A bubble in an acoustic field experiences a net 'Bjerknes' force from the nonlinear coupling of its radial oscillations with the oscillating buoyancy force. It is typically assumed that the bubble's net terminal velocity can be found by considering a spherical bubble with the imposed 'Bjerknes stresses'. We have analyzed the motion of such a bubble using a rigorous perturbation approach and found that one must include a term involving an effective mass flux through the bubble that arises from the time average of the second-order nonlinear terms in the kinematic boundary condition. The importance of this term is governed by the dimensionless parameter {alpha} = R{sup 2} {phi}/R{sup 2} {phi} {nu}.-{nu}, where R is the bubble radius, {phi} is the driving frequency, and {nu} is the liquid kinematic viscosity. If {alpha} is large, this term is unimportant, but if {alpha} is small, this term is the dominant factor in determining the terminal velocity.
Simulations are in excellent agreement with experiments: structure - Matzke, shear modulus - Princen and Kiss E = 3.30 {sigma}/R{sub 32} = 5.32/(1 + p) {sigma}/(V){sup 1/2}, G {approx} 0.155 E = 0.512 {sigma}/R{sub 32}. IPP theory captures dependence of cell geometry on V and F. Future challenges are: simulating simple shearing flow is very expensive because of frequent topological transitions. Random wet foams require very large simulations.
A series of experiments has been performed to allow observation of the foaming process and the collection of temperature, rise rate, and microstructural data. Microfocus video is used in conjunction with particle image velocimetry (PIV) to elucidate the boundary condition at the wall. Rheology, reaction kinetics and density measurements complement the flow visualization. X-ray computed tomography (CT) is used to examine the cured foams to determine density gradients. These data provide input to a continuum level finite element model of the blowing process.
Many weapons components (e.g. firing sets) are encapsulated with blown foams. Foam is a strong lightweight material--good compromise between conflicting needs of structural stability and electronic function. Current foaming processes can lead to unacceptable voids, property variations, cracking, and slipped schedules which is a long-standing issue. Predicting the process is not currently possible because the material is polymerizing and multiphase with changing microstructure. The goals of this project is: (1) Produce uniform encapsulant consistently and improve processability; (2) Eliminate metering issues/voids; (3) Lower residual stresses, exotherm to protect electronics; and (4) Maintain desired properties--lightweight, strong, no delamination/cracking, and ease of removal. The summary of achievements in the first year are: (1) Developed patentable chemical foaming chemistry - TA; (2) Developed persistent non-curing foam for systematic evaluation of fundamental physics of foams--Initial testing of non-curing foam shows that surfactants very important; (3) Identified foam stability strategy using a stacked reaction scheme; (4) Developed foam rheology methodologies and shear apparatuses--Began testing candidates for shear stability; (5) Began development of computational model; and (6) Development of methodology and collection of property measurements/boundary conditions for input to computational model.
The cell structure and rheology of gas-liquid foams confined between parallel plates depend on the ratio H/R, where H is the plate spacing and R is the (equivalent spherical) bubble radius. We consider ordered three-dimensional foams that consist of 1-3 layers of bubbles. In the 'dry' limit, where the gas fraction is unity, one confined layer is composed of hexagonal cylinders; two layers contain Fejes Toth cells; and three or more layers are modeled as Kelvin cells sandwiched between Fejes Toth cells. We also consider wet foams where all of the liquid is assumed to be located in either conventional Plateau borders or wall Plateau borders adjacent to the plates. The Surface Evolver is used to calculate the foam structure and stress as a function of H/R, which enables us to evaluate elastic behavior. A relationship between the two-dimensional structure at the wall and bubble size has application to foam characterization.
This report documents the results for the FY07 ASC Integrated Codes Level 2 Milestone number 2354. The description for this milestone is, 'Demonstrate level set free surface tracking capabilities in ARIA to simulate the dynamics of the formation and time evolution of a weld pool in laser welding applications for neutron generator production'. The specialized boundary conditions and material properties for the laser welding application were implemented and verified by comparison with existing, two-dimensional applications. Analyses of stationary spot welds and traveling line welds were performed and the accuracy of the three-dimensional (3D) level set algorithm is assessed by comparison with 3D moving mesh calculations.
As part of an effort to reduce costs and improve quality control in encapsulation and potting processes the Technology Initiative Project ''Defect Free Manufacturing and Assembly'' has completed a computational modeling study of flows representative of those seen in these processes. Flow solutions are obtained using a coupled, finite-element-based, numerical method based on the GOMA/ARIA suite of Sandia flow solvers. The evolution of the free surface is solved with an advanced level set algorithm. This approach incorporates novel methods for representing surface tension and wetting forces that affect the evolution of the free surface. In addition, two commercially available codes, ProCAST and MOLDFLOW, are also used on geometries representing encapsulation processes at the Kansas City Plant. Visual observations of the flow in several geometries are recorded in the laboratory and compared to the models. Wetting properties for the materials in these experiments are measured using a unique flowthrough goniometer.
The properties of solid foams depend on their structure, which usually evolves in the fluid state as gas bubbles expand to form polyhedral cells. The characteristic feature of foam structure-randomly packed cells of different sizes and shapes-is examined in this article by considering soap froth. This material can be modeled as a network of minimal surfaces that divide space into polyhedral cells. The cell-level geometry of random soap froth is calculated with Brakke's Surface Evolver software. The distribution of cell volumes ranges from monodisperse to highly polydisperse. Topological and geometric properties, such as surface area and edge length, of the entire foam and individual cells, are discussed. The shape of struts in solid foams is related to Plateau borders in liquid foams and calculated for different volume fractions of material. The models of soap froth are used as templates to produce finite element models of open-cell foams. Three-dimensional images of open-cell foams obtained with x-ray microtomography allow virtual reconstruction of skeletal structures that compare well with the Surface Evolver simulations of soap-froth geometry.
The microrheology of dry soap foams subjected to large, quasistatic, simple shearing deformations 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 calculating foam structures that minimize total surface area at each value of strain. The minimal surfaces are computed with the Surface Evolver program developed by Brakke. The foam geometry and macroscopic stress are piecewise continuous functions of strain. The stress scales as T/V{sup 1/3} 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 point, which can lead to symmetry-breaking bifurcations. In general, the new foam topology associated with each stable solution branch results from a cascade of local topology changes called T1 transitions. Each T1 cascade produces different cell neighbors, reduces surface energy, and provides an irreversible, film-level mechanism for plastic yield behavior. Stress-strain curves and average stresses are evaluated by examining foam orientations that admit strain-periodic behavior. For some orientations, the deformation cycle includes Kelvin cells instead of the original TCP structure; but the foam does not remain perfectly ordered. Bifurcations during subsequent T1 cascades lead to disorder and can even cause strain localization.
The microrheology of liquid foams is discussed for two different regimes: static equilibrium where the capillary number Ca is zero, and the viscous regime where viscosity and surface tension are important and Ca is finite. The Surface Evolver is used to calculate the equilibrium structure of wet Kelvin foams and dry soap froths with random structure, i.e., topological disorder. The distributions of polyhedra and faces are compared with the experimental data of Matzke. Simple shearing flow of a random foam under quasistatic conditions is also described. Viscous phenomena are explored in the context of uniform expansion of 2D and 3D foams at low Reynolds number. Boundary integral methods are used to calculate the influence of Ca on the evolution of foam microstructure, which includes bubble shape and the distribution of liquid between films, Plateau borders, and (in 3D) the nodes where Plateau borders meet. The micromechanical point of view guides the development of structure-property-processing relationships for foams.
Foam evokes many different images: waves breaking at the seashore, the head on a pint of Guinness, an elegant dessert, shaving, the comfortable cushion on which you may be seated... From the mundane to the high tech, foams, emulsions, and cellular solids encompass a broad range of materials and applications. Soap suds, mayonnaise, and foamed polymers provide practical motivation and only hint at the variety of materials at issue. Typical of mukiphase materiaIs, the rheoIogy or mechanical behavior of foams is more complicated than that of the constituent phases alone, which may be gas, liquid, or solid. For example, a soap froth exhibits a static shear modulus-a hallmark of an elastic solid-even though it is composed primarily of two Newtonian fluids (water and air), which have no shear modulus. This apparent paradox is easily resolved. Soap froth contains a small amount of surfactant that stabilizes the delicate network of thin liq- uid films against rupture. The soap-film network deforms in response to a macroscopic strain; this increases interracial area and the corresponding sur- face energy, and provides the strain energy of classical elasticity theory [1]. This physical mechanism is easily imagined but very challenging to quantify for a realistic three-dimensional soap froth in view of its complex geome- try. Foam micromechanics addresses the connection between constituent properties, cell-level structure, and macroscopic mechanical behavior. This article is a survey of micromechanics applied to gas-liquid foams, liquid-liquid emulsions, and cellular solids. We will focus on static response where the foam deformation is very slow and rate-dependent phenomena such as viscous flow can be neglected. This includes nonlinear elasticity when deformations are large but reversible. We will also discuss elastic- plastic behavior, which involves yield phenomena. Foam structures based on polyhedra packed to fill space provide a unify- ing geometrical theme. Because a two-dimensional situation is always easier to visualize and usually easier to analyze, the roots of foam micromechanics lie in the plane packed with polygons. There are striking similarities as well as obvious differences between 2D and 3D.
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
A boundary layer theory for the flow of power-law fluids in a converging planar channel has been developed. This theory suggests a Reynolds number for such flows, and following numerical integration, a boundary layer thickness. This boundary layer thickness has been used in the generation of a finite element mesh for the finite element code FIDAP. FIDAP was then used to simulate the flow of power-law fluids through a converging channel. Comparison of the analytic and finite element results shows the two to be in very good agreement in regions where entrance and exit effects (not considered in the boundary layer theory) can be neglected. 6 refs., 8 figs., 1 tab.