Abstract
This paper presents a novel formulation of the completed indirect boundary-element method to study the shrinkage of air bubbles on a slow viscous flow in a bounded region subject to surface tension. The formulation has application to viscous sintering, a process for manufacturing high-quality glass by means of sol-gel processing. The theoretical background is explained in detail, including mathematical proofs of existence and uniqueness of solutions. Numerical results are included and compared to analytical and previous numerical solutions.
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References
H. K. Kuiken, Viscous sintering: the surface-tension-driven flow of a liquid form under the influence of curvature gradients at its surface. J. Fluid Mech. 214 (1990) 503-515.
H. K. Kuiken, Deforming surfaces and viscous sintering. In: D. G. Dritschel and R. J. Perkins (eds), The Mathematics of Deforming Surfaces. Oxford: Clarendon Press (1996) 75-97.
G. A. L. van de Vorst, R. M. M. Mattheij and H. K. Kuiken, A boundary element solution for two-dimensional viscous sintering. J. Comp. Phys. 100 (1992) 50-63.
G. A. L. van de Vorst and R. M. M. Mattheij, Numerical analysis of a two-dimensional viscous sintering problem with non-smooth boundaries. Computing 49 (1992) 239-263.
H. A. Lorentz, A general theorem on the motion of a fluid with friction and a few results derived from it (in Dutch). Versl. Acad. Wetensch 5 (1896) 168-175. Translated and reprinted by H. K. Kuiken, J. Eng. Math. 30 (1996) 19-24.
H. Power and G. Miranda, Second kind integral equation formulation of Stokes flows past a particle of arbitrary shape. SIAM J. Appl. Math. 47 (1987) 689-698.
H. Power, Second kind integral equation solution of Stokes flows past n bodies of arbitrary shape. In: C. A. Brebbia, W. L. Wendland and G. Kuhn (eds), Boundary Elements IX, Vol. 3. Berlin: Springer-Verlag (1987) 477-488.
S. J. Karrila, Y. O. Fuentes and S. Kim, Parallel computational strategies for hydrodynamic interactions between rigid particles of arbitrary shape in a viscous fluid. J. Rheology 33 (1989) 913-947.
H. Power and G. Miranda, Integral equation formulation for the creeping flow of an incompressible viscous fluid between two arbitrary closed surfaces, and a possible mathematical model for the brain fluid dynamics. J. Math. Anal. Appl. 137 (1989) 1-16.
S. J. Karrila and S. Kim, Integral equations of the second kind for Stokes flow: direct solution for physical variables and removal of inherent accuracy limitations. Chem. Eng. Commun. 82 (1989) 123-161.
H. Power, Matched-asymptotic analysis of low Reynolds number flow past a cylinder of arbitrary cross-sectional shape. Math. Appl. Comp. 9 (1990) 111-122.
H. Power, The low Reynolds number deformation of a gas bubble in shear flow: a general approach via integral equations. Eng. Anal. with Boundary Elements 16 (1992) 61-74.
H. Power, The completed double layer boundary integral equation method for two-dimensional Stokes flow. IMA J. Appl. Math. 51 (1993) 123-145.
S. G. Mikhlin, Integral Equations and their Application to Certain Problems in Mechanics, Mathematical Physics and Technology. London: Pergamon Press (1957) 338pp.
S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications. London: Butterworth-Heinemann (1991) 507pp.
C. Pozrikidis, Boundry Integral and Singularity Methods for Linearized Viscous Flow. Cambridge: Cambridge University Press (1992) 259pp.
H. Power and L. C. Wrobel, Boundary Integral Methods in Fluid Mechanics. Southampton: Computational Mechanics Publications (1995) 330pp.
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach (1963) 185pp.
R. Courant and D. Hilbert, Methods of Mathematical Physics. London: Interscience (1992) Vol. 2, 830pp.
F. Hartmann, Introduction to Boundary Elements. Berlin: Springer-Verlag (1989) 416pp.
C. A. Brebbia, J. C. F. Telles and L. C. Wrobel, Boundary Element Techniques. Berlin: Springer-Verlag (1984) 464pp.
G. A. L. van de Vorst, Integral method for a two-dimensional Stokes flow with shrinking holes applied to viscous sintering. J. Fluid Mech. 257 (1993) 667-689.
A. R. M. Primo, Novel Boundary Integral Formulations for Slow Viscous Flow with Moving Boundaries, Ph.D. Thesis, Brunel University, UK (1998) 169pp.
J. E. Gómez and H. Power, A multipole direct and indirect BEM for two-dimensional cavity flow at low Reynolds number. Eng. Anal. with Boundary Elements 19 (1997) 17-31.
I. D. Chang and R. Finn, On the solution of a class of equations occurring in continuum mechanics, with application to the Stokes paradox. Arch. Rational Mech. Anal. 7 (1961) 388-401.
C. Pozrikidis, The instability of a moving viscous drop. J. Fluid Mech. 210 (1989) 1-21.
E. Goursat, A Course in Mathematical Analysis. New York: Dover Publications (1964) Vol. II, 389pp.
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Primo, A., Wrobel, L. & Power, H. An indirect boundary-element method for slow viscous flow in a bounded region containing air bubbles. Journal of Engineering Mathematics 37, 305–326 (2000). https://doi.org/10.1023/A:1004631329906
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DOI: https://doi.org/10.1023/A:1004631329906