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

We probe the flow of two-dimensional (2D) 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 adapt a simple model [E. Janiaud, D. Weaire, and S. Hutzler, Phys. Rev. Lett. 97, 038302 (2006)] based on balancing the averaged drag forces between the bubbles and the top plate (F) over bar (bw) and the averaged bubble-bubble drag forces (F) over bar (bb) by assuming that (F) over bar (bw) similar to v(2/3) and (F) over bar (bb) similar to (partial derivative(y)v)(beta), where v and (partial derivative(y)v) denote average bubble velocities and gradients. This model captures the observed rate-dependent flows for beta approximate to 0.36, and the rate independent flows for beta approximate to 0.67. Third, we perform independent rheological measurements of (F) over bar (bw) and (F) over bar (bb), both for ordered and disordered systems, and find these to be fully consistent with the forms assumed in the simple model. Disorder thus leads to a modified effective exponent beta. Fourth, we vary the packing fraction phi of the foam over a substantial range and find that the flow profiles become increasingly shear banded when the foam is made wetter. Surprisingly, the 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 that measures the ratio of the prefactors of the viscous drag laws is seen to vary with packing fraction. We find that k similar to (phi-phi(c))(-1), where phi(c) approximate to 0.84 corresponds 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.