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Integral method for a two-dimensional Stokes flow with shrinking holes applied to viscous sintering

Published online by Cambridge University Press:  26 April 2006

G. A. L. van de Vorst
G. A. L. van de Vorst
Affiliation:
Department of Mathematics and Computing Science, University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
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Abstract

An integral method is developed to solve the two-dimensional Stokes problem with Neumann boundary conditions for multiply connected domains in which the inside hole area can shrink and disappear. The method is applied to simulate viscous sintering. In particular the sintering of glasses can be modelled as such, i.e. a viscous incompressible Newtonian volume flow driven solely by surface tension. A Boundary Element Method is applied to solve the integral equations of Stokes flow involved, and the time integration is carried out by a variable-step, variable-order Backward Differences Formulae method. The derived numerical algorithm is demonstrated for several arbitrarily shaped multiply connected sintering domains. In particular some cylindrical packings are considered. The latter simulations provide a justification for the use of ‘unit problems’ in the theory of sintering.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 257 , December 1993 , pp. 667 - 689
Copyright
© 1993 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Hopper, R. W. 1990 Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349–375.Google Scholar
Hopper, R. W. 1991 Plane Stokes flow driven by capillarity on a free surface. Part 2. Further developments. J. Fluid Mech. 230, 355–364.Google Scholar
Hopper, R. W. 1992 Stokes flow of a cylinder and half-space driven by capillarity. J. Fluid Mech. 243, 171–181.Google Scholar
Hsiao, G. C. & Kress, R. 1985 On an integral equation for the two-dimensional exterior Stokes problem. Appl. Numer. Math 1, 77–93.Google Scholar
Jagota, A. & Dawson, P. R. 1988a Micromechanical modeling of powder compacts-I. Unit problems for sintering and traction-induced deformation. Acta. Metall. 36, 2551–2561.Google Scholar
Jagota, A. & Dawson, P. R. 1988b Micromechanical modeling of powder compacts-II. Truss formulation of discrete packings. Acta. Metall. 36, 2563–2573.Google Scholar
Jagota, A. & Dawson, P. R. 1990 Simulation of the viscous sintering of two particles. J. Am. Ceram. Soc. 73, 173–177.Google Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Korwin, D. M., Lange, S. R., Eaton, W. C., Joseph, I. & Pye, L. D. 1992 A study of the sintering behavior of glass in two geometric configurations. In Proc. 16th Intl Congress on Glass, published as a special issue of Boletián de la Sociedad Española de Cerámica y Virdo.
Kuiken, H. K. 1990 Viscous sintering: the surface-tension-driven flow of a liquid form under the influence of curvature gradients at its surface. J. Fluid Mech. 214, 503–515.CrossRefGoogle Scholar
Kreyszig, E. 1978 Introductory Functional Analysis with Applications. Wiley.
Kress, R. 1989 Linear Integral Equations. Springer-Verlag.
Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.
Lorentz, H. A. 1896 Eene Algemeene Stelling omtrent de Beweging eener Vloeistof met Wrijving en eenige daaruit afgeleide Gevolgen. Versl. Acad. Wetensch. Amsterdam 5, 168–175 (Transl. in Collected Papers 4, pp. 7–14, Martinus Nijhoff (1937)).Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Richardson, S. 1992 Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Eur. J. Appl. Maths. 3, 193–207.Google Scholar
Ross, J. W., Miller, W. A. & Weatherly, G. C. 1981 Dynamic computer simulation of viscous flow sintering kinetics. J. Appl. Phys. 52, 3884–3888.Google Scholar
Scherer, G. W. 1984 Viscous sintering of a bimodal pore-size distribution. J. Am. Ceram. Soc. 67, 709–715.Google Scholar
Sōmiya, S. & Moriyoshi, Y. (ed.) 1990 Sintering Key Papers Elsevier.
Tanzosh, J., Manga, M. & Stone, H. A. 1992 Boundary integral methods for viscous free-boundary problems: deformation of single and multiple fluid-fluid interfaces. In Proc. Conf. on Boundary Element Technology VII (ed. C. A. Brebbia & M. S. Ingber), pp. 19–39. Computational Mechanics Publications, Southampton.
Vorst, G. A. L. van de & Mattheij, R. M. M. 1991 Implementing the boundary element method for 2-D viscous sintering. In Proc. Conf. on Computational Modelling of Free and Moving Boundary Problems, Vol.1 Fluid Flow (ed. L. C. Wrobel & C. A. Brebbia), pp. 341–355. Computational Mechanics Publications, Southampton.
Vorst, G. A. L. van de, Mattheij, R. M. M. & Kuiken, H. K. 1992 A boundary element solution for two-dimensional viscous sintering. J. Comput. Phys. 100, 50–63.Google Scholar
Vorst, G. A. L. van de & Mattheij, R. M. M. 1992a A BDF-BEM scheme for modelling viscous sintering. In Proc. Conf. on Boundary Element Technology VII (ed. C. A. Brebbia & M. S. Ingber), pp. 59–74. Computational Mechanics Publications, Southampton.
Vorst, G. A. L. van de & Mattheij, R. M. M. 1992b Numerical analysis of a 2-D viscous sintering problem with non smooth boundaries. Computing 49, 239–263.Google Scholar
Weinbaum, S., Ganatos, P. & Yan, Z. Y. 1990 Numerical multipole and boundary integral equation techniques in Stokes flow. Ann. Rev. Fluid Mech. 22, 275–316.Google Scholar