Rheology of foams: I. Theory for dry foams

A two-dimensional model for foams having gas volume fraction approaching unity has been developed. A general expression for the stress tensor is obtained which gives the total stress in terms of the shape of the cells, interfacial tension, the initial cell orientation, and the rate of deformation in the liquid. A formalism for describing cell deformation is also presented. By assuming hexagonal, monodisperse foam cells, we are able to obtain analytic expressions for stresses for small shearing and elongational deformations (below the yield strain) as well as for steady shear flow. For strains below the yield point, the stress-strain relation is independent of initial cell orientation. However, the critical strain varies with orientation, and therefore, the yield stress, τ0, is a function of orientation. In steady shear flow, the shear stress has a contribution from the liquid which is proportional to the shear rate. Thus, τyx = τ0 − Cμγ., where C is a constant determined from viscous dissipation in the thin liquid films, μ is the liquid film viscosity, and γ. is the shear rate. The yield stress is directly proportional to the liquid surface tension and inversely proportional to cell size.