The Saffman–Taylor instability: From the linear to the circular geometry

Abstract

The Saffman–Taylor fingers are studied in cells that have the form of sectors of a disk. The less viscous fluid can be injected at the apex (divergent flow) or at the periphery (convergent flow). As in the linear geometry, at large velocities, a unique finger tends to occupy a well determined fraction λ of the cell angular width. This fraction is a function of the angle of the cell, being larger than 0.5 in the divergent case and smaller in the convergent case. In both cases these fractions tend linearly toward λ=0.5 when the angle of the cell tends to zero. In support of recent theories, these results show how the selection is changed when the geometry induces an increase or a decrease of the curvature of the profiles. The formation of fingers in the circular geometry is revisited. In a divergent flow, the circular front appears to break into independent parts so that each finger grows as if it were contained in a sector shaped cell. The rate of occupancy of the cell by one of the fluids as a function of the distance to the center is then discussed. Finally, the existence of the mathematical counterpart to the well-known Saffman–Taylor finger solutions in a nonparallel cell is discussed in the Appendix.

Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations

“…The mechanics of pulmonary airway reopening is governed by an interaction between viscous, capillary and elastic forces, within the more complex geometry of a network of collapsed tubes (Heap & Juel 2008;Heil & Hazel 2011). Recent work on the injection of an air bubble into an elastic-walled, liquid-filled Hele-Shaw cell showed that the well-known fingering instability that occurs when a gas displaces a viscous fluid in a rigid Hele-Shaw cell (Saffman & Taylor 1958;Paterson 1981;Homsy 1987;Chen 1989;Thomé et al 1989;Miranda & Widom 1998) can be delayed or even suppressed through changes in cell geometry due to elastohydrodynamic interactions (Pihler-Puzović et al 2012AlHousseiny et al 2013). Related fluid-structure interaction problems include the peeling of lubricated flexible sheets from viscous substrates (McEwan & Taylor 1966;Hosoi & Mahadevan 2004) and suppression of ribbing instabilities by the use of elastomer-coated rolls; see, e.g., Carvalho & Scriven (1997.…”

Analysis of a model for foam improved oil recovery

“…Normally the injected fluid is less viscous (i.e. more mobile) than the oil which it drives out, which can lead to fingering phenomena (Ashoori et al 2010;Casademunt & Magdaleno 2000;Cottin et al 2010a,b;Couder et al 1986;Kessler & Levine 1986a,b;Moore et al 2002;Pitts 1980;Saffman & Taylor 1958;Sahimi 2011;Tabeling et al 1987;Thome et al 1989): fingers of the driving fluid penetrate and break through the oil, after which additional injected fluid will continue to follow the already established flow paths, without displacing any more oil. This viscous instability initially follows, and greatly magnifies, heterogeneities in permeability within the reservoir (Lake 2010).…”

Diffusion-limited aggregation in channel geometry

“…These stationary translating solutions -called SaffmanTaylor fingers [10]-have been studied in viscous fingering experiments in Hele-Shaw cells [11]. In this paper we compare them with average profiles of DLA clusters.…”