Capillary rarefaction

Capillary rarefaction. The continuous fluid phase filling the porous space between deformed bubbles has a common concave boundary with these bubbles. It follows that the local pressure Pi in the liquid phase is less than the pressure Pg in the gaseous phase and these variables are related by... 

In contrast with the capillary pressure APg, the capillary rarefaction APi is a local variable. The distribution of APi in the continuous phase determines all interior flows of liquid in foam. At the same time, the averaged integral capillary rarefaction determines the capability of foam to absorb liquid, and it presents certain strength to the foam body. 

Sometimes, the term osmotic dispersion pressure [378, 383] is used instead of capillary rarefaction. The osmotic pressure is defined as the excessive external pressure that must be applied to the semipermeable membrane interface between foam and fluid to stop the flux of the fluid sucked into the foam from the free volume. In this case, it is assumed that foam cell faces are flat, and therefore, the capillary pressure in foam bubbles is zero. 

According to (7.1.15), the capillary rarefaction depends on the mean curvature s of the internal foam surface at the nodes of the foam structure. This variable was calculated in [378] for a monodisperse foam with cells modeled by pentagonal dodecahedra ... 

It follows from [378, 383] that the formula for the osmotic pressure differs from the corresponding formula for the capillary rarefaction only by the coefficient g(< ) = (1 - 1.83 1 - )2 which characterizes the fraction of the membrane area adjacent to flat faces of foam cells. In the case of polydisperse foam, it is also expedient to use formula (7.1.16) with a replaced by the Sauter mean radius (7.1.9). 

Although, according to (7.1.15), the capillary rarefaction is a local variable, its values at nodes and in Plateau borders are equalized rather rapidly due to the flow of liquid, which allows one to assume that the capillary rarefaction at nodes and in Plateau borders is a unified integral characteristic of foam. This means that the mean curvature of menisci at nodes and borders is the same. However, since nodes possess a spherical curvature and Plateau borders a cylindrical curvature, the radius of curvature of the latter must be two times less than that of the nodal menisci,... 

As regards such a geometric parameter as the film thickness h, in actual foam, the film thickness gradually decreases due to slow pressing-out of liquid from films into Plateau borders. This process is virtually terminating when the pressure difference between the phases (equal to the sum of the capillary pressure and the capillary rarefaction) is compensated for by the disjoining pressure II(/i), which arises in thin films under the interaction of adsorption layers [116] ... 

The outflow of a liquid from foam under the action of gravity field was considered in [15] and termed syneresis. Later on, syneresis was attributed to capillary effects, primarily due to the gradients of capillary rarefaction. The importance of these effects was already mentioned in [266], but the gradient of capillary rarefaction was rightly set to be zero, since only steady-state flows of liquids through foam layer were considered there. The fundamental role of the gradient of capillary rarefaction in the process of evolution of a foam layer in syneresis was also pointed out in [215, 335]. 

Indeed, as fluid flows, foam channels closed above grow in thickness at the bottom, thus creating an increasing counteraction to the gravitational force, which slows down the outflow until equilibrium is attained [214]. It should be noted that this effect is possible only in closed deformable channels with negative curvature, which are typical of foam. According to [324], the capillary rarefaction is a characteristic of the foam compressibility and determines its elastic resistance to the strain caused by the liquid redistribution. 

Here p and p are the liquid and gas densities, respectively, g is the vector of the gravitational acceleration, and AP is the capillary rarefaction given by (7.1.10) and (7.1.15). The kinetic coefficient H was called the coefficient of hydroconductivity and calculated for polyhedral foam models [245, 246]. Generally speaking, the variable H is a tensor, but usually the isotropic approximation is used, where this parameter is a scalar. Various expressions for the coefficient H were proposed and made more precise in [125, 214, 245]. Thus, different approaches used to calculate the coefficient of hydroconductivity were analyzed in [488]. For example, the structure of spherical and cellular foam was studied under the assumption that liquid flows through a porous layer according... 

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