Simulations of two-dimensional foam rheology: localization in linear Couette flow and the interaction of settling discs

Micro-mechanical prediction of the effect of surfactant concentration and external friction on the visco-elasto-plastic response of an aqueous foam

We apply a combination of the Viscous Froth model and a surfactant transfer model [Zaccagnino et al., Phys. Rev. E, 2018, 98, 022801] to predict the rheological response of a two-dimensional dry aqueous foam. The model includes both the effect of friction between the foam and the boundaries of the container and also the dissipative effects on the film interfaces caused by surfactant motion. These dynamics are characterized by two free parameters: the Gibbs elasticity, relating surfactant concentration to interfacial tension, and the mobility of the surfactant molecules on the interfaces. We employ numerical simulations to evaluate the static shear modulus, yield stress and the storage and loss moduli of a foam and investigate the effect of our free parameters on these rheological properties.

Universal Features of Metastable State Energies in Cellular Matter

Mechanical equilibrium states of cellular matter are overwhelmingly metastable and separated from each other by topology changes. Using theory and simulations, it is shown that for a wide class of energy functionals in 2D, including those describing tissue cell layers, local energy differences between neighboring metastable states as well as global energy differences between initial states and ground states are governed by simple, universal relations. Knowledge of instantaneous length of an edge undergoing a T1 transition is sufficient to predict local energy changes, while the initial edge length distribution yields a successful prediction for the global energy difference. An analytical understanding of the model parameters is provided.

A geometric exploration of stress in deformed liquid foams

We explore an alternate way of looking at the rheological response of a yield stress fluid: using discrete geometry to probe the heterogeneous distribution of stress in soap froth. We present quasi-static, uniaxial, isochoric compression and extension of three-dimensional random monodisperse soap froth in periodic boundary conditions and examine the stress and geometry that result. The stress and shape anisotropy of individual cells is quantified by Q, a scalar measure derived from the interface tensor that gauges each cell's contribution to the global stress. Cumulatively, the spatial distribution of highly deformed cells allows us to examine how stress is internally distributed. The topology of highly deformed cells, how they arrange relative to one another in space, gives insight into the heterogeneous distribution of stress.

Detailed Structural and Mechanical Response of Wet Foam to the Settling Particle

Liquid foam, as a complex fluid, provides an observable prototype for studying a discrete fluid system. In this work, a numerical study on the settling behavior of a round particle in wet polydisperse foam has been conducted on the bubble scale. The local and nonuniform distribution of bubble pressure, as well as the localized plastic events, is presented. It shows a foam region of higher pressure in front of the settling particle due to the extrusion deformation of the bubbles applied by the particle. Additionally, the forces exerted on the particle by the disordered wet foam are measured during the sedimentation. It exhibits in particular a power-law dependence of the drag force caused by the bubble as a function of the foam quality. Moreover, sedimentation experiments are demonstrated to verify this power-law relation. The evolution of the components of drag force is demonstrated when a plastic event occurs in front of the settling particle. The result shows that both the contributions of the pulling force of foam films and the bubble pressure to the drag force decrease in that case. Likewise, the variation of both these contributions to the drag force is illustrated as well when a bubble in the wake detaches from the particle. These results assist in understanding the mesoscopic response of wet foam to a settling particle.

Networklike propagation of cell-level stress in sheared random foams

Quasistatic simple shearing flow of random monodisperse soap froth is investigated by analyzing surface evolver simulations of spatially periodic foams. Elastic-plastic behavior is caused by irreversible topological rearrangements (T1s) that occur when Plateau's laws are violated; the first T1 determines the elastic limit and frequent T1 avalanches sustain the yield-stress plateau at large strains. The stress and shape anisotropy of individual cells is quantified by Q, a scalar derived from an interface tensor that gauges the cell's contribution to the global stress. During each T1 avalanche, the connected set of cells with decreasing Q, called the stress release domain, is networklike and nonlocal. Geometrically, the networklike nature of the stress release domains is corroborated through morphological analysis using the Euler characteristic. The stress release domain is distinctly different from the set of cells that change topology during a T1 avalanche. Our results highlight the connection between the unique rheological behavior of foams and the complex large-scale cooperative rearrangements of foam cells that accompany distinctly local topological transitions.

Faraday instability at foam-water interface

A nearly two-dimensional foam is generated inside a Hele-shaw cell and left at rest on its liquid bath. The system is then vertically shaken and, above a well-defined acceleration threshold, surface waves appear at the foam-liquid interface. Those waves are shown to be subharmonic. The acceleration threshold is studied and compared to the common liquid-gas case, emphasizing the energy dissipation inside the foam. An empirical model is proposed for this energy loss, accounting for the foam characteristics such as the bubble size but also the excitation parameter, namely the linear velocity.

Analytical results for size-topology correlations in 2D disk and cellular packings

Random tilings or packings in the plane are characterized by a size distribution of individual elements (domains) and by the statistics of neighbor relations between the domains. Most systems occurring in nature or technology have a unimodal distribution of both areas and number of neighbors. Empirically, strong correlations between these distributions have been observed and formulated as universal laws. Using only the local, correlation-free granocentric model approach with no free parameters, we construct accurate analytical descriptions for disk crystallization, size-topology correlations, and Lemaître's law.

Elastoplastic flow of a foam around an obstacle

We simulate quasistatic flows of an ideal two-dimensional monodisperse foam around different obstacles, both symmetric and asymmetric, in a channel. We record both pressure and network contributions to the drag and lift forces and study them as a function of obstacle geometry. We show that the drag force increases linearly with the cross section of an obstacles. The lift on an asymmetric aerofoil-like shape is negative and increases with its arc length, mainly due to the pressure contribution.

The continuum theory of shear localization in two-dimensional foam

We review some recent advances in the rheology of two-dimensional liquid foams, which should have implications for three-dimensional foams, as well as other mechanical systems that have a yield stress. We focus primarily on shear localization under steady shear, an effect first highlighted in an experiment by Debrégeas et al. A continuum theory which incorporates wall drag has reproduced the effect. Its further refinements are successful in matching results of more extensive observations and making interesting predictions regarding experiments for low strain rates and non-steady shear. Despite these successes, puzzles remain, particularly in relation to quasistatic simulations. The continuum model is semi-empirical: the meaning of its parameters may be sought in comparison with more detailed simulations and other experiments. The question of the origin of the Herschel-Bulkley relation is particularly interesting.

Foam as a complex system

What is a 'complex system'? The two-dimensional foam, as originally popularized by Cyril Stanley Smith, provides an ideal context in which to explore this question.

Comparison of low-amplitude oscillatory shear in experimental and computational studies of model foams

A fundamental difference between fluids and solids is their response to applied shear. Solids possess static shear moduli, while fluids do not. Complex fluids such as foams display an intermediate response to shear with nontrivial frequency-dependent shear moduli. In this paper, we conduct coordinated experiments and numerical simulations of model foams subjected to boundary-driven oscillatory planar shear. Our studies are performed on bubble rafts (experiments) and the bubble model (simulations) in two dimensions. We focus on the low-amplitude flow regime in which T1 events, i.e., bubble rearrangement events where originally touching bubbles switch nearest neighbors, do not occur, yet the system transitions from solid- to liquidlike behavior as the driving frequency is increased. In both simulations and experiments, we observe two distinct flow regimes. At low frequencies omega, the velocity profile of the bubbles increases linearly with distance from the stationary wall, and there is a nonzero total phase shift between the moving boundary and interior bubbles. In this frequency regime, the total phase shift scales as a power law Delta approximately omegan with n approximately 3. In contrast, for frequencies above a crossover frequency omega>omegap, the total phase shift Delta scales linearly with the driving frequency. At even higher frequencies above a characteristic frequency omeganl>omegap, the velocity profile changes from linear to nonlinear. We fully characterize this transition from solid- to liquidlike flow behavior in both the simulations and experiments and find qualitative and quantitative agreements for the characteristic frequencies.

Viscous shear banding in foam

Shear banding is an important feature of flow in complex fluids. Essentially, shear bands refer to the coexistence of flowing and nonflowing regions in driven material. Understanding the possible sources of shear banding has important implications for a wide range of flow applications. In this regard, quasi-two-dimensional flow offers a unique opportunity to study competing factors that result in shear bands. One proposal for interpretation and analysis is the competition between intrinsic dissipation and an external source of dissipation. In this paper, we report on the experimental observation of the transition between different classes of shear bands that have been predicted to exist in cylindrical geometry as the result of this competition [R. J. Clancy, E. Janiaud, D. Weaire, and S. Hutzlet, Eur. J. Phys. E 21, 123 (2006)].

Rheological properties of the soft-disk model of two-dimensional foams

The soft-disk model previously developed and applied by Durian [D. J. Durian, Phys. Rev. Lett. 75, 4780 (1995)] is brought to bear on problems of foam rheology of longstanding and current interest, using two-dimensional systems. The questions at issue include the origin of the Herschel-Bulkley relation, normal stress effects (dilatancy), and localization in the presence of wall drag. We show that even a model that incorporates only linear viscous effects at the local level gives rise to nonlinear (power-law) dependence of the limit stress on strain rate. With wall drag, shear localization is found. Its nonexponential form and the variation of localization length with boundary velocity are well described by a continuum model in the spirit of Janiaud etal [Phys. Rev. Lett. 97, 038302 (2006)]. Other results satisfactorily link localization to model parameters, and hence tie together continuum and local descriptions.