Velocity dependence of shear localisation in a 2D foam

Abstract

We give an exact formula for the velocity profile of shear localisation in a 2D foam, represented by a continuum model that incorporates a Herschel-Bulkley constitutive relation and wall drag. A more approximate treatment provides a relation between the localisation length and the boundary velocity as a power law whose exponent is explicitly determined by the input parameters of the model. This is corroborated, and its conditions for validity are clarified, by the analysis of various expansions of the exact solution. The general consequences are consistent with the recent findings of Katgert and co-workers (G. Katgert, M. Mo¨bius, and M. van Hecke, available from http://arxiv.org/abs/0711.4024 [cond-mat.soft].

The role of surfactant type and bubble surface mobility in foam rheology

“…For example, the values of the flow indexes and the character of shear-localization phenomena were shown to be affected strongly by the covering solid wall (present in most of the experiments with 2D-foams) which creates additional bubble-wall friction force without direct analog in 3D-foams. [76][77][78][79][80] Even when the solid wall is absent, the measured flow indexes are often different from those measured with 3D-foams of the same surfactants (for example, cf. the results with Dawn presented in section 5.3 below to those presented in ref.…”

“…79,81 On this basis, a hypothesis was put forward that these results should be representative for the typical polydisperse foams. [76][77][78]81 To clarify the role of bubble polydispersity in foam viscous friction, we performed additional experiments with monodisperse 3D-foams. In Fig.…”

“…[76][77][78][79][80][81][93][94][95][96][97][98][99][100] These experiments proved to be very valuable in analyzing several important phenomena (e.g., those related to non-homogeneous flows) and to clarify some common features of the different types of ''yield stress systems'', such as soft glassy materials, granular materials, foams, and emulsions. Also, the theoretical modelling of 2D-foams is technically much simpler, as compared to the 3D-foams, so that the theoretical concepts could be more easily tested with 2D-foams.…”

“…At high shear rates, dynamic foam and emulsion films are formed between the neighboring bubbles and drops-the thickness of these films must depend on the applied shear rates and, therefore, one of the main assumptions of the theoretical models in ref. [76][77][78][79][80][81] is not justified. Also, direct optical observations show that the 3D-foam flow at high shear rates is rather homogeneous and regular, independently of foam polydispersity.…”

“…in the regime of well developed, regular foam flow. Therefore, we suppose that the theoretical models predicting n z 1/2 at constant film thickness [76][77][78][79][80][81] could be representative only for foams and emulsions sheared at low rates, close to the jamming transitions, when the flow of polydisperse foams is non-homogeneous. At high shear rates, dynamic foam and emulsion films are formed between the neighboring bubbles and drops-the thickness of these films must depend on the applied shear rates and, therefore, one of the main assumptions of the theoretical models in ref.…”