No abstract

Our experiments on viscous (Saffman-Taylor) fingering in Hele-Shaw channels reveal finger width fluctuations that were not observed in previous experiments, which had lower aspect ratios and higher capillary numbers Ca. These fluctuations intermittently narrow the finger from its expected width. The magnitude of these fluctuations is described by a power law, Ca −0.64 , which holds for all aspect ratios studied up to the onset of tip instabilities. Further, for large aspect ratios, the mean finger width exhibits a maximum as Ca is decreased instead of the predicted monotonic increase. PACS: 47.20.Ma, 47.54.+r, 47.20.Hw, When a less viscous fluid displaces a more viscous fluid in a Hele-Shaw channel (a quasi-2D geometry in which the width w is much greater than the channel thickness b), the interface between the fluids forms a pattern of growing "fingers". A single finger forms at low flow rates; more complex branched patterns evolve at high flow rates. This phenomenon is the prototypical example of moving interface problems, such as dendritic growth and flame propagation, and thus continues to receive attention for the insight it provides into these important problems [1]. The problem was studied first by Saffman and Taylor (1958) [2], who injected air into oil in a Hele-Shaw cell and observed the formation of a single, steadily moving finger whose width decreased monotonically to 1/2 of the channel width as the finger speed was increased. In subsequent experimental [3], numerical [4], and theoretical [5] work, the ratio of finger width to channel width, λ, was found to depend on a modified capillary number, 1/B = 12 (w/b) 2 Ca, which combines the aspect ratio, w/b, and the capillary number, Ca = µV /σ, where µ is the dynamic viscosity of the liquid, V is the velocity of the tip of the finger, and σ is the surface tension. For large 1/B values, a transition to complex patterns of tip-splitting occurs [3,6,7].Our experiments reveal fluctuations in the width of the evolving viscous fingers that have not been reported in previous experiments [2,3,6] nor predicted theoretically. The fluctuations intermittently narrow the fingers from their expected width and are largest at low Ca, falling off as a power law with increasing Ca. Further, at large aspect ratios, the fluctuations are accompanied by substantial departures from the monotonic dependence of the finger width on 1/B found previously [3][4][5]: for aspect ratios w/b 250, which were not examined in previous work, we find that the mean finger width no longer scales monotonically with 1/B; for smaller aspect ratios, our finger width measurements are in accord with previous results. Because large aspect ratios should more closely approach the ideal of two-dimensional flow, our observations pose a challenge to the assumptions underlying theoretical analyses of viscous fingering.Experimental Methods. We conducted experiments in a 254 cm long channel formed of 1.9 cm thick glass plates. The spacers between the glass plates were stainless steel strips with thicknesses...

We have measured the coarsening due to surface tension of radially grown fractal viscous fingering patterns. The patterns at late times depend on the structural form at the onset of coarsening, providing information on the age of the fractal. The coarsening process is not dynamically scale invariant, exhibiting two dynamic length scales that grow as L1(t) approximately t(0.22+/-0.02) and L2(t) approximately t(0.31+/-0.02). The measured exponents are in agreement with the results of recent numerical studies of diffusion-controlled coarsening of a diffusion-limited aggregation fractal [Phys. Rev. E 65, 050501 (2002)]].

Abstract.Our experiments on viscous (Saffman-Taylor) fingering in Hele-Shaw channels reveal several phenomena that were not observed in previous experiments. At low flow rates, growing fingers undergo width fluctuations that intermittently narrow the finger as they evolve. The magnitude of these fluctuations is proportional to Ca −0.64 , where Ca is the capillary number, which is proportional to the finger velocity. This relation holds for all aspect ratios studied up to the onset of tip instabilities. At higher flow rates, finger pinch-off and reconnection events are observed. These events appear to be caused by an interaction between the actively growing finger and suppressed fingers at the back of the channel. Both the fluctuation and pinch-off phenomena are robust but not explained by current theory.Viscous fingering occurs when a less viscous fluid displaces a more viscous fluid in a Hele-Shaw channel (a quasi-2D geometry in which the width w is much greater than the channel thickness b); the interface between the fluids is unstable and forms a growing pattern of "fingers". A single finger forms at low flow rates; more complex branched patterns evolve at high flow rates. This phenomenon is the simplest example of the class of interfacial pattern forming systems which includes dendritic growth and flame propagation. Viscous fingering thus continues to receive attention for the insight it provides into these important problems [1,2,3,4].Saffman and Taylor first studied the problem in 1958 [5] by injecting air into oil in a Hele-Shaw cell. They observed the formation of a single, steadily moving finger whose width decreased monotonically to 1/2 of the channel width as the finger speed was increased. Subsequent experimental [6], numerical [7], and theoretical [8,9] work found that the ratio of finger width to channel width, λ depended on a modified capillary number, 1/B = 12 (w/b) 2 Ca, which combines the aspect ratio, w/b, and the capillary number, Ca = µV /σ , where µ is the dynamic viscosity of the liquid, V is the velocity of the tip of the finger, and σ is the surface tension. A transition to complex patterns of tip-splitting occurs at large 1/B values [6,10,11,12]. Our experiments have revealed two phenomena that were not reported in prior experiments [5,6,10,11] or predicted theoretically: fluctuations in the width of the evolving viscous fingers [13], and finger pinch-off events.