Flow in linearly sheared two-dimensional foams: From bubble to bulk scale

Abstract

We probe the flow of two dimensional foams, consisting of a monolayer of bubbles sandwiched between a liquid bath and glass plate, as a function of driving rate, packing fraction and degree of disorder. First, we find that bidisperse, disordered foams exhibit strongly rate dependent and inhomogeneous (shear banded) velocity profiles, while monodisperse, ordered foams are also shear banded, but essentially rate independent. Second, we introduce a simple model based on balancing the averaged drag forces between the bubbles and the top plate,F bw and the averaged bubblebubble drag forcesF bb , and assume thatF bw ∼ v 2/3 andF bb ∼ (∂yv) β , where v and (∂yv) denote average bubble velocities and gradients. This model captures the observed rate dependent flows for β ≈ 0.36, and the rate independent flows for β ≈ 0.67. Third, we perform independent rheological measurements ofF bw andF bb , both for ordered and disordered systems, and find these to be fully consistent with the forms assumed in the simple model. Disorder thus modifies the exponent β. Fourth, we vary the packing fraction φ of the foam over a substantial range, and find that the flow profiles become increasingly shear banded when the foam is made wetter. Surprisingly, our model describes flow profiles and rate dependence over the whole range of packing fractions with the same power law exponents -only a dimensionless number k which measures the ratio of the pre-factors of the viscous drag laws is seen to vary with packing fraction. We find that k ∼ (φ − φc)−1 , where φc ≈ 0.84, corresponding to the 2d jamming density, and suggest that this scaling follows from the geometry of the deformed facets between bubbles in contact. Overall, our work shows that the presence of disorder qualitatively changes the effective bubble-bubble drag forces, and suggests a route to rationalize aspects of the ubiquitous Herschel-Bulkley (power law) rheology observed in a wide range of disordered materials.

Understanding and predicting viscous, elastic, plastic flows

“…On the other hand, it can be progressively enriched by incorporating additional non-linearities, for instance related with the physico-chemistry of foams. When the velocity or the rigidity of the surfactant layer increases, shear thinning effects [16] could be introduced by an extended version of the VEP model [53] based on an Herschel-Bulkley viscoplasticity [27] instead of a Bingham one, as observed in [17,54]. The friction on walls too scales non-linearly with high velocity [16].…”

Cyclic deformation of bidisperse two-dimensional foams

“…While the deformation of monodisperse or polydisperse foams has been studied extensively, there has been less work on the shear of bidisperse foams [7,8,9,10]. In fact, bidisperse foams were found to behave differently to monodisperse foams when subjected to shear flow [7,8,9]. This difference in mechanical properties induced by the distribution of bubble sizes, and the relevance of foam flow to industry, makes this subject of particular importance.…”

“…The deformation or flow of foams is of extreme importance in industry, with applications in enhanced oil recovery, ore separation and personal care products, among many others. While the deformation of monodisperse or polydisperse foams has been studied extensively, there has been less work on the shear of bidisperse foams [7,8,9,10]. In fact, bidisperse foams were found to behave differently to monodisperse foams when subjected to shear flow [7,8,9].…”

The jamming scenario—an introduction and outlook

“…As a result, models of frictionless spheres or discs provide a reference frame to study the effect of shear. Indeed, recent experiments on sheared foams [100,101] hold the promise for testing some of the predictions of the sheared bubble model that were discussed in section 1.3.3.…”