Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T09:02:48.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Fluid flow in a medium distorted by a quasiconformal map can produce fractal boundaries

Published online by Cambridge University Press:  26 September 2008

Olli Martio  and
Bernt Øksendal
Olli Martio
Affiliation:
Department of Mathematics, P.O. Box 4, FIN-00014University of Helsinki, Finland
Bernt Øksendal
Affiliation:
Department of Mathematics/VISTA, University of Oslo, Box 1053 Blindern, N-0316 Oslo, Norway
Get access Rights & Permissions [Opens in a new window]

Abstract

Physical experiments indicate that when an expanding fluid flows through a porous rock then the boundary between the wet and the dry region can be very irregular (e.g. see [OMBAFJ] and the references therein). In fact, it has been conjectured that this boundary is a fractal with Hausdorff dimension about 2.5. The (one-phase) fluid flow in a porous medium can be modelled mathematically by a system of partial differential equations, which, under some simplifying assumptions, can be reduced to a family of semi-elliptic boundary value problems involving the (unknown) pressure p(x) of the fluid (at the point x and at t) and the (unknown) wet region Ut at time t. (See equations (1.5)–(1.7) below). This set of equations, called the moving boundary problem involves the permeability matrix K(x) of the medium at x. A question which has been debated is whether this relatively simple mathematical model can explain such a complicated fractal nature of ∂Ut. More precisely, does there exist a symmetric non-negative definite matrix K(x) such that the solution Ut of the corresponding (expanding) moving boundary problem has a fractal boundary? The purpose of this paper is to prove that this is indeed the case. More precisely, we show that a porous medium which produce fractal wet boundaries can be obtained by distorting a completely homogeneous medium by means of a quasiconformal map.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 1 , February 1996 , pp. 1 - 10
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[BG1] Begehr, H. & Gilbert, R. P. 1986 Hele–Shaw type flows in Rn. Nonlinear Analysis; Theory, Methods and Applic. 10: 65–85.Google Scholar
[BG2] Begehr, H. & Gilbert, R. P. 1987 Non-Newtonian Hele-Shaw flows in n ≥ 2 dimensions. Nonlinear Analysis; Theory, Methods and Applic. 11: 17–47.Google Scholar
[EJ] Elliott, C. M. & Janovsky, V. 1981 A variational inequality approach to Hele–Shaw flow with a moving boundary. Proc. Royal Soc. Edinburgh 88A: 93–107.CrossRefGoogle Scholar
[F] Fukushima, M. 1980 Dirichlet Forms and Markov Processes. North-Holland/Kodansha.Google Scholar
[FKS] Fabes, E. B., Kenig, C. E. & Serapioni, R. P. 1982 The local regularity of solutions of degenerate elliptic equations. Comm. PDE 7: 77–116.CrossRefGoogle Scholar
[G] Gustafsson, B. 1985 Applications of variational inequalities to a moving boundary problem for Hele–Shaw flows. SIAM J. Math. Anal. 16: 279–300.CrossRefGoogle Scholar
[GV] Gehring, F. W. & Väisälä, J. 1973 Hausdorff dimension and quasiconformal mappings. J. London Math. Soc. 6: 504–512.CrossRefGoogle Scholar
[HKM] Heinonen, J., Kilpeläinen, T. & Martio, O. 1993 Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press.Google Scholar
[LØU] Lindstrøm, T., Øksendal, B. & Ubøe, J. 1991 Stochastic modelling of fluid flow in porous media. In: Control Theory, Stochastic Analysis and Applications (Chen, S. & Yong, J., eds), pp. 156–172. World Scientific.Google Scholar
[LV] Lehto, O. & Virtanen, K. I. 1973 Quasiconformal Mappings in the Plane, Springer-Verlag.CrossRefGoogle Scholar
[M] Meakin, P. 1988 The growth of fractal aggregates and their fractal measures. In: Phase Transitions (Domb, C. & Lebowitz, J. L., Eds), pp. 335–489. Academic Press.Google Scholar
[MFJ] Måløy, K. J., Feder, J. & Jøssang, T. 1985 Phys. Rev. Lett. 55: 2688–2691.CrossRefGoogle Scholar
[Ø] Øksendal, B. 1990 A stochastic approach to moving boundary problems. In: Diffusion Processes and Related Problems in Analysis, Vol. I (Pinsky, M., ed.) pp. 201–218. Birkhäuser.Google Scholar
[OMBAFJ] Oxaal, U., Murat, M., Boger, F., Aharony, A., Feder, J. & Jøssang, T. 1987 Viscous fingering on percolation clusters. Nature 329: 32–37.CrossRefGoogle Scholar
[R] Reichelt, Y. Moving boundary problems arising from degenerate elliptic equations (to appear).Google Scholar
[V] Väisälä, J. 1972 Modulus and capacity inequalities for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. AI 509: 1–14.Google Scholar
[WS] Witten, T. A. & Sander, L. M. 1983 Diffusion-limited aggregation: A kinetic critical phenomenon. Phys. Rev. Lett. 47: 1400–1403. (See also Phys. Rev. B27, 1983: 5686–5697.)CrossRefGoogle Scholar