N12, 1, p.69-84, 1993.
There are no such interactions in the classical theoactions in the classical theory of capillarity (unless the notion of disjointing pressure is introduced, i.e., long-range forces between the wall and the liquid-vapor interface). The gravity is then the only parameter which influences the equilibrium: It is the Rayleigh-Taylor instability [Taylor, 1950]. Here, our goal is to study these interactions within the framework of Cahn and Hilliard theory [Cahn, Hilliard, 1959],[Casal, 1972],[Gatignol; Seppecher, 1986]. The Cahn and Hilliard model treats both phases (vapor and liquid) as a unique fluid. Its free energy density depends not only on the mass density and the temperature but also on the gradient of the mass density. We will not introduce any long range force.
This model is of mathematical interest: (problems of minimizing surfaces as limits of more regular problems [Evans; al.,1992],[Modica, 1987b], study of non-convex functional, application of G-convergence [Bouchitte, 1990]). It is also of mechanical importance: As the compatibility of this model with the thermodynamic second principle is not obvious, it has been shown that we must add an unusual energy flux in the energy balance law [Dunn, Serrin, 1985] and that a suitable description of the strengths in such a fluid is performed by using the virtual power principle in the case of the second gradient theory [G, S, 1986], [Germain, 1973]. In this theory the stress tensor is no longer sufficient to describe the strengths, an extra stress tensor (of(of order three) appears. This extra stress tensor is not an intuitive concept. Boundary conditions in second gradient theory are very complex in the general case [G, 1973]. They can be summarized in case of a Cahn and Hilliard fluid on a rigid wall, to the classical stick condition and an unusual condition: the normal derivative of the mass density is given on the boundaries [Seppecher, 1989]. The last condition may be viewed as an information about the interactions between the fluid and the wall. It is connected with the wetting properties of the fluid on the wall: when a liquid-vapor interface is in contact with the wall the contact angle q formed at the common line is associated to this data [Cahn, 1977],[Modica, 1987a],[S, 1989]. The practical significance of the model is not clear. The exponential convergence of the mass density to its values in the phases has been criticized [De Gennes, 1985]. The accuracy of the approximation of continuum mechanics in such thin layers as interfaces is not obvious. This objection is irrelevant when the fluid is close to its critical point [Rowlinson, Widom, 1984]. On an other hand, the coefficients of the model are not known. For example, the capillarity coefficient l (cf. equation 1) is assumed to be constant because of its mathematical simplicity. Nevertheless this model is the simplest one describing interfaces.
The dependence of the free energy density upon the gradient of mass density in introduce a small characteristic length L (the characteristic thickness of the liquid-vapor interface). It has been shown that, for a fixed domain, the problem of the equilibrium of a Cahn and Hilliard fluid converges, as L tends to zero, to the classical problem of equilibrium with interfaces (Plateau problem) [M, 1987a]. So, we may presume new behaviours only when this limit procedure cannot be performed, i.e., if an other characteristic length is very small. For examples, if the wall is not plane but oscillates with a small wavelength we may expect a hysteresis behaviour [Bouchitte, Seppecher, 1992], when studying the vicinity of a moving contact line we may expect the disappearance of the dissipation singularity [Seppecher, 1991]. In the problem we deal with here, we may presume some new behaviour when the film is very thin. We emphasize that the study of problems which may mark the difference with the classical model of capillarity is the only way to form an opinion of the Cahn and Hilliard model. In the first section we recall the monodimensional solution for the equilibrium of a Cahn and Hilliard fluid [C, 1977] and what are the dimensionless variables of this equilibrium. Three of them are especially significant: the first denoted by e, is the ratio of gravity forces to capillarity strengths inside the interface, the second, denoted by a, is the ratio of the thickness of the liquid film to the thickness of the interface, the lasst, denoted by m characterize the wetting properties of the fluid at the wall. The sign of m is particularly significant. If m is positive the fluid will be called by us a wetting fluid, if m is negative it will be called a slightly wetting fluid. These two cases correspond to a contact angle smaller or larger than p/2. The case where m is close to zero, i.e. if the contact angle is close to p/2, is called the neutral case. In the second section we study the linear stability of this equilibrium in three cases. In the first case the liquid film is much thicker than the interface. We show then that the first approximation for the critical wavenumber is the classical wavenumber given by the Rayleigh-Taylor theory. In the two other cases the thickness of the liquid film is finite and there is no gravity. We show that the equilibrium is unstable if m<0, i.e., if the fluid is slightly wetting. The equilibrium is stable if m>0. The influence of the wall decreases exponentially when the thickness of the liquid film increases.
In the last part we compare the effects of the gravity to the effects of the wall. We show that, in usual conditions, even in micro-gravity conditions, the effects of the wall are insignificant. These effects may only become important for dreadfully thin films (some hundred Angstroms). However, when the temperature is close to the critical temperature of the fluid, these effects may become important even for thick fi films.