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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: he@math.psu.edu
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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

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