mixture, corresponding to quenches within the unstable range of its phase diagram. Spinodal decomposition patterns for off-critical mixtures are studied numerically in two dimensions in the creeping flow limit and for a large Lewis number, together with their dependence on the fluidity coefficient. Our numerical results reproduce the large-scale unidirectional migration of phase-separating droplets that was observed experimentally by Califano et al. ͓"Large-scale, unidirectional convection during phase separation of a density-matched liquid mixture," Phys. Fluids 17, 094109 ͑2005͔͒, who measured typical speeds that are quite larger than the Marangoni velocity. To understand this finding, we then studied the temperature-gradient-induced motion of an isolated droplet of the minority phase embedded in a continuous phase, showing that when the drop is near local equilibrium, its speed is of the same order as the Marangoni velocity, i.e., it is proportional to the unperturbed temperature gradient and the fluidity coefficient. However, far from local equilibrium, i.e., for very large unperturbed temperature gradients, the drop first accelerates to a speed that is larger than the Marangoni velocity, then, later, it decelerates, exhibiting an increase-decrease behavior, as described by Yin et al. ͓"Thermocapillary migration of nondeformable drops," Phys. Fluids 20, 082101 ͑2008͔͒. Such behavior is due to the large nonequilibrium, Korteweg-driven convection, which at first accelerates the droplets to relatively large velocities, and then tends to induce an approximately uniform inside temperature distribution so that the drop experiences an effective temperature gradient that is much smaller than the unperturbed one and, consequently, decelerates.

Coalescence and breakup of large droplets in turbulent channel flow

“…13 The result reported here can be generalized to physical systems with non-uniform surface tension: in that case, the Marangoni force 30 term is recovered by introducing a non-constant Cahn number. 31,32…”

Dissolution or Growth of a Liquid Drop via Phase-Field Ternary Mixture Model Based on the Non-Random, Two-Liquid Equation

“…In conclusion, it is worth reiterating that Equations ( 7 ) and ( 8 ) constitute a system of fourth-order equations, which represents a generalization of the classical Cahn-Hilliard equation to describe phase separation in binary mixtures [ 11 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 ].…”

Triphase Separation of a Ternary Symmetric Highly Viscous Mixture

“…These relations, together with the equalities and , define all chemical potential differences for a ternary system. Finally, it is worth reiterating that Equations ( 4 ) and ( 5 ) constitute a system of fourth-order equations, which represents a generalization (specifically, a ternary version of Model B in the taxonomy of Hohenberg and Halperin [ 51 ]) of the classical Cahn–Hilliard equation to describe phase separation in binary mixtures [ 33 , 34 , 37 , 38 , 39 , 41 , 52 , 53 , 54 ].…”