Tangentially-Immobile Films

Each of the constituent terms of Equation 11.1 represents a distinct force field. From left to right the terms represent the contributions of viscous forces, surface tension forces due to the curvature at the free interface (Laplace pressure), and the excess intermolecular forces (disjoining pressure) respectively [37, 38, 65, 67]. The viscous force in no way influences the stability as it merely controls the dynamics of the system. For tangentially immobile films, the prefactor of the viscous term 3 is replaced by 12 [38, 65]. The Laplace pressure arising from surface tension has a stabilizing influence, as already discussed. Thus, the only term that may induce an instability in the system is the one representing the excess intermolecular interactions [37,38,65]. 

In the case when two plane-parallel ellipsoidal disks of tangentially immobile surfaces are moving against each other under the action of an external force, F,, from Equations 5.255 and 5.256, we can derive the Reynolds equation for the velocity of film thinning ... 

The final thickness, hp may coincide with the critical thickness of film rupture. Equation 5.273 is derived for tangentially immobile interfaces from Equation 5.259 at a fixed driving force (no disjoining pressure). 

As discussed in Sections 5.5.2.1 and 5.5.3.2, a fluid particle in the presence of high surfactant concentration can be treated as a deformable particle of tangentially immobile surfaces. Such a particle deforms when pressed against a solid wall (see the inset in Figure 5.49). To describe the drag due to the film intervening between the deformed particle and the wall, we may use the... 

In a first approximation, one can assume that the viscous dissipation of kinetic energy happens mostly in the thin liquid film intervening between two drops. (In reality, some energy dissipation happens also in the transition zone between the film and the bulk continuous phase.) If the drop interfaces are tangentially immobile (owing to adsorbed surfactant), then the velocity of approach of the two drops can be estimated by meanss of the Reynolds formula for the velocity of approach of two parallel solid disks of radius R, equal to the film radius (142) ... 

Equation (82) shows that the disjoining pressure significantly influences the transitional thickness The effect of surface mobility is characterized by the parameter d, see Eq. (53) in particular, d = 0 for tangentially immobile interfaces. Equation (82) is valid for H < 2o/u, i.e., when the film thins and ruptures before reaehing its equilibrium thickness, eorresponding to H = 2o/a [cf Eqs (42), (43), and (59)]. 

In the speeial ease of tangentially immobile interfaces and large film (negligible effeet of die transition zone) one has Oy(h) = 1, and the integration in Eq. (84) ean be earried out... 

To develop a quantitative model, the problem was approached in stages. The first model assumed that the surface of the film was wedge-shaped the narrowing with time was predicted in the tangentially-immobile case [51]. The film was then modelled with a deforming surface the film still thins, but it takes on more complicated shapes predicted by the solutions of non-linear partial differential equations. These generalisations have resulted in a series of models that gradually incorporate more of the experimental behaviour. A few details and some results from these models will now be discussed. 

Figure 5.11 Film shape at t = 16 for several values of S (shown in the upper right). The film is much thinner for small values of S because the free surface is mobile. For large values of S, the free surface becomes tangentially immobile.

Eq. (8.6) applies for the following conditions (1) The liquid flows between parallel plane surfaces (2) film surfaces are tangentially immobile and (3) the rate of thinning due to evaporation is negligible compared with the thinning due to drainage. 

There are various cases of particle-interface interactions, which require separate theoretical treatment. The simpler case is the hydrodynamic interaction of a solid particle with a solid interface. Other cases are the interactions of fluid particles (of tangentially mobile or immobile interfaces) with a solid surface in these cases, the hydrodynamic interaction is accompanied by deformation of the particle. On the other hand, the colloidal particles (both solid and fluid) may hydrodynamically interact with a fluid interface, which thereby undergoes a deformation. In the case of fluid interfaces, the effects of surfactant adsorption, surface diffusivity, and viscosity affect the hydrodynamic interactions. A special class of problems concerns particles attached to an interface, which are moving throughout the interface. Another class of problems is related to the case when colloidal particles are confined in a restricted space within a narrow cylindrical channel or between two parallel interfaces (solid and/or fluid) in the latter case, the particles interact simultaneously with both film surfaces. 

FIGURE 5.49 Deformed fluid particle (the inset) moving tangentially to an immobile solid snrface plot of the dimensionless drag coefficient, fy, vs. the dimensionless film thickness, hlR for three valnes of the dimensionless film radius, R/R (see Equation 5.303). 

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