We formulate and apply a continuum model that incorporates elasticity, yield stress, plasticity, and viscous drag. It is motivated by the two-dimensional foam rheology experiments of Debregeas et al. [Phys. Rev. Lett. 87, 178305 (2001)] and Wang et al. [Phys. Rev. E 73, 031401 (2006)], and is successful in exhibiting their principal features, which are an exponentially decaying velocity profile and strain localization. Transient effects are also identified. DOI: 10.1103/PhysRevLett.97.038302 PACS numbers: 83.80.Iz, 82.70.Rr, 83.10.Ff While initially two-dimensional (2D) foams were introduced only as a simple model system for numerical and theoretical studies [1,2], recent years have also seen a variety of rheological experiments on so-called quasi-2D foams, i.e., foams consisting of a single layer of bubbles [3][4][5][6][7][8]. Using bubbles trapped between two glass plates (Hele-Shaw cell) in a cylindrical Couette geometry (the foam is contained between two concentric cylinders), Debrégeas et al. found that the flow of the foam localizes near the inner moving wall with an exponential velocity profile, forming shear bands [4]. While quasistatic cellular simulations [9,10] showed some agreement with the results, they continue to excite debate [7], especially in regard to the localization of shear and deformation [6], which is the salient feature of the experiment. Recently, Wang et al. have extended shear experiments to the simpler planar geometry [8]. While their experiments using bubbles between a liquid pool and a glass plate showed the formation of shear bands with an exponential velocity profile, a nearly linear velocity profile was obtained for a bubble floating on the liquid (bubble raft or Bragg raft). This has evidenced the crucial role played by the method used to confine the bubbles and indicates that the nonuniform stress imposed by the Couette geometry is not sufficient to explain the formation of shear bands with exponential decaying velocity.In this Letter, we introduce an elementary continuum model for the analysis of rheological properties of a twodimensional foam. It includes a viscous drag that has no counterpart in conventional 3D foam rheology. Our model is therefore closely related to the 2D viscous froth model [11] which was designed to enable dynamic simulations to be undertaken with the full cellular structure of the foam and included just such a viscous drag. Here the viscous drag will enter as a term in the continuum description, depending on a local average of the boundary velocity. Experiment and theory have already addressed this force as it arises in the flow of bubbles in cylindrical tubes and in narrow channels [5]. It is often associated with the name of Bretherton, who showed that the force varies with twothirds the power of velocity [12]. In some circumstances, a power law of one-half is suggested [13]. Nevertheless, as in the case of the 2D viscous froth, we adopt a linear form in order to keep the model and the analysis simple, in a search for a qualitative and semiquan...

We study the equilibrium energies of two-dimensional (2D) noncoarsening fluid foams, which consist of bubbles with fixed areas. The equilibrium states correspond to local minima of the total perimeter. We present a theoretical derivation of energy minima; experiments with ferrofluid foams, which can be either highly distorted, locally relaxed, or globally annealed; and Monte Carlo simulations using the extended large-Q Potts model. For a dry foam with small size variance we develop physical insight and an electrostatic analogy, which enables us to (i) find an approximate value of the global minimum perimeter, accounting for (small) area disorder, the topological distribution, and physical boundary conditions; (ii) conjecture the corresponding pattern and topology: small bubbles sort inward and large bubbles sort outward, topological charges of the same signs "repel" while charges of the opposite signs "attract;" (iii) define local and global markers to determine directly from an image how far a foam is from its ground state; (iv) conjecture that, in a local perimeter minimum at prescribed topology, the pressure distribution and thus the edge curvature are unique. Some results also apply to 3D foams.