• miscele
• modello ad interfacce diffuse
• separazione di fase

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].