# Mathematical Modeling of Some Glass Problems

• K. Laevsky
• R. M. M. Mattheij
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

## Abstract

In studying glass morphology one often uses models, which describe it as a strongly viscous Newtonian fluid. In this chapter we shall study two types of problems encountered in glass technology. One is dealing with so-called sintering, which plays a role in producing high-quality glasses, for example, and the other with producing packing glass by so-called pressing. We give a Stokes model to describe these processes and discuss various aspects of the evolution of both forming problems. The sintering problem is solved by a boundary element method, for which we use an interesting analytical tool to avoid numerical instabilities. The pressing problem actually deals with the morphology of a bottle or jar. Here we focus on simulating the glass flow. We first show how to deal with the temperature separately, by a suitable dimension analysis. Then we consider the flow of the glass in a domain with a partially free and partially moving boundary. We give a number of numerical examples to sustain our result.

## Keywords

Complex Flow Contact Radius Excess Free Energy Liquid Glass Contact Conductance
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Chandra, T.D., and Rienstra, S. W., Analytical approximation to the viscous glass flow problem in the mould-plunger pressing process, RANA 97–08, Technical University of Eindhoven (1997).Google Scholar
2. 2.
Doyle, P.J., Glass Making Today, R.A.N. Publisher, Ohio (1994).Google Scholar
3. 3.
Hopper, R.W., Plane Stokes flow driven by capillarity on a free surface, J. Fluid Mech., 213, (1990), 349–75.
4. 4.
Hopper, R.W., Plane Stokes flow driven by capillarity on a free surface, 2: Further developments, J. Fluid Mech., 230, (1991), 355–64.
5. 5.
Hsiao, G.C., Kopp, P., and Wendland, W.L., Some applications of a Galerkin-collocation method for boundary integral equations of the first kind, Math. Methods in the Appl. Sciences, 6, (1984), 280–325.
6. 6.
Kuiken, H.K., Mattheij, R.M.M., and van de Vorst, G.A.L., A boundary element solution for 2-dimensional viscous sintering, J. Comput. Phys.,100 (1992), 50–63.
7. 7.
Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, Godon and Beach, New York (1963).
8. 8.
Mattheij, R.M.M., van de Vorst, G.A.L., Mathematical modelling and numerical simulation of viscous sintering processes, Surv. Math. Ind., 7 (1998), 255–81.
9. 9.
Mulder, C.A.M., van Lierop, J.G., and Frens, G., Densification of SiO2-xerogels to glass by Ostwald ripening, J. Noncryst. Solids, 82 (1986), 92–6.
10. 10.
Rawson, H., Properties and applications of glass, Glass Sci. and Tech., 3, Elsevier (1980).Google Scholar
11. 11.
Reed, J.S., Introduction to the Principles of Ceramic Processing, Wiley-Interscience, Chichester (1988).Google Scholar
12. 12.
Sómiya, S., and Moriyoshi, Y., Eds., Sintering Key Papers, Elsevier Applied Science, London (1990).Google Scholar
13. 13.
Stokes, Y.M., Very viscous flows driven by gravity, with particular application to slumping of molten glass, Ph.D. Thesis, University of Adelaide (1998).Google Scholar
14. 14.
Uhlmann, D.R., and Kreidl, V.J., eds., Glass Science and Technology, Academic Press, London (1986).Google Scholar
15. 15.
van de Vorst, G.A.L., Integral method for a two-dimensional Stokes flow with shrinking holes, applied to viscous sintering, J. Fluid Mech., 257 (1993). 667–89.
16. 16.
van de Vorst, G.A.L., Modelling and numerical simulation of axisymmetric viscous sintering, Ph.D. Thesis, Eindhoven (1994).Google Scholar