Growth of radial viscous fingers in a Hele-Shaw cell

• Published in: Journal of fluid mechanics 1989-04, Vol.201 (1), p.223-242
• Main Author: Chen, Jing-Den
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Nonhomogeneous flows; Physics
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112089000911

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 at a constant volumetric flow rate, q, to radially displace a much more viscous liquid (glycerine. The data presented in this paper are taken mostly at 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 × 105−7.0 × 106. The miscible patterns are insensitive to dispersion for large q/Db. For immiscible fingers λ/b scales with Ca−0.31 for capillary number Ca ranging from about 8 × 10−4 to 0.05. For miscible fingers, λ/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 λ the splitting wavelength.