Foam–wall friction: Effect of air volume fraction for tangentially immobile bubble surface

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Wall-slip is an important phenomenon for the rheological behavior of various dispersions, including foams. In a previous study [N.D. Denkov et al., Coll. Surf. A 263 (2005) 129] we showed experimentally that the viscous stress between a foam and sliding solid wall is proportional to Ca 1/2 for a foam generated from potassium carboxylate surfactants at air volume fraction of Φ 3D = 90% ( Ca = μV 0/ σ is the capillary number). This experimental finding was explained by a theoretical model, whose main assumptions were: (1) the foam bubbles had tangentially immobile surface, and (2) the viscous friction occurred only in the dynamic wetting films formed between the bubbles and the solid wall. The current study is a continuation of our previous work, with the major aim to clarify theoretically the effect of air volume fraction, Φ, on the foam–wall friction (the variation of Φ is equivalent to varying the ratio of the film to Plateau border radii, R F/ R P). By using the lubrication approximation, we show with numerical calculations that for bubbles with tangentially immobile surfaces, more generally, the bubble–wall friction force could be represented as a superposition of two components: friction inside the wetting film, which is proportional to Ca 1/2 and friction in the transition zone film-meniscus, which is proportional to ( A 1 Ca 3/4 − A 2 Ca), where A 1,2 are numerical constants depending on Φ. The numerical calculations show also that the term describing the friction in the transition zone film-meniscus, is very well approximated by A 3 Ca 2/3 in the range of capillary numbers 10 −6 ≤ Ca ≤ 10 −3. An explicit expression is proposed, which describes rather well the numerical results for the friction force. The calculations show that the relative contribution of the friction inside the film dominates at Φ 3D ≥ ca. 90%. However, at lower air volume fractions (i.e., at smaller ratio R F/ R P) the friction in the transition zone film-meniscus could be significant and the effective power law index could have a value close to 2/3, as predicted in Bretherton's model [F.P. Bretherton, J. Fluid Mech. 10 (1961) 166]. The predictions of the current model are verified with literature data about the bubble–wall friction force, as well as with original experimental results about the thickness of the dynamic wetting films, formed between bubbles and moving solid wall. The obtained results could be useful in the analysis of the particle–wall friction in other systems, such as emulsions and dispersions of gel particles. The theoretical approach could be modified to describe various phenomena, such as motion of drops/bubbles in capillaries (including microfluidic applications) and hydrodynamic lubrication of deformable solid particles.