Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-01T04:58:05.671Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial effects on viscous fingering in the complex plane

Published online by Cambridge University Press:  26 January 2011

ANDONG HE  and
ANDREW BELMONTE
ANDONG HE*
Affiliation:
W. G. Pritchard Laboratories, Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
ANDREW BELMONTE
Affiliation:
W. G. Pritchard Laboratories, Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a nonlinear unsteady Darcy's equation which includes inertial effects for flows in a porous medium or Hele-Shaw cell and discuss the conditions under which it reduces to the classical Darcy's law. In the absence of surface tension we derive a generalized Polubarinova–Galin equation in a circular geometry, using the method of conformal mapping. The linear stability of the base-flow state is examined by perturbing the corresponding conformal map. We show that inertia always has a tendency to stabilize the interface, regardless of whether a less viscous fluid is displacing a more viscous fluid or vice versa.

JFM classification

Interfacial Flows (free surface): Fingering instability Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 668 , 10 February 2011 , pp. 436 - 445
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Acheson, D. 1990 Elementary Fluid Dynamics. Oxford University Press.Google Scholar
Aitchison, J. & Howison, S. 1985 Computation of Hele-Shaw flows with free boundaries. J. Comput. Phys. 60, 376–390.CrossRefGoogle Scholar
Bender, C. & Orszag, S. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Bensimon, D. 1986 Stability of viscous fingering. Phys. Rev. A 33, 1302–1308.Google Scholar
Chevalier, C., Amar, M., Bonn, D. & Lindner, A. 2006 Inertial effects on Saffman–Taylor viscous fingering. J. Fluid Mech. 552, 83–97.CrossRefGoogle Scholar
Crowdy, D. 2006 Exact solutions to the unsteady two-phase Hele-Shaw problem. Q. J. Mech. Appl. Math. 59, 475–485.CrossRefGoogle Scholar
Crowdy, D. & Cloke, M. 2002 Stability analysis of a class of two-dimensional multipolar vortex equilibria. Phys. Fluids 14, 1862–1876.Google Scholar
Dai, W.-S., Kadanoff, L. & Zhou, S.-M. 1991 Interface dynamics and the motion of complex singularities. Phys. Rev. A 43, 6672–6682.Google Scholar
Dawson, S. & Mineev-Weinstein, M. 1998 Dynamics of closed interfaces in two-dimensional Laplacian growth. Phys. Rev. E 57, 3063–3072.Google Scholar
Gondret, P. & Rabaud, M. 1997 Shear instability of two-fluid parallel flow in a Hele-Shaw cell. Phys. Fluids 9 (11), 3267–3274.CrossRefGoogle Scholar
Gustafsson, B. & Vasil'ev, A. 2006 Conformal and Potential Analysis in Hele-Shaw Cells, Birkhauser.Google Scholar
Hele-Shaw, H. S. 1899 The flow of water. Nature 59, 222–223.CrossRefGoogle Scholar
Howison, S. 1986 Fingering in Hele-Shaw cells. J. Fluid Mech. 167, 439–453.Google Scholar
Howison, S. 1992 Complex variable methods in Hele-Shaw moving boundary problems. Euro. J. Appl. Math. 3, 209–224.Google Scholar
Howison, S. 2000 A note on the two-phase Hele-Shaw problem. J. Fluid Mech. 409, 243–249.CrossRefGoogle Scholar
Hunt, J. 1999 Pattern formation in solidification. Mater. Sci. Tech. 15, 9–15.Google Scholar
Langer, J. 1980 Instabilities and pattern formation in crystal growth. Rev. Mod. Phys. 52, 1–28.Google Scholar
Lemaire, E., Levitz, P., Daccord, G. & Damme, H. 1991 From viscous fingering to viscoelastic fracturing. Phys. Rev. Lett. 67, 2009–2012.CrossRefGoogle ScholarPubMed
Li, S., Lowengrub, J., Fontana, J. & Palffy-Muhoray, P. 2009 Control of viscous fingering patterns in a radial Hele-Shaw cell. Phys. Rev. Lett. 102, 174501.CrossRefGoogle Scholar
McLean, J. & Saffman, P. 1980 The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 455–469.Google Scholar
Matsushita, M., Sano, M., Hayakawa, Y., Honjo, H. & Sawada, Y. 1984 Fractal structures of zinc metal leaves grown by electrodeposition. Phys. Rev. Lett. 53, 286–289.Google Scholar
Meiron, D., Saffman, P. & Schatzman, J. 1984 The linear two-dimensional stability of inviscid vortex streets of finite-cored vortices. J. Fluid Mech. 147, 187–212.Google Scholar
Mora, S. & Manna, M. 2010 Saffman–Taylor instability of viscoelastic fluids: from viscous fingering to elastic fractures. Phys. Rev. E 81, 026305.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell. J. Fluid Mech. 113, 513–529.Google Scholar
Podgorski, T., Sostarecz, M. C., Zorman, S. & Belmonte, A. 2007 Fingering instabilities of a reactive micellar interface. Phys. Rev. E. 76, 016202.CrossRefGoogle ScholarPubMed
Ruyer-Quil, C. 2001 Inertial corrections to the Darcy law in a Hele-Shaw cell. C. R. Acad. Sci., Paris IIb 329, 337–342.Google Scholar
Saffman, P. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar