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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 a t a constant volumetric flow rate, q, to radially displace a much more viscous liquid (glycerine, 1050 cP) in the cell. Oil is immiscible with and water is miscible with glycerine. The data presented in this paper are taken mostly a t 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 x lo5-7.0 x lo6. The miscible patterns are insensitive to dispersion for large q/Db. For immiscible fingers h / b scales with for capillary number Ca ranging from about 8 x to 0.05. For miscible fingers, h / 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 h the splitting wavelength.

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When a small drop or bubble is driven through a liquid phase to a fluid-fluid interface, a thin liquid film which forms between them drains, until an instability forms and coalescence occurs. Lin and Slattery (1982b) developed a hydrodynamic theory for the first portion of this coalescence process: the drainage of the thin liquid film which occurs while it is sufficiently thick that the effects of London-van der Waals forces and electrostatic forces can be ignored. Here we extend their theory to include the effects of the London-van der Waals forces. To simplify the analysis, we follow the suggestion of Lipkina (1975,1978) in developing an expression for the rate of thinning at the rim or barrier ring of the draining film, A linear stability analysis permits us to determine the coalescence time or the elapsed time between the formation of a dimpled film and its rupture at the rim.For comparison, this same linear stability analysis is applied to the thinning equations developed by MacKay and Mason (1963) for the plane parallel disc model and by Hodgson and Woods (1969) for the cylindrical drop model.For all three models, our linear stability estimate for the coalescence time t, is in better agreement with the available experimental data than is the elapsed time tm between the formation of a dimpled film and its drainage to zero thickness at the rim in the absence of instabilities. SCOPEThe rate at which drops or bubbles suspended in a liquid coalesce is important to the preparation and stability of emulsions, of foams and of dispersions, to liquid-liquid extraction, to the formation of an oil bank during the displacement of oil from a reservoir rock, and to the displacement of an unstable foam used for mobility control in a tertiary oil recovery process.On a smaller scale, when two drops (or bubbles) are forced to approach one another in a liquid phase or when a drop is driven through a liquid phase to a fluid-fluid interface, a thin J. Chen is presently with Schlumberger-Doll Research, Old Quarry Road, Ridgefield, CT 06877.liquid film forms between the two interfaces and begins to drain. As the thickness of the draining film becomes sufficiently small (about 1,000 A), the effects of the London-van der Waals forces and of any electrostatic double layer become significant. Depending upon the sign and the magnitude of the disjoining pressure attributable to the London-van der Waals forces and the repulsive force of any electrostatic double layer, there may be a critical thickness at which the film becomes unstable, ruptures and coalescence occurs.Lin and Slattery (1982b) considered the early stage of this coalescence process, when the draining film is sufficiently thick that the effects of the London-van der Waals forces and of any electrostatic double layer can be neglected. To simplify the

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