Integral formulation to simulate the viscous sintering of a two-dimensional lattice of periodic unit cells

In this paper a mathematical formulation is presented which is used to calculate the flow field of a two-dimensional Stokes fluid that is represented by a lattice of unit cells with pores inside. The formulation is described in terms of an integral equation based on Lorentz's formulation, whereby the fundamental solution is used that represents the flow due to a periodic lattice of point forces. The derived integral equation is applied to model the viscous sintering phenomenon, viz. the process that occurs (for example) during the densification of a porous glass heated to such a high temperature that it becomes a viscous fluid. The numerical simulation is carried out by solving the governing Stokes flow equations for a fixed domain through a Boundary Element Method (BEM). The resulting velocity field then determines an approximate geometry at a next time point which is obtained by an implicit integration method. From this formulation quite a few theoretical insights can be obtained of the viscous sintering process with respect to both pore size and pore distribution of the porous glass. In particular, this model is able to examine the consequences of microstructure on the evolution of pore-size distribution, as will be demonstrated for several example problems

Numerical analysis of a 2-D viscous sintering problem with non smooth boundaries

By viscous sintering it is meant the process of bringing a granular compact to a temperature at which the viscosity of the material becomes low enough for surface tension to cause the particles to deform and coalesce, whereby the material transport can be modelled as a viscous incompressible newtonian volume flow. Here a two-dimensional model is considered. A Boundary Element Method is applied to solve the governing Stokes creeping flow equations for an arbitrarily initial shaped fluid region. In this paper we show that the viscous sintering problem is well-conditioned from an evolutionary point of view. However as boundary value problem at each time step, the problem is ill-conditioned when the contact surfaces of the particles are small, i.e. in the early stages of the coalescence. This is because the curvature of the boundary at those places can be very large. This ill-conditioning is illustrated by an example: the coalescence of two equal circles. This example demonstrates the main evolutionary features of the sintering phenomenon very well. A numerical consequence of this ill-conditioning is that special care has to be taken for distributing and redistributing the nodal points at these boundary parts. Therefore an algorithm for this node redistribution is outlined. Several numerical examples sustain the analysis. Viskoses Sintern ist ein Prozeß, bei dem ein Granulat auf eine Temperatur gebracht wird, bei der die Viskosität des Materials niedrig genug ist, um eine Verformung und ein Zusammenfließen der Partikel zu ermöglichen; die Materialbewegung kann dabei als viskose inkompressible Newtonsche Strömung modelliert werden. Für den hier betrachteten zweidimensionalen Fall wird zur Lösung der Stokesschen Kriechflußgleichungen eine Randelementmethode für eine beliebig geformte Anfangskonfiguration verwendet. In der Arbeit zeigen wird, daß das viskose Sinterproblem als Evolutionsproblem gut konditioniert ist. Das bei jedem Zeitschritt auftretende Randwertproblem ist dagegen während der anfänglichen Stufen des Zusammenfließens schlecht konditioniert, wenn nämlich die Kontaktflächen zwischen den Partikeln noch klein sind. Die Ursache ist die möglicherweise große Krümmung der Berandung an diesen Stellen. Diese schlechte Kondition wird an einem Beispiel, dem Zusammenfließen zweier Kreise, veranschaulicht, das die wesentlichen Entwicklungsstufen des Sinterphänomens gut zeigt. Als Folge dieser schlechten Kondition muß numerisch der Verteilung und Neuverteilung der Knotenpunkte in diesen Randabschnitten besondere Sorgfalt geschenkt werden. Es wird deshalb ein Algorithmus für diese Knotenverteilung skizziert. Mehrere numerische Beispiele bestätigen die Analyse

Mathematical modelling and numerical simulation of viscous sintering processes

The objective of this research is to develop reliable numerical methods to predict the deformation of an incompressible Newtonian viscous fluid region (Stokes flow) driven by the surface tension. In particular this mathematical model describes the physical processes that appear when a compact of glassy particles is heated to such a high temperature that the glass becomes a viscous creeping fluid. As a result the particles are joining together so that the cohesion of the compact is increasing with time. This phenomenon is usually called viscous sintering and appears, e.g . in the production of high-quality glasses. From the methods developed, theoretical insights can be obtained about the densification kinetics of such a compact. Therefore, a numerical simulation program is developed which calculates the deformation of a representative two-dimensional or an axisymmetric unit cell of the compact. A boundary element method is applied to solve the integral equations arising frorn the Stokes problem and the time integration is carried out by a variable-step, variable-order backward differences formulae method

A boundary element solution for two-dimensional viscous sintering

By viscous sintering is meant the process in which a granular compact is heated to a temperature at which the viscosity of the material under consideration becomes low enough for surface tension to cause the powder particles to deform and coalesce. For the sake of simplicity this process is modeled in a two-dimensional space. The governing (Stokes) equations describe the deformation of a two-dimensional viscous liquid region under the influence of the curvature of the outer boundary. However, some extra conditions are needed to ensure that these equations can be solved uniquely. A boundary element method is applied to solve the equations for an arbitrarily initial-shaped fluid region. The numerical problems that can arise in computing the curvature, in particular when this is varying rapidly, are discussed. A number of numerical examples are shown for simply connected regions which transform themselves into circles as time increases