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

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.

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

“…These general considerations are substantiated in the very recent paper by Talon et al 6 While these authors include the effect of unstable viscosity ratios, the findings in their work that are most relevant here are as follows. Using direct numerical simulation of the Stokes equations for a HeleShaw flow, they find that the density profiles produced by the parallel flow and depicted schematically in Fig.…”

“…Results of experiments in rectilinear displacements could also be easier to compare with theoretical stability analyses or full numerical solutions of the relevant equations. 6,7 Simulations could moreover test the influence of the different parameters of the problem on the evolution of the pattern in 3D. …”

“…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. The influence of density differences across the miscible interface on the viscous fingering pattern has been recently addressed by both linear stability analysis 6 and nonlinear simulations 7 of the full 3D Stokes problem. It has been shown that a buoyancy-driven instability can act within the gap of the cell on the upper side of the finger.…”

Schlieren imaging of viscous fingering in a horizontal Hele-Shaw cell

“…Field of view: 9 cm x 9 cm Fig. 8 Schematic of a less viscous fluid displacing a more viscous one (possibly of different density) in a horizontal HS cell taken from the stability analysis of Talon et al (2013) results in the formation of thin little stripes developing in the invading fluid and perpendicular to the front, as seen using a dye in a viscously stable case (Haudin et al 2014). These stripes result from streamwise, counter-rotating vortices, such as can be seen in Figs.…”

Capillary Model of Gravity Elution of High Viscosity Liquid from a Porous Bed with a Low-Viscosity Liquid

“…However, in the immiscible flow only the dynamics of flowing liquids are considered. The phenomena accompanying miscible flow in a single channel were discussed and supported by relevant numerical simulations provided by Sahu et al (2009a, b), Talon and Meiburg (2011), Talon et al (2013), Taghavi et al (2012). The various systems of channels with different initial conditions were analyzed there.…”