The Surface Evolver was used to calculate the equilibrium microstructure of random monodisperse soap froth, starting from Voronoi partitions of randomly packed spheres. The sphere packing has a strong influence on foam properties, such as E (surface free energy) and (average number of faces per cell). This means that random foams composed of equal-volume cells come in a range of structures with different topological and geometric properties. Annealing-subjecting relaxed foams to large-deformation, tension-compression cycles-provokes topological transitions that can further reduce E and . All of the foams have

Abstract. The steady-state shape of a finger penetrating into a region filled with a viscous fluid is examined. The two-dimensional and axisymmetric problems are solved using Stokes equations for low Reynolds number flow. To solve the equations, an assumption for the shape of the finger is made and the normal-stress boundary condition is dropped. The remaining equations are solved numerically by covering the domain with a composite mesh composed of a curvilinear grid which follows the curved interface, and a rectilinear grid parallel to the straight boundaries. The shape of the finger is then altered to satisfy the normal-stress boundary condition by using a nonlinear least squares iteration method. The results are compared with the singular perturbation solution of Bretherton (J. Fluid Mech., 10 (1961), pp. 166-188). When the axisymmetric finger moves through a tube, a fraction m of the viscous fluid is left behind on the walls of the tube. The fraction m was measured experimentally by Taylor (J.

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The Surface Evolver was used to compute the equilibrium microstructure of dry soap foams with random structure and a wide range of cell-size distributions. Topological and geometric properties of foams and individual cells were evaluated. The theory for isotropic Plateau polyhedra describes the dependence of cell geometric properties on their volume and number of faces. The surface area of all cells is about 10% greater than a sphere of equal volume; this leads to a simple but accurate theory for the surface free energy density of foam. A novel parameter based on the surface-volume mean bubble radius R32 is used to characterize foam polydispersity. The foam energy, total cell edge length, and average number of faces per cell all decrease with increasing polydispersity. Pentagonal faces are the most common in monodisperse foam but quadrilaterals take over in highly polydisperse structures.

A mean-field theory for the geometry and diffusive growth rate of soap bubbles in dry 3D foams is presented. Idealized foam cells called isotropic Plateau polyhedra (IPPs), with F identical spherical-cap faces, are introduced. The geometric properties (e.g., surface area S, curvature R, edge length L, volume V) and growth rate G of the cells are obtained as analytical functions of F , the sole variable. IPPs accurately represent average foam bubble geometry for arbitrary F ≥ 4, even though they are only constructible for F = 4, 6, 12. While R/V 1/3 , L/V 1/3 and G exhibit F 1/2 behavior, the specific surface area S/V 2/3 is virtually independent of F. The results are contrasted with those for convex isotropic polyhedra with flat faces.