Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-21T05:54:06.823Z Has data issue: false hasContentIssue false
  • English
  • Français

The nonlinear growth of a gravitationally unstable interface in a Hele-Shaw cell

Published online by Cambridge University Press:  21 April 2006

T. Maxworthy
T. Maxworthy
Affiliation:
Departments of Mechanical and Aerospace Engineering, University of Southern California, Los Angeles, CA 90089–1453, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We report observations of the nonlinear growth of an unstable interface in, a large Hele-Shaw cell in which it is possible, in the most extreme case, to follow the emergence of a single ‘finger’ from approximately thirty initial wavelets. The larger ‘fingers’ that emerge undergo further instability to form a highly contorted interface. We have measured a number of the averaged characteristics of this interface and can relate the various cases studied through length and time scales constructed from the independent parameters of the problem. We show that the most distorted interfaces may have a fractal dimension at scales larger than the instability wavelength and discuss the significance of this possibility.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 177 , April 1987 , pp. 207 - 232
Copyright
© 1987 Cambridge University Press

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

Aref, H. & Tryggvason, G. 1984 Vortex dynamics of passive and active interfaces. Physica 120, 59–70.Google Scholar
Bensimon, D., Kadanoff, L. P., Liang, S., Schraiman, B. I & Tang, C. 1985 Viscous flow in two dimensions. Preprint. J. Franck Inst., University of Chicago.
Caperan, P. & Maxworthy, T. 1985 Experiments on two dimensional turbulence in a stratified fluid. Proc. French Physical Society, Nice, Sept.
Degregoria, A. J. & Schwartz, L. W. 1986 A boundary-integral method for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 164, 383–400.Google Scholar
Emmons, H. W., Chang, C. T. & Watson, B. C. 1960 Taylor instability of finite surface waves. J. Fluid Mech. 7, 177–193.Google Scholar
Liang, S. 1986 Random walk simulation in Hele-Shaw cells. Phys. Rev. A 33, 2663–2674.Google Scholar
Liu, Y. N., Maxworthy, T. & Spedding G. 1985 The collapse of a turbulent front in a stratified fluid. Part 1. Nominally two dimensional evolution in a narrow tank. IUTAM Mtg on ‘Turbulence in Stratified Fluids’, Margaret River, W.A., Sept. 1985. Also submitted to J. Geophys. Res.Google Scholar
Maher, J. V. 1985 Development of viscous fingering patterns. Phy. Rev. Lett. 54, 1498–1501.Google Scholar
Mandelbrodt, B. 1983 The Fractal Geometry of Nature.. W. H. Freeman and Co..
Maxworthy, T. 1985 A note on the non-linear growth of a gravitationally unstable interface in a Hele-Shaw cell. Phys. Fluids 28, 2637.Google Scholar
Mclean, J. W. & Saffman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 455–469.Google Scholar
Nittman, J., Daccord, G. & Stanley, H. E. 1985 Fractal growth of viscous fingers: quantitative characterisation of a fluid instability phenomenon. Nature, 314, 141–144.Google Scholar
Park, C.-W., Gorell, S. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: experiments on viscously driven instabilities. J. Fluid Mech. 141, 257–287.Google Scholar
Park, C.-W. & Homsy, G. M. 1984 Two phase displacements in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Park, C.-W. & Homsy, G. M. 1985 The instability of long fingers in Hele-Shaw flows. Phys. Fluids 28, 1583–1585.Google Scholar
Reinelt, D. A. & Saffman, P. G. 1985 The penetration of a finger into viscous fluid in a channel or a tube. SIAM. J. Sci. Stat. Comput. 6, 542–561.Google Scholar
Robinson, A. R. 1985 Fractal fingers in viscous fluids. Science 228, 1077–1080.Google Scholar
Saffman, P. G. 1959 Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium. Q. J. Mech. Appl. Maths 12, 146–150.Google Scholar
Saffman, P. G. & Taylor, G. L. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid. Proc. Roy. Soc. Lond. A 245, 312–329.Google Scholar
Tabeling, P. & Libchaber, A. 1986 Film draining and the Saffman–Taylor problem. Phys. Rev. A33, 794–796.Google Scholar
Tabeling, P., Zocchi, G. & Libchaber, A. 1986 An experimental study of the Saffman-Taylor problem. Preprint, James Franck and E. Fermi Insts., University of Chicago.
Taylor, G. I. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Q. J. Mech. Appl. Maths. 12, 265–279.Google Scholar
Tryggvason, G. & Aref, H. 1983 Numerical experiments on Hele-Shaw flow with a sharp interface. J. Fluid Mech., 136, 1–30.Google Scholar
Wooding, R. A. & Morel-Seytoux, H. J. 1976 Multiphase fluid flow through porous media. Ann. Rev. Fluid Mech. 8, 233–274.Google Scholar