Miscible displacement in a Hele-Shaw cell at high rates

Abstract

We study experimentally and theoretically the downward vertical displacement of one miscible fluid by another lighter one in the gap of a Hele-Shaw cell at sufficiently high velocities for diffusive effects to be negligible. Under certain conditions on the viscosity ratio, M, and the normalized flow rate, U, this results in the formation of a two-dimensional tongue of the injected fluid, which is symmetric with respect to the midplane. Thresholds in flow rate and viscosity ratio exist above which the two- dimensional flow destabilizes, giving rise to a three-dimensional pattern. We describe in detail the two-dimensional regime using a kinematic wave theory similar to Yang & Yortsos (1997) and we delineate in the (M, U)-plane three different domains, characterized respectively by the absence of a shock, the presence of an internal shock and the presence of a frontal shock. Theoretical and experimental results are compared and found to be in good agreement for the first two domains, but not for the third domain, where the frontal shock is not of the contact type. An analogous treatment is also applied to the case of axisymmetric displacement in a cylindrical tube.

Double diffusive effects on pressure-driven miscible displacement flows in a channel

“…Their nonlinear simulations of the Stokes equations predict that increasing the unstable density stratification and decreasing diffusion increase the front velocity. The flow fields obtained by these simulations are qualitatively similar to those observed in the experiment of Petitjeans & Maxworthy (1996) in capillary tubes and in the theoretical predictions of Lajeunesse et al (1999) for Hele-Shaw cells. The study of Petitjeans & Maxworthy (1996) discussed the formation and propagation of a single finger for a miscible fluid in a capillary tube.…”

“…Experimental studies in miscible core-annular flows (Taylor 1961;Cox 1962;Chen & Meiburg 1996;Petitjeans & Maxworthy 1996;Kuang, Maxworthy & Petitjeans 2003) have focused on analysing the thickness of the more viscous fluid layer left on the pipe walls and the speed of the propagating 'finger' tip. The development of different instability patterns, like axisymmetric 'corkscrew' patterns, in miscible flows has also been investigated (Lajeunesse et al 1997(Lajeunesse et al , 1999Scoffoni, Lajeunesse & Homsy 2001;Cao et al 2003;Gabard & Hulin 2003). Axisymmetric 'pearl' and 'mushroom' patterns were observed in neutrally buoyant core-annular horizontal pipe flows at high Schmidt number and Reynolds number in the range 2 < Re < 60 (d 'Olce et al 2008).…”

Miscible displacements in Hele–Shaw cells: Nonmonotonic viscosity profiles

“…Generally poor agreement is observed between the Stokes results and corresponding findings for Darcy flows [4]. [5] extend this investigation to variable density displacements in vertical Hele-Shaw cells and find excellent agreement with the experimental data of [6] regarding the most amplified wavelength. These results establish Stokes based direct simulations and accompanying linear stability analyses as powerful tools for comparing Hele-Shaw displacements with their Darcy based counterparts.…”

Fingering versus stability in the limit of zero interfacial tension

“…We investigate the growth of patterns formed in the small-wavelength limit using pairs of miscible fluids, where the interfacial tension s is negligible. For large enough Peclet numbers, where advection dominates over diffusion, the inter-diffusion of the fluids is negligible so that the fluids remain separated by a well-defined interface [20][21][22] . This is the situation we investigate here.…”