Viscous Fingering Patterns and Evolution of Their Fractal Dimension

We focus here on the fractal dimension of the patterns and how it varies with the lifting pressure in the Hele-Shaw cell. The fractal dimension of the...
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Ind. Eng. Chem. Res. 2009, 48, 8837–8841

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Viscous Fingering Patterns and Evolution of Their Fractal Dimension Suparna Sinha and Sujata Tarafdar* Condensed Matter Physics Research Centre, Physics Department JadaVpur UniVersity, Kolkata 700032, India

We present a study of viscous fingering patterns which arise because of the instability at the interface of two fluids of dissimilar viscosities. Viscous fingering experiments were performed in a lifting Hele-Shaw cell with a Newtonian defending fluid, castor oil or olive oil, while the invading fluid is air. We focus here on the fractal dimension of the patterns and how it varies with the lifting pressure in the Hele-Shaw cell. The fractal dimension of the fluid-air interface is measured by the divider step method. It is seen that the dimension changes from a value close to one, indicating no instability, to a value around 1.6 as the pressure is increased. A change in fractal dimension with time is also observed as the pattern evolves. Measurement of the fractal dimension offers a method for quantifying the instability. 1. Introduction A wide variety of pattern formation processes involving complex fluids have been studied1,2 in the last few decades. An interesting aspect of many of these patterns is that they exhibit self-similarity which can be described through a fractal approach. The fractal picture is not only an elegant way of describing the patterns but offers a means of quanitifying the instability leading to the self-similarity, through a number of measurable parameters such as the fractal dimension, chemical distance exponent, Hurst exponent, lacunarity, and so on.2,3 These quantities are not only of academic interest but now are routinely being used in diverse fields such as medicine, geology, chemical engineering, and many others.4,5 For example the Hurst exponent in heartbeat patterns and the fractal dimension in brain scans are now used as diagnostic tools.6 The present study belongs to the family of pattern-forming instability in fluid-fluid interfaces. The underlying phenomenon behind the formation of the static or dynamic interfaces may be evaporation, mixing, drying, chemical reactions, temperature gradient, surface tension, viscosity, and so on. The interfacial instability which develops when a low viscosity fluid is pushed into a high viscosity fluid in a quasitwo-dimensional geometry is commonly referred to as viscous fingering (VF). The development of VF patterns has been observed from very early times.7 VF patterns exhibit two types of morphologies: a single smooth finger referred to as the Saffman-Taylor finger, on one hand,8,9 and highly branched fractal fingers, on the other hand.10,11 Different modifications of the Hele-Shaw (HS) cell including the lifting Hele-Shaw cell (LHSC) have been widely used to study VF patterns.12 The lifting Hele-Shaw cell (LHSC) is applicable in the practical problem of adhesion13,14 and thus is of special interest. The instability developed here depends on several parameters such as the viscosity and surface tension of the fluids, the lifting force, and substrate properties. We show that measurement of the fractal dimension df under different conditions is a meaningful way of quantifying the strength of the instability.

under study. The plates are of about 1 cm thickness and 10 cm in diameter. In the lifting version (LHSC), the plates are separated normally with a constant force15 or at a constant velocity.13,16 This is essentially the same setup as a probe-tack experiment,17 used to study adhesion. Initially a small volume of fluid measured accurately with the help of an acu-pipet is placed at the center of the stationary lower plate. On lifting the upper plate, low pressure, less than the surrounding atmospheric pressure, is created because of a suction effect, and the surrounding low viscosity fluid moves in, displacing the defending fluid within the plates. When the interface of the trapped fluid with air or any other less viscous fluid adjacent to it becomes unstable, the so-called viscous fingers (VF) are formed. Figure 1 shows a schematic diagram of our LHSC setup. We use transparent plates of glass or perspex, so that the pattern formation can be observed and recorded on video from below. A pneumatic cylinder operated by an air compressor lifts the upper plate at a constant force, which can be controlled. 3. Measuring the Fractal Dimension Certain conditions must be met in order for the instability of the fluid-fluid interface to develop. The condition is usually described through a dimensionless capillary number.2 Nca ) µVint/S

(1)

2. The LHSC Setup The Hele-Shaw cell (HSC)7 consists of two plane plates separated by a small gap, which contains the high viscous fluid * To whom correspondence should be addressed. E-mail: sujata@ phys.jdvu.ac.in. Phone: +91 33 24146666 (extn. 2760). Fax: +91 33 24146584.

Figure 1. A schematic of the lifting Hele-Shaw cell used in the experiments described here.

10.1021/ie801836r CCC: $40.75  2009 American Chemical Society Published on Web 04/21/2009

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Here µ and S are the fluid viscosity and surface tension, respectively, and Vint is the interface velocity. A high capillary number favors viscous fingering. For low Nca, the interface moves inward without fingering. However, it has been shown14 that the properties of the substrate are important, too; this is not reflected in the capillary number. A fingering velocity parameter Vfin was suggested by Sinha et al.,14 involving fluid and solid dielectric constants, for a more complete picture. S 1 Vfin ) (2) µ (ε ε )2 s f Here εs and εf are dielctric constants of the solid substrate and fluid, respectively. For a given fluid substrate combination, the determining parameter for fingering is the lifting force, which determines the interface velocity. In patterns involving only Newtonian fluids, the viscous fingers may not always be fractal; nonlinearity in some form, as with non-Newtonian fluids, or a tortuous space as in porous rocks, leads easily to highly branched fractal structures. However, even for Newtonian fluids, with a strong lifting pressure a fractal interface is produced. An interesting aspect of the fractal nature of LHSC patterns is the following: when the process starts, the inVading fluid has a branched structure with a nontrivial fractal dimension, while toward the end of the process the defending fluid pattern is fractal.18 So there is a crossover for the mass fractal dimension of the invading fluid from fractal-nonfractal and vice versa for the defending fluid. In the present paper we do not attempt to study this crossover; rather, we focus on the interface fractal dimension which should have a value between 1 and 2. We study two aspects of variation in the fractal dimension df: (1) variation of df for the fully developed fingering pattern as the lifting force in the LHSC is varied and (2) the variation in df with time for a single lifting experiment at a constant pressure. The fluids used in the experiment are castor oil (Jyoti chemicals) and olive oil (Bertolli), which are Newtonian. The air pressure Pr in the compressor is recorded from the pressure gauge. A proportional force acts on the fluid as the upper plate is lifted by the pneumatic cylinder arrrangement. Our results are reported with Pr as the control parameter. The dynamic profile of the moving fluid-air interface is recorded by a CCD camera, placed below the lower glass plate. 3.1. Image Analysis. Dimension of the Evolving Patterns. The software Image-Pro Plus 6.0 is used for image analysis. The images are captured at the rate of 4 frames/s. The captured image is initially acquired by the computer from the NTSC video camera and enhanced or filtered to increase the visibility. The frames are separated to identify precisely the different stages of pattern formation up to the time of separation of the plates. It is noted whether viscous fingers appear at any point during the process, and the force at which fingering starts, if at all. The initial blob of fluid has an almost circular boundary, when lifting begins. The circular boundary either shrinks uniformly (in the case of no fingering, for low Pr) or shows an instability with the boundary becoming increasingly irregular, with increasing pressure. When there is no instability, as lifting continues, the fluid blob becomes very small, and a bridge of fluid still connects the two plates. On further lifting, the bridge forms a narrow neck, which finally breaks. Now the outline of the small blob of fluid starts to increase again. In this case, of course, the boundary is always smooth, so a trivial value of df ) 1 is obtained. In the present study we are interested in the case where the interface becomes unstable at some point and fingers of air,

Figure 2. A typical photograph of the viscous fingering pattern obtained with olive oil in the setup described.

Figure 3. The log-log plot for interface length versus length scale, from which the fractal dimension is determined. Data for high and low pressures are shown.

which may branch repeatedly, enter the fluid, forming a treelike pattern (Figure 2). In this case, toward the end of the fingering process, a number of necks connect the two plates until complete separation. The number of such necks increases with Pr. We analyze the fully developed fractal pattern visible just before the plates separate completely. The fractal dimension of the frames of interest in the video clip are analyzed using the divider-step method. In this method we set a minimum scale such as l0 with which we measure the length L0 of the interface. Next we find the length of the interface with a larger scale l1. The process is repeated for larger and larger scales, until at the nth step the measuring scale ln becomes of the order of the size of the whole pattern. The results Ln are plotted on a log-log plot against ln, and a straight line is obtained, showing that the interface is indeed self-similar (Figure 3). The slope of the straight line gives df, with an accuracy to the second decimal place. The variation in fractal dimension with lifting pressure, for the fully developed pattern has been measured, as well as the variation in df for a particular pattern evolving in time. The results are given below.

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Figure 4. The patterns obtained at different lifting forces (the corresponding air pressure in the compressor is shown in the graph). Photographs show the VF patterns for castor oil and air. The interface changes from smooth to fingered as pressure increases. The plate separation time is plotted in the graph, with a power-law fit.

Figure 5. The variation of fractal dimension with lifting force.

4. Results Figure 3 shows typical data for fractal dimension determination for two different values of lifting pressure. Measured length Ln for scale ln is plotted on a logarithmic plot, and a linear behavior is obtained over a 20× variation in length scale. 4.1. Variation of Fractal Dimension with Lifting Pressure. The variation in the appearance of the fully developed fractal VF patterns for different lifting pressures in the range 7000 to 28 000 kg/m2 is shown in Figure 4. The initial film thickness varies between 5 and 10 µm. for this range of pressures. The symbols in the graph show variation of plate separation time with lifting force (proportional to Pr), which follows a power law with exponents close to 0.5. This is discussed in detail in the report from Sinha et al.14 The development of the instability is clearly seen from the photographs of the patterns. For the lowest pressure, the fluid-air interface is a smooth circular outline, but as Pr increases, more and more feathery ramified fingers develop. The fractal dimension accordingly increases from 1 toward 1.6. The variation in df with Pr is shown in Figure 5. The variation is roughly linear.

Figure 6. The interface length measured using different scales for lifting with a force of 28 000 kg/m2, at different times. The upper lines on the logarithmic plot are for intermediate times, when df is maximum.

Points on the graph are averages over four different patterns under the same conditions. 4.2. Variation of Fractal Dimension of a VF Pattern with Time. The values of df shown in Figure 5 have been calculated for the fully developed VF pattern, before the plates separate. However, there is some variation in df during the evolution of the pattern in time as well. Interestingly, df first increases somewhat and then falls with time. This behavior is understood as follows. Just as lifting starts, the interface is smooth. The fingering develops in time; after a point, however, as the process moves toward plate separation, the fingers start to smooth out again. The data points for the divider-step calculation of df are shown in Figure 6. Figure 7 for Pr ) 28 000 kg/m2 shows the photographs corresponding to data in Figure 6, and the variation in the fractal dimesion. Though the variation is small, it is unmistakable. For pressures higher than 22 500 kg/m2, there is development of cavitation in the pattern. This is also seen in Figure 7. Initially the fluid blob was free of air bubbles, but cavities appear on lifting at high pressure. Development of cavitation is discussed in detail by Poivet et al.19,20

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Figure 7. The patterns obtained at different times during finger evolution for a lifting force of 28 000 kg/m2. Photographs show the VF patterns for olive oil and air. The fractal dimension first increases and then decreases. The range of df varies from about 1.5 to 1.65, which is outside error limits.

It was shown earlier15 that the finger velocities first increase and then decrease with time; it is possible that the maximum in fractal dimension coincides with the maximum in finger velocity. This conjecture has to be verified through more experiments. 5. Discussion A careful observation of Figure 4 shows the difference in the patterns for different lifting force. At the lowest pressures the outline of the fluid blob is a smooth circle, which becomes smaller and smaller as lifting progresses. After separation of course, the Newtonian fluids spread again, and the outline again increases. This observation is useful to identify the plate separation time, which has been studied in detail previously.14 With increasing Pr, the fingers start to appear, but initially they are small undulations, which die out later. The fractal dimension now starts to depart from 1. For still larger Pr, the fingers persist and grow. The nature of variation in df with time is rather surprising, since the value of df decreases after reaching a peak. The general appearance of the central photo in Figure 7 corresponding to the peak in df does not indicate that df corresponding to the last photo has a much lower value. But there are finer branchings in the central photo which appear on magnification and lead to a higher df. However, more observations are needed to corroborate whether this is a general feature seen at all Pr. In the present preliminary study, we have used two vegetable oils, castor oil and olive oil. These are seen to be Newtonian for moderate shear rates,21 and their surface tension and viscosity are suitable for producing prominent viscous fingers with reproducible characteristics. Here we report pressure variation for castor oil and time variation in df for olive oil. We expect similar behavior for the opposite case; however, more complete studies with these and other Newtonian and non-Newtonian fluids for pressure as well as time variation are in progress. The range of variation shown in Figure 4 is from 1 for the stable interface to ∼1.6, which is close to that for typical diffusion-

limited aggregates (DLA). For miscible fluids in the normal Hele-Shaw cell, DLA-like values are well-known.2 In the lifting Hele-Shaw cell, the fingers start to develop at the periphery and move inward, but toward the end of the fingering process, the pattern formed by the defending fluid does indeed resemble the DLA, though the formation process of the two patterns in quite different. We expect results for the immiscible fluids we are studying to range up to DLA-like values, irrespective of precise details such as plate separation and size or surface tension and viscosity of the fluids. However, more experimental work is needed to establish this. Though the original Hele-Shaw cell7 was introduced more than a century ago, viscous fingering patterns and their relevance to the problem of adhesion is still a topic of intense activity.14,16,17 Different aspects of the lifting Hele-Shaw cell, in particular, are gaining importance. While Poivet at al.19,20 focus on cavitation, Ben Amar and Bonn22 have shown recently that the three-dimensional character of the problem and nonlinear effects must be considered to get a true understanding of the experimental results. We conclude by saying that VF patterns and their fractal character have been known for a long time,2 but the transition from a stable to unstable interface is usually considered to be a discrete change. This work shows that the instability sets in gradually, and one can quantify the strength of the instability through the interface fractal dimension. We hope such studies may pave the way for more useful applications of fractal measurements in practical problems. This may be expected from the interdisciplinary nature of fields where fractal concepts have emerged in recent times. Acknowledgment This work has been supported by DST through R/P No SR/ S2/CMP-22/2004. S.S. is grateful to DST for financial support.

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Literature Cited (1) Daoud, M.; Van Damme, H. Fractals in Soft Matter Physics; Daoud, M., Williams, C. E., Eds.; Springer-Verlag: Berlin, 1999. (2) Vicsek T. Fractal Growth Processes; World Scientific: Singapore, 1989. (3) Bunde, A.; Havlin, S. Fractals and Disordered Systems; SpringerVerlag: Secaucus, NJ, 1995. (4) Havlin, S.; Buldyrev, S. V.; Bunde, A.; Goldberger, A. L.; Ivanov P Ch Peng, C.-K. Stanley H E Scaling in nature: from DNA to heartbeats to weather. Physica A 1999, 46. (5) Turcotte, D. L. Fractals and Chaos in Geology and Geophysics, 2nd ed.; Cambridge Univ. Press: New York, 1997. (6) Klonowski, W. From conformons to human brains: an informal overview of nonlinear dynamics and its application in biomedicine. Nonlinear Biomed. Phys. 2007, http://dx.doi.org/10.1186/1753-4631-1-5. (7) Hele-Shaw, H. J. S. On the motion of a viscous fluid between two parallel plates. Nature (London) 1898, 58, 34. (8) Saffman, P. G.; Taylor, G. I. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. London Ser. A 1958, 245, 312. (9) Hong, H. C.; Langer, J. S. Analytic theory of the selection mechanism of the Saffman-Taylor problem. Phys. ReV. Lett. 1986, 56, 2036. (10) Paterson, L. Radial fingering in a Hele-Shaw cell. J. Fluid Mech. 1981, 113, 513. (11) Combescot, R.; Ben Amar, M. Selection of Saffman-Taylor fingers in the sector geometry. Phys. ReV. Lett. 1991, 6. (12) McCloud, K. V.; Maher, J. V. Experimental perturbation to Saffman-Taylor flow. Phys. Rep. 1995, 260, 139.

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(13) Gay, C.; Leibler, L. On stickiness. Phys. Today 1999, NoV, 48–52. (14) Sinha, S.; Dutta, T.; Tarafdar, S. Adhesion and fingering in the lifting Hele-Shaw cell: Role of the substrate. Eur. Phys. J. E 2008, 25, 267. (15) Kabiraj, S. K.; Tarafdar, S. Finger velocities in the lifting HeleShaw cell. Physica A 2003, 328, 305. (16) Lindner, A.; Derks, D.; Shelley, M. J. Stretch flow of thin layers of Newtonian fluids: Fingering patterns and lifting forces. Phys. Fluids 2005, 17, 072107. (17) Tirumkudulu, M.; Russel, W. B. On the measurement of ‘tack’ for adhesives. Phys. Fluids 2003, 5, 6. (18) Tarafdar, S.; Roy, S. Tree patterns on viscous paste surfaces. Fractals 1995, 3, 99. (19) Poivet, S.; Nallet, F.; Gay, C.; Fabre, P. Cavitation induced force transition in confined viscous liquids under traction. Europhys. Lett. 2003, 62, 244. (20) Poivet, S.; Nallet, F.; Gay, C.; Teisseire, J.; Fabre, P. Force response of a viscous liquid in probe-tack geometry: Fingering versus cavitation. Eur. Phys. J. E 2004, 15, 97. (21) www.engineeringtoolbox.com. (22) Ben Amar, M.; Bonn, D. Fingering instabilities in adhesive failure. Physica D 2005, 209, 1.

ReceiVed for reView December 1, 2008 ReVised manuscript receiVed March 7, 2009 Accepted April 1, 2009 IE801836R