Mixing and Phase Separation of Fluid Mixtures

During the three years of the PhD project we extended the di®use interface (DI) method and apply it to engineering related problems, particularly re- lated to mixing and demixing of two °uids. To do that, ¯rst the DI model itself was validated, showing that, in agreement with its predictions, a single drop immersed in a continuum phase moves whenever its composition and that of the continuum phase are not at mutual equilibrium [D. Molin, R. Mauri, and V. Tricoli, "Experimental Evidence of the Motion of a Single Out-of-Equilibrium Drop," Langmuir 23, 7459-7461 (2007)]. Then, we de- veloped a computer code and validated it, comparing its results on phase separation and mixing with those obtained previously. At this point, the DI model was extended to include heat transport e®ects in regular mixtures In fact, in the DI approach, convection and di®usion are coupled via a nonequi- librium, reversible body force that is associated with the Kortweg stresses. This, in turn, induces a material °ux, which enhances both heat and mass transfer. Accordingly, the equation of energy conservation was developed in detail, showing that the in°uence of temperature is two-folded: on one hand, it determine phase transition directly, as the system is brought from the single-phase to the two-phase region of its phase diagram. On the other hand, temperature can also change surface tension, that is the excess free en- ergy stored within the interface at equilibrium. These e®ects were described using the temperature dependence of the Margules parameter. In addition, the heat of mixing was also taken into account, being equal to the excess free energy. [D. Molin and R. Mauri, "Di®use Interface Model of Multiphase Fluids," Int. J. Heat Mass Tranf., submitted]. The new model was applied to study the phase separation of a binary mixture due to the temperature quench of its two con¯ning walls. The results of our simulations showed that, as heat is drawn from the bulk to the walls, the mixture phase tends to phase separate ¯rst in vicinity of the walls, and then, deeper and deeper within the bulk. During this process, convection may arise, due to the above mentioned non equilibrium reversible body force, thus enhancing heat transport and, in particular increasing the heat °ux at the walls [D. Molin, and R. Mauri, "Enhanced Heat Transport during Phase Separation of Liquid Binary Mix- tures," Phys. Fluids 19, 074102-1-10 (2007)]. The model has been extended then and applied to the case where the two phases have di®erent heat con- 3 ductivities. We saw that heat transport depends on two parameters, the Lewis number and the heat conductivity ratio. In particular, varying these parameters can a®ect the orientation of the domains that form during phase separation. Domain orientation has been parameterized using an isotropy coe±cient », varying from -1 to 1, with » = 0 when the morphology is isotropic, » = +1 when it is composed of straight lines along the transversal (i.e. perpendicular to the walls) direction, and » = ¡1 when it is composed of straight lines along the longitudinal (i.e. parallel to the walls) direction [D. Molin, and R. Mauri, "Spinodal Decomposition of Binary Mixtures with Composition-Dependent Heat Conductivities," Int. J. Engng. Sci., in press (2007)]. In order to further extend the model, we removed the constraint of a constant viscosity, and simulated a well known problem of drops in shear °ows. There we found that, predictably, below a certain threshold value of the capillary number, the drop will ¯rst stretch and then snap back. At lager capillary numbers, though, we predict that the drop will stretch and then, eventually, break in two or more satellite drops. On the other hand, applying traditional °uid mechanics (i.e. with in¯nitesimal interface thick- ness) such stretching would continue inde¯nitely [D. Molin and R. Mauri, " Drop Coalescence and Breakup under Shear using the Di®use Interface Model," in preparation]. Finally, during a period of three months at the Eindhoven University, we extended the DI model to a three component °uid mixture, using a di®erent form of the free energy, as derived by Lowengrub and Coworkers.. With this extension, we simulated two simple problems: ¯rst, the coalescence/repulsion of two-component drops immersed in a third component continuum phase; second, the e®ect of adding a third component to a separated two phase system. Both simulations seem to capture physical behaviors that were observed experimentally [D. Molin, R. Mauri and P. Anderson, " Phase Separation and Mixing of Three Component Mixtures," in preparation].