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Simple shearing flow of a 3D foam
Foams, like most highly structured fluids, exhibiting rheological behavior that is both fascinating and complex. We have developed microrheological models for uniaxial extension and simple shearing flow of a `dry`, perfectly ordered, three-dimensional foam composed of thin films with uniform surface tension T and negligible liquid content. We neglect viscous flow in the thin films and examine large elastic-plastic deformations of the foam. The primitive undeformed foam structure is composed of regular space-filling tetrakaidecahedra, which have six square and eight hexagonal surfaces. This structure possesses the film-network topology that is necessary to satisfy equilibrium: three films meet at each edge, which corresponds to a Plateau border, and four edges meet at vertex. However, to minimize surface energy, the films must meet at equal angles of 120{degrees} and the edges must join at equal tetrahedral angles of cos{sup {minus}1}({minus}1/3) {approx} 10.947{degree}. No film in an equilibrium foam structure can be a planar polygon because no planar polygon has all angles equal to the tetrahedral edge. In the equilibrium foam structure known as Kelvin`s minimal tetrakaidecahedron, the `squares` are planar quadrilateral surfaces with curved edges and the `hexagons` are non-planar saddle surfaces with zero mean curvature. As the foam structure evolves with the macroscopic flow, each film maintains zero mean curvature because the pressure is the same in every bubble. In general, the shape of each thin film, defined by z = h(x,y), satisfies R{sub 1}/1 + R{sub 2}/1 = {del}{center_dot} (1 + {vert_bar}{del}h{vert_bar}){sup {1/2}} = O where R{sub 1}{sup {minus}1} and A{sub 2}{sup {minus}1} are the principal curvatures. The appropriate boundary conditions correspond to three films meeting at equal angles. For the homogeneous deformations under consideration, the center of each film moves affinely with the flow. 5 refs