Wall slip and viscous dissipation in sheared foams: Effect of surface mobility
Wall-slip is a general phenomenon in the rheological behavior of foams and has to be considered explicitly in the description of foam flow through pipes and orifices, upon spreading on surfaces, and in the rheological measurements. On the other hand, the wall-slip, occurring between a plug of foam and smooth wall, is an appropriate phenomenon for experimental and theoretical study of the viscous friction in liquid films, because the corresponding viscous stress, which is amenable to experimental measurement, does not interfere with the foam elastic stress. The current paper presents a theoretical model and experimental results about the viscous friction between foam and smooth wall. First, the lubrication model is used to calculate the friction force between a single bubble and the wall, in the case of bubbles with tangentially immobile surfaces. Next, the functions introduced by Princen and Kiss [H.M. Princen, A.D. Kiss, Langmuir 3 (1987) 36] to relate the micro-structure of the foam (bubble and film radii, bubble capillary pressure) with the foam macroscopic properties (air volume fraction and foam osmotic pressure) are used to estimate the average, experimentally accessible wall stress, ÏW, from the friction force of individual bubbles. The model predicts ÏW ∝ (Ca*)1/2 where Ca* = (Î¼V0/Ï) is the capillary number, defined with respect to the relative velocity of the foam and wall, V0 (Î¼ is the liquid viscosity and Ï the surface tension). This prediction differs from the classical result, ÏW ∝ (Ca*)2/3, derived by Bretherton [F.P. Bretherton, J. Fluid Mech. 10 (1961) 166]. The analysis shows that the two theoretical models correspond to two limiting cases, governed mainly by the surface mobility of the bubbles. These limiting cases are verified experimentally by measuring the viscous stress in the foam/wall region with properly chosen surfactant solutions, which ensure tangentially mobile or immobile surface of the bubbles. Furthermore, it is shown experimentally that the effect of bubble surface mobility affects strongly the viscous friction inside sheared foams. The viscous stress in continuously sheared foam is described very well with a power law, ÏV ∝ Can, where Ca is the capillary number defined here with respect to the shear rate inside the foam. The power index n was determined experimentally to be equal to 0.25 ± 0.02 for tangentially immobile and to 0.42 ± 0.02 for tangentially mobile bubble surfaces, respectively, at air volume fraction of 90%.