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Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis

Published online by Cambridge University Press:  13 March 2013

L. Talon ,
N. Goyal  and
E. Meiburg
L. Talon
Affiliation:
Laboratoire Fluides Automatique et Systèmes Thermiques, Universités P. et M. Curie, CNRS (UMR 7608) Bâtiment 502, Campus Universitaire, 91405 Orsay Cedex, France Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
N. Goyal
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
E. Meiburg*
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: meiburg@engineering.ucsb.edu
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Abstract

A computational investigation of variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells is presented. As a first step, two-dimensional base states are obtained by means of simulations of the Stokes equations, which are nonlinear due to the dependence of the viscosity on the local concentration. Here, the vertical position of the displacement front is seen to reach a quasisteady equilibrium value, reflecting a balance between viscous and gravitational forces. These base states allow for two instability modes: first, there is the familiar tip instability driven by the unfavourable viscosity contrast of the displacement, which is modulated by the presence of density variations in the gravitational field; second, a gravitational instability occurs at the unstably stratified horizontal interface along the side of the finger. Both of these instability modes are investigated by means of a linear stability analysis. The gravitational mode along the side of the finger is characterized by a wavelength of about one half to one full gap width. It becomes more unstable as the gravity parameter increases, even though the interface is shifted closer to the wall. The growth rate is largest far behind the finger tip, where the interface is both thicker, and located closer to the wall, than near the finger tip. The competing influences of interface thickness and wall proximity are clarified by means of a parametric stability analysis. The tip instability mode represents a gravity-modulated version of the neutrally buoyant mode. The analysis shows that in the presence of density stratification its growth rate increases, while the dominant wavelength decreases. This overall destabilizing effect of gravity is due to the additional terms appearing in the stability equations, which outweigh the stabilizing effects of gravity onto the base state.

JFM classification

Convection: Buoyancy-driven instability Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 721 , 25 April 2013 , pp. 268 - 294
Copyright
©2013 Cambridge University Press

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