Fractal Viscous Fingering in Thick Oil

Chishty S.Q.¹* Mazahar Farooqui², Syed Abed³, Mohd.Khizar⁴

¹Department of Physics, Dr. Rafiq Zakaria College for Women, Aurangabad

²Department of Chemistry, Dr. Rafiq Zakaria College for Women, Aurangabad

³Department of Chemistry, Govt. College of Arts Science, Aurangabad

⁴Department of Physics, Kohinoor College of Arts and Science, Khultabad, Aurangabad.

*Corresponding Author E-mail: chishtysq@gmail.com

ABSTRACT:

The study of irregular shapes their self-similarity and randomness is seriously taken by the scientists. The formation of such patterns and parameters affecting their shapes are necessary to study. Such studies provide “bridge between Euclidean geometry and fractal geometry. The processes giving rise to such shapes are usually complex. In the present studies a Hele- Shaw cell is designed to obtain viscous fingering patterns. A SLR camera is used to record complete growth process. Images are selected and digitized for the calculation of fractal dimensions arid results. are presented.

KEYWORDS: Fractal, Hele Shaw Cell Viscous Fingering.

1.0 INTRODUCTION:

Viscous fingering is a phenomenon in which less viscous fluid is injected into more viscous fluid under controlled conditions, which leads to development of fingered interface. The process is governed by Navier-Stokes equation, which in turn can be reduced to a problem of Laplacian growth. In such phenomena fluid flows through narrow path. Viscous fingering is also seen[1]. When air is injected into a thin layer of viscous fluid. The Hele- Shaw cell is a good example of such a system where the interface is allowed to grow between two closely placed surfaces. In the radial type of Hele- Shaw cell, the viscous fluid is placed in the cavity between the two plates and less viscous fluid is injected through a hole at the center. It has been shown that at low pressures the growth exhibits fractal character obeying power law with an exponent of 1.8 for the radius of gyration. At higher pressures, the growth is more compact and dense and the fractal dimension based on radius of gyration approaches 2 indicating non-fractal homogeneous growth[2]. The present work is carried out using Hele Shaw cell; the viscous fluid used is Castor oil. Under low and intermediate pressure conditions the growth of the Fingering pattern is less compact as compared to one at .high pressures[3].

The shapes exhibit self-similarity and scaling over wide range of distance. The fractal dimensions obtained using box counting technique show strong evidence of presence of scaling over a wide range of length scale.

2.0: EXPERIMENTAL:

Hele- Shaw cell was used to study viscous fingering[4]. The cell was made using high quality 9 rnm thick glass plates. The two plates were provided with a collar of 25 mm along the periphery to contain the fluid. The lower plate was bigger than the upper plate to accommodate the upper plate. The spacing between the two plates can be adjusted using spacers of desired thickness. 2.5 mm hole was drilled at ·the center of upper plate to inject air inside the cell[5]. Arrangement was made to control the pressure and the rate of injection of air. The viscous fingering patterns were photographed using SLR camera with zoom lens. Illumination control becomes delicate while handling colour-less fluids to obtain good contrast. Typical patterns obtained using thick oil as viscous fluid are shown in fig. 1 a and b for low and high pressure respectively. The photographs are digitized and converted to a matrix form for processing[6]. The fractal dimensions obtained using box counting technique are presented[7]. The shapes are also analyzed for structure and texture of the boundary using Richardson plot technique Fig. 1 (a) is an open structure growing radially with increasing channel diameter as is the characteristic of the phenomenon, the shape has limited structural has almost no fine texture. This yields a fractal dimension of about 1.6 on the basis of box counting technique.

The box counting fractal dimensions [8]of pattern for high pressure, Fig. 1 (b) comes out to be 1.80. A plot of log (N) versus log(r) is shown in Fig. 2, the points plotted are actual points and the line joining them is the least square fit to the data point giving R square of 0.9996. The scaling exists over four orders as is seen from the graph[9]. At intermediate pressures the fingering obtained deviates from the one shown in Fig. 1 (a) and tends to approach shapes as shown in Fig. 1 (b) where the increase in diameter of the channels with increasing radius of the shape gradually disappears[10]. At high pressure this effect simply disappears and the effect is seen localized to small regions in the secondary branches only.

Fig (1): Actual practical results when air is injected into thick oil (a) at low pressure (b) at high pressure

Following figure shows digitized images for computerized box counting.

3.0: CONCLUSION AND DISCUSSION:

While studying viscous fingering in a Hele Shaw type cell we found that almost for all pressure ranges the patterns obtained exhibit fractal character. Scaling is observed at almost all the length scales. The Box counting dimension gradually increases with increase .in pressure. At low pressures the shapes resemble more with the Laplacian growth patterns. At higher pressures the shapes deviate from the Laplacian growth patterns giving rise to branching and fine structure. The patterns are also dense with higher fractal dimensions.

4.0: REFERENCES:

1. B.B. Mandlebrot, 1983, Fractal Geometry of Nature, Freeman Sanfransisco,

2. T. Vicsek, 1992b Fractal Growth Phenomena, World Scientific Publishing Co. Pvt. Ltd, Singapore.

3: Vicsek, T et al 1988a,; Euro phys. News 19, 24.

4: Markus Alber et al 1998 phy. rev. E. 57, 5

5: Witten T.Aet al 1981 ‘Fractals and Choas III Fractals’ phy. rev. lett. 47,1400-1403 .

6: Bensimon, et al, 1986., Rev. Mod. Phys. 58, 977.

7. S.Buczkowski,1998 Physica, 252 ,23-24.

8. David C. Caccia,1997 Physica, A 246, 609-632.

9. Rauseo,S.N., et al ,1987., Phys. Rey. A 35,1245.

10 S. Tarafdar.et al,1996 phy. rev. E. 54, 6 .

Received on 16.02.2014 Modified on 05.03.2014

Asian J. Research Chem. 7(4): April 2014; Page 377-378