3D Instability of Miscible Displacements in a Hele-Shaw Cell

Abstract

We study the downward miscible displacement of a fluid by a lighter and less viscous one in the gap of a Hele-Shaw cell. For sufficiently large velocities, a well-defined interface separates the two fluids. As long as the velocity or the viscosity ratio are below a critical value, the interface has the shape of a tongue symmetric across the gap. For viscosity ratios larger than a critical value, estimated at 1.5, there exists a critical velocity, above which the interface becomes unstable, leading to a new 3D pattern involving regularly spaced fingers of wavelength about 5 times the cell thickness. We delineate the stability diagram. [S0031-9007(97)04791-1] PACS numbers: 47.20.Gv, 47.20.Bp, 83.85.Pt A key issue in interface dynamics is the understanding of pattern selection. Of particular interest is the pattern selection related to the 2D Saffman-Taylor finger (ST) [1], the selection rule of which has raised a great amount of interest [1][2][3][4][5]. ST dynamics govern a variety of seemingly different physical phenomena, such as viscous fingering [2], directional solidification [3], or thermal plume [4]. We recall that the well-studied ST finger results from the displacement of a viscous fluid by an immiscible, less viscous one in a Hele-Shaw cell (consisting of two parallel plates L 3 W , separated by a small gap b). In this immiscible displacement problem, the interface in the gap consists of a nearly hemispherical meniscus, which completely spans the cell gap at the edges of the finger [6], provided that the capillary number ͑Ca hq͞g, with h the viscosity, g the surface tension, and q the fluid velocity) is sufficiently small. The patterns resulting from the balance between capillary and viscous forces are by necessity quasi-2D (namely, in the L 3 W plane). The removal of surface tension, which makes the coexistence of the fluids across the gap possible, provides the opportunity to obtain 3D patterns, also extending across the gap. The existence of such patterns has an interest of its own, as well as in connection to the 2D selection rule discussed above. 3D patterns can be achieved using two miscible fluids, provided that the displacement velocity (i.e., the Peclet number, Pe qb͞D m , where D m is the molecular diffusion coefficient) is sufficiently large for diffusion effects to be negligible [7,8], so that a sharp fluid interface can be defined. So far, only two specific experiments have addressed miscible displacements in a Hele-Shaw cell: gravity-driven flow in a rectangular geometry [9] and viscous fingering in a horizontal radial geometry (Paterson [10]). However promising Paterson's flower pattern might have been, his pioneering work was surprisingly not resumed even for the simple geometry of a rectangular cell.This Letter presents experimental results of miscible displacements in a vertical planar Hele-Shaw cell in the high Pe regime and emphasizes the role that displacement features along the gap have on the overall pattern. Depending on the velocity and viscosity ratio M (M h 1 ͞h 2 . 1, whe...

Fingering versus stability in the limit of zero interfacial tension

“…The removal of interfacial tension allows for three dimensional (3D) structures, 'tongues' of one fluid penetrating into the other [19][20][21]24 . In regime III, the tip of the tongue is rounded as shown in the upper panel in Fig.…”

“…Further, the boundary coincides with a change in the 3D shape of the 'tongues' from rounded to flat fronts, as mentioned previously. Lajeunesse et al 19,20 have shown that in the limit of long times and zero diffusion, the transition between a smooth profile and a shock front is analytically predicted to occur at Z in /Z out | c ¼ 0.67. This value is higher than that observed in our experiments: Z in /Z out | c E0.…”

“…This result is highly counter intuitive-it implies that the interface becomes stabilized when interfacial tension, which is expected to stabilize the interface, is removed. They further demonstrated that the onset of the instability at lower viscosity ratio coincides with a change in the profile of the interface across the gap: in the range of viscosity ratios where the interface remained stable the profile is quadratic near its tip, whereas in the unstable regime it is characterized by a flat shock front 19,20 .…”

“…They reported a complete suppression of the instability for fluids with DZ40 when the viscosity ratio Z in /Z out was above a certain threshold value 19,20 . This result is highly counter intuitive-it implies that the interface becomes stabilized when interfacial tension, which is expected to stabilize the interface, is removed.…”

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

“…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).…”