Growth of radial viscous fingers in a Hele-Shaw cell

Abstract

A series of experiments are performed in a Hele-Shaw cell, consisting of two parallel closely spaced glass plates. A liquid (oil or water, both of viscosity of 1.0 cP) is injected a t a constant volumetric flow rate, q, to radially displace a much more viscous liquid (glycerine, 1050 cP) in the cell. Oil is immiscible with and water is miscible with glycerine. The data presented in this paper are taken mostly a t late stages of the fingering process, when the pattern has multiple generations of splitting. Correlations with time are obtained for the finger length and the overall pattern density. The time-and lengthscales have been found for the immiscible case. At the same dimensionless time, immiscible patterns are similar and have the same generation of splitting. The overall density of each pattern decreases with time. The pattern shows fractal behaviour only after a certain number of generations of splitting. The fractal dimension of the immiscible pattern decreases from 1.9 to 1.82 when the pattern goes from the third to the sixth generation of splitting. The fractal dimension of the miscible pattern reaches a constant value after about ten generations of splitting and the fractal dimension ranges from 1.50 to 1.69 for q/Db = 4.8 x lo5-7.0 x lo6. The miscible patterns are insensitive to dispersion for large q/Db. For immiscible fingers h / b scales with for capillary number Ca ranging from about 8 x to 0.05. For miscible fingers, h / b is insensitive to dispersion and ranges from 5 to 10 for large q/Db. Here D is the molecular diffusion coefficient in glycerine, b the cell gap width and h the splitting wavelength.

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.…”

Fingering versus stability in the limit of zero interfacial tension

“…Such unstable interfaces are important in applications, significantly for sugar refining, oil recovery, hydrology and carbon sequestration [2][3][4][5][6] and much recent work has focused on how to control the instabilities [7][8][9][10] . Viscous fingering plays a central role in our understanding of pattern formation in part because it is amenable to both theory and experiment 1,6,[11][12][13][14] . Of particular importance is the limit where the characteristic finger width, set by the mostunstable wavelength, l c , approaches zero.…”

Minimization of Viscous Fluid Fingering: A Variational Scheme for Optimal Flow Rates

“…The radial flow setup of the Saffman-Taylor problem [8][9][10][11] has provided important insights into fundamental aspects of branching patterns. It takes place when the less viscous fluid is radially injected under constant injection rate, producing fingerlike structures which tend to split at their tips.…”