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## Master part 1

This following posts present some of the work I’ve done in my Master thesis under the coordination of Dorin Bucur and Edouard Oudet and it will present a mathematical approach to the study the equilibrium configurations of immiscible fluids. The mathematical model of the energy of a configuration of immiscible fluids goes back to Massari [1] and takes into account all the important factors which exist in the physical configuration: the interfacial tension, the contact with the container and the gravitational potential energy.

The interfacial tension between two immiscible fluids comes as a consequence of the different structure of the molecules of the two fluids and the lack of attraction between molecules of different kinds. One way to explain the interfacial tension is that molecules in the bulk of the fluid are surrounded in all sides by similar molecules, and even though the attraction force between molecules of the same kind is large, the fact that these attraction forces pull equally in every direction gives some kind of equilibrium. On the other hand, molecules situated at the boundary of the fluid have only half of the neighbors that a bulk molecule has, and as a consequence the boundary molecules are pulled inwards into the liquid. Since the fluid doesn’t shrink indefinitely due to this inward attraction of boundary molecules, it follows that there is another force which balances this inward attraction force, and this force acts on the boundary of the fluid. This is the interfacial tension. The interfacial tension is measured in force per unit length, so it does not depend on the size of the interface of the fluids.

Another explanation for the existence of the interfacial tension is that when we want to increase the boundary surface between two immiscible fluids we consume energy, since we replace the strong bounds of the molecules of one of the fluids with weak bounds of molecules between different fluids, and we get nothing in return. This means that there is a force which resists the increase of the surface of the boundary, and that is the interfacial tension.

Although the interfacial tension is an intermolecular force, some of its effects can be seen on a macroscopic scale: water strider can walk on water, some needles and coins can float, etc.

One of the properties of the surface tension of three immiscible fluids ${E_1,E_2,E_3}$ is that

$\displaystyle \sigma_{12}+\sigma_{23} > \sigma_{13}$

where ${\sigma_{ij}}$ represents the interfacial tension between fluids ${E_i}$ and ${E_j}$. For proofs and justifications of this inequality see [2] and [3]. The idea is that a triangle with sides ${\sigma_{12},\sigma_{23}}$ and ${\sigma_{13}}$ can be formed due to the connection of the sizes of ${\sigma_{ij}}$ with the angles formed at the interface of three fluids. This hypothesis on the values of the surface tensions is essential in the proof of the existence of the optimal configuration.

The contact with the walls of the container is almost negligible for large containers. In fact there exists a particular length ${\kappa^{-1}=\sqrt{\gamma/\rho g}}$ which is referred to as thecapillary length and physical experiments show that for sizes of order smaller than ${\kappa^{-1}}$ the gravity can be defeated by the capillary effects. This can be seen as liquids rise in thin tubes, but not in thick tubes. When the size of the configuration is larger than ${\kappa^{-1}}$ then the gravitational energy prevails. Still it can be observed that at a distance of ${\kappa^{-1}}$ near the wall of the container or near a small floating object, the surface of the fluid is perturbed by the wall or by the floating object.

Still, if we denote by ${\beta_i}$ the interfacial tension of the fluid ${E_i}$ with walls of the container ${\Omega}$ the angle which the interface between fluids ${E_i,E_j}$ makes with the boundary of the container satisfies the relation

$\displaystyle \cos\alpha = \frac{\beta_i-\beta_j}{\sigma_{ij}}$

which readily implies the inequality ${|\beta_i-\beta_j|\leq \sigma_{ij}}$ which is considered as a hypothesis in the existence theorem. The relation above shows that this hypothesis is not artificial. For the proof the reader may consult [2].

Gravity has a great influence on the configuration of immiscible fluids. As can be seen in some pictures taken in space, in zero gravity water bulks takes the shape of balls, proving again that fluids left free tend to minimize their surface, whereas in the presence of gravity only small drops of water have spherical form. Gravity makes the heavier fluids travel to the bottom of the container, while the easier fluids go towards the upper part of the container. Its effects are important and cannot be neglected if we want to propose a realistic model.

Once the model has been proposed, we are interested in two aspects of the study: the existence of the optimal configuration, which minimizes the energy of the system, and the determination or approximation of the equilibrium configuration. For the existence results we use the classical methods of the calculus of variations, i.e. we prove that the functional is bounded from below, that any minimizing sequence contains a subsequence which is convergent and that the energy functional is lower semicontinuous with respect to a good topology, facts which combined prove that a minimizer exists.

The only aspect of the ones presented above which presents some difficulties is the proof of the lower semicontinuity of the energy functional. In the case of two or three fluids the terms containing the surface tension are lower semicontinuous, and the gravity terms are continuous (we consider the container to be bounded), so the terms which give rise to problems are the terms which model the contact with the walls of the container. To go beyond this difficulty we have two approaches. The first one, adapted after the ideas of Massari [1], uses some trace inequalities for ${BV}$ functions from the article of Anzellotti and Giaquinta [4], which bounds the wall contact energy in terms of the relative perimeter and the volume of the fluid. The second one, adapted after the ideas of Baldo [5] uses the fact that the ${\Gamma}$-limit is already lower semicontinuous, so if we prove that our functional is the ${\Gamma}$-limit of a sequence of well chosen functionals then we are done. One great advantage is that the ${\Gamma}$-convergence is stable under continuous perturbations, so the gravity term doesn’t pose any problems in our study.

This last approach gives also an approximation result for the equilibrium configuration by ${\Gamma}$-convergence and it works for an arbitrary finite number of fluids, not only for two or three fluids. Numeric results using ${\Gamma}$-convergence have been obtained for optimal partitions by \’E. Oudet [6]. In the last part we will present some numerical approximations of the optimal configuration using ${\Gamma}$-convergence.

[1] Massari, U. The parametric problem of capillarity: The case of two and three fluids

[2] Rowlinson, J.S. and Widom, B. Molecular Theory of Capillarity

[3] Cahn, John W., Critical point wetting

[4] Anzellotti, G. and Giaquinta, M., BV functions and traces

[5] Sisto Baldo, Minimal Interface Criterion for Phase Transitions in Mixtures of Cahn-Hilliard fluids

[6] Oudet, Edouard, Approximation of partitions of least perimeter by ${\Gamma}$-convergence: around Kelvin’s conjecture