Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

Abstract

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

Effect of diffusing layer thickness on the density-driven natural convection of miscible fluids in porous media: Modeling of mass transport

“…the local concentration of NaI in fingers was higher when the initial interface thickness was thin. The experimental results were consistent with the LSA results for a Hele-Shaw cell (Fernandez et al 2002;Graf et al 2002;Lajeunesse et al 1997) and viscous fingering (Manickam and Homsy 1995;Tan and Homsy 1986); the wavelength and amplification factor decreased with interface thickness. The case of thicker initial interface, for example = 23.4 mm, only one finger occurred and extended downward because of the scale of the packed bed.…”

“…Heller (1966) demonstrated that the amplification factor decreased with Rayleigh number Raδ with the thickness of interface as characteristic length by a LSA for porous media. Using LSA, Graf et al (2002) showed that the amplification factor (the maximum eigenvalue) for a Hele-Shaw cell decreased with increasing interface thickness. Tan and Homsy (1986) used QSSA to determine that the amplification factor and wave number decreased with time (i.e., diffusion of the interface for viscous fingering).…”

“…At low Rayleigh numbers, Ra, where a span-wise Hele-Shaw mode dominates, the dominant wavelength scales with Ra −1 (Fernandez et al 2002) where Ra is defined as the ratio of the mass transport by convection to that of diffusion. In contrast, at high Rayleigh numbers, the wavelength saturates at approximately five times the cell thickness, which is unique to the Hele-Shaw cell and characterized by an instability across the gap (Fernandez et al 2001(Fernandez et al , 2002Graf et al 2002;Lajeunesse et al 1997). Graf et al (2002) performed a linear stability analysis (LSA) based on the full three-dimensional Stokes equations and found that the optimally amplified wavelength is approximately five times the gap width, with the exact value depending on the interface thickness.…”

“…In contrast, at high Rayleigh numbers, the wavelength saturates at approximately five times the cell thickness, which is unique to the Hele-Shaw cell and characterized by an instability across the gap (Fernandez et al 2001(Fernandez et al , 2002Graf et al 2002;Lajeunesse et al 1997). Graf et al (2002) performed a linear stability analysis (LSA) based on the full three-dimensional Stokes equations and found that the optimally amplified wavelength is approximately five times the gap width, with the exact value depending on the interface thickness. The most amplified wavenumber increases qualitatively with Ra and decreases with interface thickness.…”

Experimental study of a buoyancy-driven instability of a miscible horizontal displacement in a Hele-Shaw cell

“…1 In a gravity field, if the more dense fluid lies on top of the less dense one, the horizontal interface is buoyantly unstable because of a RayleighTaylor instability and related "density fingering" can be observed. [2][3][4] In vertical displacements, such viscous and buoyancy effects can either cooperate to stabilize or destabilize the flow 5 or, on the contrary, compete, in which case there is a critical injection velocity above which there is an instability. 1 In horizontal systems, the density jump is in the direction perpendicular to gravity and the stability of the interface is classically considered to depend only on the viscosity difference: viscous fingering can be observed when the less viscous fluid is injected into the more viscous one while the reverse displacement is viscously stable.…”