Mean Flow Velocity of Antifoam Entities

We will suppose that a foam film drains according to lubrication theory, which means that the drainage is described by the Reynolds equation [33]. For such a cylindrical film of radius Vf and thickness h with two immobile surfaces, the fluid flow velocity profile u(z) is given [33,42] 

As illustrated in Hgure 5.7, the excluded volume effect means that spherical antifoam entities of radius r p can only experience flow between z = r p and z = h- r p. 

FIGURE 5.8 Schematic illustration of drainage in cylindrical, plane-parallel film with immohile air-water surfaces. 

Here we suppose that the concentration of antifoam entities is sufficiently high so that entities sample the complete velocity profile. If then we neglect interactions between antifoam entities and their interactions with the air-water surface (including electrostatic interactions), we can write for the mean velocity Uf,p of the antifoam entities 

2 Effect of Excluded Volume on Antifoam Concentration in a Draining Film 

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