When a viscous fluid filling the voids in a porous medium is driven forwards by the pressure of another driving fluid, the interface between them is liable to be unstable if the driving fluid is the less viscous of the two. This condition occurs in oil fields. To describe the normal modes of small disturbances from a plane interface and their rate of growth, it is necessary to know, or to assume one knows, the conditions which must be satisfied at the interface. The simplest assumption, that the fluids remain completely separated along a definite interface, leads to formulae which are analogous to known expressions developed by scientists working in the oil industry, and also analogous to expressions representing the instability of accelerated interfaces between fluids of different densities. In the latter case the instability develops into round-ended fingers of less dense fluid penetrating into the more dense one. Experiments in which a viscous fluid confined between closely spaced parallel sheets of glass, a Hele-Shaw cell, is driven out by a less viscous one reveal a similar state. The motion in a Hele-Shaw cell is mathematically analogous to two-dimensional flow in a porous medium.
Analysis which assumes continuity of pressure through the interface shows that a flow is possible in which equally spaced fingers advance steadily. The ratio λ = (width of finger)/(spacing of fingers) appears as the parameter in a singly infinite set of such motions, all of which appear equally possible. Experiments in which various fluids were forced into a narrow Hele-Shaw cell showed that single fingers can be produced, and that unless the flow is very slow λ = (width of finger)/(width of channel) is close to , so that behind the tips of the advancing fingers the widths of the two columns of fluid are equal. When λ = 1/2 the calculated form of the fingers is very close to that which is registered photographically in the Hele-Shaw cell, but at very slow speeds where the measured value of λ increased from 1/2 to the limit 1.0 as the speed decreased to zero, there were considerable differences. Assuming that these might be due to surface tension, experiments were made in which a fluid of small viscosity, air or water, displaced a much more viscous oil. It is to be expected in that case that λ would be a function of μU/T only, where μ is the viscosity, U the speed of advance and T the interfacial tension. This was verified using air as the less viscous fluid penetrating two oils of viscosities 0.30 and 4.5 poises.

The Penetration of a Fluid into a Porous Medium or Hele-Shaw Cell Containing a More Viscous Liquid

Rayleigh–Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II

Slug flow belongs to a class of intermittent flows that has very distinctive features Even if the inlet conditions are stationary, the flow as seen by an observer, is an unsteady phenomenon, dispersed flow appearing alternately with separated flow These two states follow in a random-like manner, inducing pressure and velocity fluctuations In vertical flow, the large bUbbles-typically longer than, say, one pipe diameter-rise with a round shaped front followed by a cylindrical main body surrounded by an annu­ lar liquid film In the literature, these long bubbles are frequently referred to as Taylor bubbles or Dumitrescu bubbles In the film, gravity forces the

Modeling of two-phase slug flow

Publisher Summary Chemical engineers, metallurgists, geologists, brewers and cooks all try to understand processes in which bubbles or drops move through liquids. Because interfacial phenomena affect the motion in a number of different ways, there are many cases to consider. The simplest case is a bubble rising in a Newtonian liquid, far enough from boundaries to be treated as if isolated. It is assumed that the viscosity and density of the internal gas are negligible, and that the surface rheology reduces to the single parameter of a constant uniform surface tension. Another case is concerned with drops, whose interior viscosity and density are taken into account, but which satisfy the other conditions of Newtonian liquid. The most restrictive condition is constancy of the surface tension, for it requires exceptionally pure fluids. Experiments seldom agree with the predictions of above mentioned cases for small bubbles or drops unless great care is taken to remove impurities. These adsorb at the surface and lower the surface tension. The flow of the fluid then carries the impurities around the surface, setting up inequalities of concentration and hence of surface tension, which oppose the motion. The chapter contains an account of the motion of drops and bubbles in the ideal solutions of surfactants.

The Motion of Bubbles and Drops Through Liquids

A vortex technique capable of calculating the Rayleigh–Taylor instability to large amplitudes in inviscid, incompressible, layered flows is introduced. The results show the formation of a steady‐state bubble at large times, whose velocity is in agreement with the theory of Birkhoff and Carter. It is shown that the spike acceleration can exceed free fall, as suggested recently by Menikoff and Zemach. Results are also presented for instability at various Atwood ratios and for fluids having several layers.

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Vortex simulations of the Rayleigh–Taylor instability

This paper is concerned with the flow of an incompressible heavy liquid past a gas bubble in an infinitely long vertical tube. Attention is confined to plane flow, and it is assumed that the bubble extends downwards without limit, so that the motion is steady. A co-ordinate system attached to the bubble is chosen, with the liquid falling around the bubble, instead of the bubble rising in the liquid. It is supposed that the flow is irrotational, whence it can be described in terms of a complex potential C = 0 + i3f which is an analytic function of the complex variable z = x + iy in the physical plane. The gas in the bubble is considered to be at rest, in a state of constant pressure. The vertical tube is represented in the z plane as an infinite strip - -h < x < Ih of width h. The complex potential = (z) is normalized so that it transforms the region of flowing liquid in this tube conformally on to the infinite strip - 1 < /i < 1, with the free boundary of the bubble mapping on to a slit along the positive 0 axis. If the origin is placed in the physical plane at the vertex of the bubble, Bernoulli's equation shows that the requirement of constant pressure along the free boundary of the bubble takes the form

P. R. Garabedian

Papers

On steady-state bubbles generated by Taylor instability