Fluctuations and Pinch-Offs Observed in Viscous Fingering

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. EXPERIMENTAL METHODS AND DATA ANALYSIS We conducted experiments in a 254 cm long channel formed from 1.9 cm thick glass plates. The spacing between the glass plates was set by stainless steel strips with thicknesses b = 0.051 cm, 0.064 cm, 0.102 cm, or 0.127 cm; the channel width w between the spacers was varied between 19.9 cm and 25.1 cm. Both glass plates were supported and clamped at the sides, and the channel was illuminated from below. For aspect ratios under 150, we used a smaller channel of length 102 cm and width 7.4 cm. Interferometric measurements revealed that the root-mean-square variations in gap thickness were typically 0.6% or less in the large channel and 0.8% in the small channel. Mechanical measurements of the bending of the glass due to the imposed pressure gradient revealed that such deflections were typically 0.2% or less; the maximum deflection (2.2%) was measured in the widest channel close to the oil reservoir at the highest flow rates. Experiments were conducted with air penetrating a Dow Corning silicone oil whose surface tension and dynamic viscosity was either σ = 19.6 dyne/cm, μ = 9.21 cP or σ = 20.6 dyne/cm, μ = 50.8 cP at laboratory temperature (22◦C). The oils wet the glass completely. We withdrew oil at a uniform rate using a syringe pump attached to a reservoir at one end of the channel; an air reservoir at atmospheric pressure was attached to the other end. The entire experiment was placed on a floating optical table to minimize vibrations and allow precise levelling of the channel. We obtained images of up to 1200× 10,000 pixels at a resolution of 0.25 mm/pixel using a camera and a rotating mirror. The camera captured up to 11 overlapping frames which were then concatenated, background subtracted, and corrected for perspective effects. The interfaces were then digitally traced, yielding finger width values accurate to 0.1% in the larger channel and 0.3% in the smaller channel. For each flow rate, up to four time sequences of 20-30 digital interfaces were recorded. Finger widths determined in consecutive sequences agreed within the measurement accuracy. Mean width values agreed within 0.5% for data sets repeated after channel disassembly, cleaning, and reassembly. For each experimental run, we determined the time average 〈λ 〉 and the root-mean-square (rms) fluctuation from the mean δλ (as described in [13]). Each data set was analyzed for flow rates up to the point of tip splitting, beyond which the finger width λ was no longer well defined 1.