Andrew M. Kraynik

The foam drainage equation

The drainage of liquid in a foam may be described in terms of a nonlinear partial differential equation for the foam density as a function of time and vertical position. We review the history and recent development of this theory, analysing various exact and approximate solutions and relating them to each other.

Extensional motions of spatially periodic lattices

Abstract The behavior of microrheological models for multiphase fluids that have spatially periodic structure depends on certain kinematic properties of the unit cell. Anomalous results associated with identical objects approaching too closely during the flow can be reduced if not eliminated by satisfying lattice compatibility conditions. This is straightforward for simple shearing flow but subtle for extensional flows. Using the connection between lattice compatibility and lattice reproducibility (periodic lattice behavior with the flow) we establish sufficient conditions for compatibility of arbitrary lattices in planar extensional flow. Detailed results for square and hexagonal unit cells include: initial orientations for periodic behavior; strain periods; and minimum lattice spacings D . We identify the orientation of a square unit cell that leads to periodic behavior (with the minimum period) and the largest D of any lattice in planar extensional flow. We show that no lattice exhibits periodic behavior in uniaxial extensional flow (or biaxial extensional flow) even though Adler & Brenner have established the existence of compatibility.

Foam rheology: a model of viscous phenomena

A theoretical model for foam rheology that includes viscous forces is developed by considering the deformation of two‐dimensional, spatially periodic cells in simple shearing and planar extensional flow. The undeformed hexagonal cells are separated by thin liquid films. Plateau border curvature and liquid drainage between films is neglected. Interfacial tension and viscous tractions due to stretching lamellar liquid determine the individual film tensions. The network motion is described by a system of nonlinear ordinary differential equations for which numerical solutions are obtained. Coalescense and disproportionation of Plateau borders results in the relative separation of cells and provides a mechanism for yielding and flow. This process is assumed to occur when a film’s length reduces to its thickness. The time and position dependence of the cell‐scale dynamics are computed explicitly. The effective continuum stress of the foam is described by instantaneous and time‐averaged quantities. The capillary...

Foam and emulsion rheology: a quasistatic model for large deformations of spatially-periodic cells

A microstructural model for the rheology of large‐gas‐fraction foams and concentrated emulsions is developed. Large shear and extensional deformations of a two‐dimensional spatially‐periodic network consisting of monodisperse hexagonal cells are considered. The elastic response is determined by surface tension forces and the steric interaction of thin liquid films. Coalescence and disproportionation of Plateau borders result in the relative separation of cells and provide a basic mechanism for yielding and flow. The strain dependence of the macroscopic stresses and cell morphology is very sensitive to the initial cell orientation. The response is strain periodic for discrete values of the orientation angle; however, strain‐periodic orientations for simple shear and extension are mutually exclusive. The steady‐flow material functions are determined by averaging the instantaneous stress over the strain period. Three different physical interpretations of the yield stress are considered.

Simple shearing flow of a dry Kelvin soap foam

Simple shearing flow of a dry soap foam composed of identical Kelvin cells is analysed. An undeformed Kelvin cell has six planar quadrilateral faces with curved edges and eight non-planar hexagonal faces with zero mean curvature. The elastic-plastic response of the foam is modelled by determining the bubble shape that minimizes total surface area at each value of strain. Computer simulations were performed with the Surface Evolver program developed by Brakke. The foam structure and macroscopic stress are piecewise continuous functions of strain. Each discontinuity corresponds to a topological change (Tl) that occurs when the film network is unstable. These instabilities involve shrinking films, but the surface area and edge lengths of a shrinking film do not necessarily vanish smoothly with strain. Each Tl reduces surface energy, results in cell-neighbour switching, and provides a film-level mechanism for plastic yield behaviour during foam flow. The foam structure is determined for all strains by choosing initial foam orientations that lead to strain-periodic behaviour. The average shear stress varies by an order of magnitude for different orientations. A Kelvin foam has cubic symmetry and exhibits anisotropic linear elastic behaviour; the two shear moduli and their average over all orientations are G min = 0.5706, G max = 0.9646, and

Simple shearing flow of dry soap foams with tetrahedrally close-packed structure

\overline{G} = 0.8070

On the shearing flow of foams and concentrated emulsions

, where stress is scaled by T / V 1/3 , T is surface tension, and V is bubble volume. An approximate solution for the microrheology is also determined by minimizing the total surface area of a Kelvin foam with flat films.

Viscous effects in the rheology of foams and concentrated emulsions

The microrheology of dry soap foams subjected to quasistatic, simple shearing flow is analyzed. Two different monodisperse foams with tetrahedrally close-packed (TCP) structure are examined: Weaire-Phelan (A15) and Friauf-Laves (C15). The elastic–plastic response is evaluated by using the Surface Evolver to calculate foam structures that minimize total surface area at each value of strain. The foam geometry and macroscopic stress are piecewise continuous functions of strain. The stress scales as T/V1/3, where T is surface tension and V is cell volume. Each discontinuity corresponds to large changes in foam geometry and topology that restore equilibrium to unstable configurations that violate Plateau’s laws. The instabilities occur when the length of an edge on a polyhedral foam cell vanishes. The length can tend to zero smoothly or abruptly with strain. The abrupt case occurs when a small increase in strain changes the energy profile in the neighborhood of a foam structure from a local minimum to a saddle...

Drainage of aqueous foams: Generation-pressure and cell-size effects

Shearing flow of an idealized, two-dimensional foam with monodisperse, spatially periodic cell structure is examined. Viscous effects are modelled by the film withdrawal mechanism of Mysels, Shinoda & Frankel. The primary flow occurs where thin films with inextensible interfaces are withdrawn from or recede into quasi-static Plateau borders, film junctions that contain most of the liquid. The viscous flow induces an excess tension that varies between films and alters the foam structure. The instantaneous structure and macroscopic stress for a foam of arbitrary orientation are determined for simple shearing and planar extensional flow. As the foam flows, the Plateau borders coalesce and separate, which leads to switching of bubble neighbours. The quasi-steady asymptotic analysis of the flow is valid for small capillary numbers Ca based on the macroscopic deformation rate. This requires the foam to be wet, i.e. the liquid volume fraction must be large enough that structure varies continuously with strain. The viscous contribution to the instantaneous stress is

Networklike propagation of cell-level stress in sheared random foams.

O(Ca^{\frac{2}{3}})