The Basic Equations and Boundary Conditions

The problems that we consider in this section are closely related to those in the previous section. As shown in Fig. 6-1, we again consider a thin liquid layer that is bounded below by a solid wall, but with an upper surface that is an interface with an upper fluid that we 

One additional comment regardings Eqs. (6-2) and (6-3) is the fact that we have retained the body-force terms in spite of the fact that they are multiplied by a dimensionless parameter that involves e to the power of either 2 or 3. However, the magnitude of these terms is not necessarily small because we have not yet specified the characteristic velocity uc. Unlike the lubrication problems of the preceding chapter, in which a characteristic velocity can be specified in terms of the specified velocity (or force) at the boundaries of the thin film, here the characteristic velocity is dependent on the dominant physical mechanism that is responsible for the evolution of the film with time. One of the possible mechanisms is gravitationally driven motion. If this occurs in a film that is on a surface that has a finite angle of inclination a, then we see from (6-2) that an appropriate choice for uc would be e2i2cpg/ix, and (6-3) would then be approximated as 

On the other hand, if a = 0, and the dynamics of the film is still dominated by body forces, then it appears from (6-3) that uc = eil1cpg/ii. In other cases, however, gravitational forces may play only a secondary role in the motion of the film, which is instead dominated by capillary forces. Then the appropriate choice for uc would involve the surface tension rather than either of the choices previously listed and the body-force terms in both (6-2) and (6-3) would be asymptotically small for the limit e - 0. This then is a fundamental difference between this class of thin-film problems and the lubrication problems of the previous chapter. Here, the characteristic velocity will depend on the dominant physics, and if we want to derive general equations that can be used for more than one problem, we need to temporarily retain all of the terms that could be responsible for the film motion and only specify uc (and thus determine which terms are actually large or small) after we have decided which particular problem we wish to analyze. 

Apart from the trivial inclusion of the gravitational body-force terms in (6-2) and (6-3), the governing equations, and the analysis leading to them, are identical to the governing equations for the lubrication theory of the previous chapter. The primary difference in the formulation is in the boundary conditions, and the related changes in the physics of the thin-film flows, that arise because the upper surface is now a fluid interface rather than solid surface of known shape. The boundary conditions at the lower bounding surface are  

As discussed in Chap. 2, we can specify the position of the upper interface as corresponding to those points x for which the function F (x, f ) = z — h (x s, t ) is identically equal to zero. Then, the usual boundary conditions for a fluid-fluid interface, written in dimensional form, are 

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The basic equation and boundary conditions for the symmetrical fluctuations are the same as those for the asymmetrical fluctuations except for the superscript s. The diffusion equation is written in the form... 

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