Foam hydrodynamic model

Langevin et al. [35,71] have proposed a simplified hydrodynamic model of thinning of microscopic foam films that accounts for the influence of surface elasticity on the rate of thinning in a large range of thicknesses and Ap. However, as noted by the authors, in view of the rapid loss of surfactant molecules at the surface during film drainage, the elasticity would not correspond to the actual bulk surfactant concentration but to lower values since the system is very far from equilibrium. Frequency dependence of surface elasticity has been considered by Tambe and Sharma [72]. 

The correctness of the above hydrodynamic model of a polyhedral foam with border hydroconductivity and constant radius of border curvature can be confirmed by comparing the velocity Q calculated from Eqs. (5.9) and (5.11) with the volumetric liquid rate obtained by Krotov s theory [7]. Thus, the liquid flow under gravity at r = const is... 

The expansion ratio profile of a continuously generated foam has been computed using various hydrodynamic models [87,88] but here again several significant simplifications are introduced. For example, a model of polyhedral bubbles was employed for all foam layers situated at different levels which, however, is not the real state in the lower foam layers. 

Reynolds relation requires liquid drainage from the film to follow strictly the axial symmetry between parallel walls. Rigid surfaces ensure such drainage through their non-deformability, while non-equilibrium foam films are in fact never plane-parallel. This is determined by the balance between hydrodynamic and capillary pressure. Experimental studies have shown that only microscopic films of radii less than 0.1 mm retain their quasiparallel surfaces during thinning, which makes them particularly suitable for model... 

Using a continuum rather than cellular model implies that the growing foam is regarded as a fluid with a continuously distributed source of flow. Hence, the flow rate is also a function of position the (hydrodynamic) pressure gradient is resulted from inertia, gravity and stress mechanisms operating in the fluid. 

An approach that is almost diametrically opposed to the earlier models of Khan and Armstrong, and Kraynik and Hansen, was advanced by Schwartz and Princen (108). In this model, the films are negligibly thin, so that all the continuous phase is contained in the Plateau borders, and the surfactant tiuns the film surfaces immobile as a result of surface-tension gradients. Hydrodynamic interaction between the films and the Plateau borders is considered to be crucial. This model, believed to be more realistic for common sur factant-stabilized emulsions and foams, draws on the work of Mysels et al. (109) on the dynamics of a planar, vertical soap film being pulled out of, or pushed into, a bulk solution via an intervening Plateau border. An important result of their analysis is commonly referred to as FrankeTs law, which relates the film thickness, 2h., to the pulling velocity, U, and may be written in the form ... 

This section begins with a qualitative description of thin liquid PU films. This initial investigation had five goals in mind to confirm that stable, vertically-oriented, thin liquid films could be prepared using mixtures of ingredients designed to model a PU foam, to study the hydrodynamic phenomena in the films, to compare the physical behaviour of these films to the behaviour of the more common aqueous soap films, to observe specific surfactant effects on the properties of these films, and to extrapolate conclusions about the behaviour of these films to operational PU foam. 

Of the three mechanisms, hydrodynamic drainage due to gravity is usually the most rapid and, if the foam is particularly unstable, leads to total collapse before other mechanisms can become important. In those cases, once the loss of liquid from the lamellar layer produces a critical thickness of 5-15 nm, the liquid film can no longer support the pressure of the gas in the bubble, and film rupture occurs. As a model for gravity drainage, a film may be treated as a vertical slit of thickness S (not to be confused with the solubility parameter... 

The accumulated evidence from studies on the OFC suggests that the adsorption of C ,TABs is diffusion-controlled below the cmc. Above the cmc, there are deviations from a diffusion-controlled model for CigTAB + NaBr, which can be quantitatively explained by slow micellar breakdown kinetics. The alternative of an adsorption barrier cannot be ruled out, though there is as yet no evidence of structm es at the air-water interface akin to those observed in the SAR at the solid-liquid interface. More limited studies on other femiUes of ionic scu -factants in the OFC and MBP apparatus do not show large deviations from diffusion control. The importance of well-defined hydrodynamics and accru-ate equilibriiun adsorption isotherms cannot be overstressed in quantitative studies of adsorption mechanisms. There is still a need for measurements at higher strain rates, such as occur in tcffbulent foams, jet breakup and impacting drops, and for additional studies with micellar systems to establish quantitatively the connection between micellar breakdown kinetics and rates of adsorption. 

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