The Penetration of a Fluid into a Porous Medium or Hele-Shaw Cell Containing a More Viscous Liquid
24 Jun 1958-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (Academic Press)-Vol. 245, Iss: 1242, pp 312-329
TL;DR: In this paper, it was shown that a flow is possible in which equally spaced fingers advance steadily at very slow speeds, such that behind the tips of the advancing fingers the widths of the two columns of fluid are equal.
Abstract: 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.
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TL;DR: Uounu et al. as mentioned in this paper derived the equations governing the movement of the melt and the matrix of a partially molten material from the conservation of mass, momentum, and energy using expressions from the theory of mixtures.
Abstract: The equations governing the movement of the melt and the matrix of a partially molten material are obtained from the conservation of mass, momentum, and energy using expressions from the theory of mixtures. The equations define a length scale dc called the compaction length, which depends only on the material properties of the melt and matrix. A number of simple solutions to the equations show that, if the porosity is initially constant, matrix compaction only occurs within a distance ~<5C of an impermeable boundary. Elsewhere the gravitational forces are supported by the viscous stresses resulting from the movement of melt, and no compaction occurs. The velocity necessary to prevent compaction is known as the minimum fluidization velocity. In all cases the compaction rate is controlled by the.properties of the matrix. These results can only be applied to geological problems if the values of the permeability, bulk and shear viscosity of the matrix can be estimated. All three depend on the microscopic geometry of the melt, which is in turn controlled by the dihedral angle. The likely equilibrium network provides some guidance in estimating the order of magnitude of these constants, but is no substitute for good measurements, which are yet to be carried out. Partial melting by release of pressure at constant entropy is then examined as a means of produced melt within the earth. The principal results of geological interest are that a mean mantle temperature of 1350 °C is capable of producing the oceanic crustal thickness by partial melting. Local hot jets with temperatures of 1550 °C can produce aseismic ridges with crustal thicknesses of about 20 km on ridge axes, and can generate enough melt to produce the Hawaiian Ridge. Higher mantle temperatures in the Archaean can produce komatiites if these are the result of modest amounts of melting at depths of greater than 100 km, and not shallow melting of most of the rock. The compaction rate of the partially molten rock is likely to be rapid, and melt-saturated porosities in excess of perhaps 3 per cent are unlikely to persist anywhere over geological times. The movement of melt through a matrix does not transport major and trace elements with the mean velocity of the melt, but with a slower velocity whose magnitude depends on the distribution coefficient. This effect is particularly important when the melt fraction is small, and may both explain some geochemical observations and provide a means of investigating the compaction process within the earth. I N T R O D U C T I O N There is an obvious need for a simple physical model which can describe the generation of a partially molten rock, and the separation of the melt from the residual solid, which will be referred to as the matrix. If such a model is to be useful it must lead to differential equations which can be solved by standard methods. The principal aim of this paper is to propose such a model, derive the governing equations, and obtain some solutions for particularly simple cases. The model is concerned with the physics, rather than the chemistry, of the process, though the formulation is sufficiently general to allow the inclusion of complicated phase equilibria. Several effects whose importance is unclear have not been included, in order to obtain the simplest model which can describe the generation and extraction of magma. Generation of a magma containing few solid crystals requires two operations. A partially mohen rock must first be generated, either by supplying heat or by reducing the pressure and so changing the solidus temperature. Once such a rock has been formed, the melt must UounuJ of Petrology, Vol. 25, Pirt 3, pp. 713-765, 19841 at W asngton U niersity at St L ouis on M arch 5, 2013 http://petroxfordjournals.org/ D ow nladed from
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Abstract: The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.
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Abstract: Shear-banding is a ubiquitous plastic-deformation mode in materials. In metallic glasses, shear bands are particularly important as they play the decisive role in controlling plasticity and failure at room temperature. While there have been several reviews on the general mechanical properties of metallic glasses, a pressing need remains for an overview focused exclusively on shear bands, which have received tremendous attention in the past several years. This article attempts to provide a comprehensive and up-to-date review on the rapid progress achieved very recently on this subject. We describe the shear bands from the inside out, and treat key materials-science issues of general interest, including the initiation of shear localization starting from shear transformations, the temperature and velocity reached in the propagating or sliding band, the structural evolution inside the shear-band material, and the parameters that strongly influence shear-banding. Several new discoveries and concepts, such as stick-slip cold shear-banding and strength/plasticity enhancement at sub-micrometer sample sizes, will also be highlighted. The understanding built-up from these accounts will be used to explain the successful control of shear bands achieved so far in the laboratory. The review also identifies a number of key remaining questions to be answered, and presents an outlook for the field.
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TL;DR: In this article, it was shown that when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa.
Abstract: It is shown that, when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa. The relationship between the rate of development of the instability and the length of wave-like disturbances, the acceleration and the densities is found, and similar calculations are made for the case when a sheet of liquid of uniform depth is accelerated.
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TL;DR: In this article, the authors describe measurements of the shape and rate of rise of air bubbles varying in volume from 1·5 to 200 cm. 3 when they rise through nitrobenzene or water.
Abstract: Part I describes measurements of the shape and rate of rise of air bubbles varying in volume from 1·5 to 200 cm. 3 when they rise through nitrobenzene or water. Measurements of photographs of bubbles formed in nitrobenzene show that the greater part of the upper surface is always spherical. A theoretical discussion, based on the assumption that the pressure over the front of the bubble is the same as that in ideal hydrodynamic flow round a sphere, shows that the velocity of rise, U , should be related to the radius of curvature, R , in the region of the vertex, by the equation U = 2/3√( gR ); the agreement between this relationship and the experimental results is excellent. For geometrically similar bubbles of such large diameter that the drag coefficient would be independent of Reynolds’s number, it would be expected that U would be proportional to the sixth root of the volume, V ; measurements of eighty-eight bubbles show considerable scatter in the values of U/V 1/6 , although there is no systematic variation in the value of this ratio with the volume. Part II. Though the characteristics of a large bubble are associated with the observed fact that the hydrodynamic pressure on the front of a spherical cap moving through a fluid is nearly the same as that on a complete sphere, the mechanics of a rising bubble cannot be completely understood till the observed pressure distribution on a spherical cap is understood. Failing this, the case of a large bubble running up a circular tube filled with water and emptying at the bottom is capable of being analyzed completely because the bubble is not then followed by a wake. An approxim ate calculation shows that the velocity U of rise is U = 0·46 √( ga ), where a is the radius of the tube. Experiments with a tube 7·9 cm. diameter gave values of U from 29·1 to 30·6 cm./sec., corresponding with values of U /√( ga ) from 0·466 to 0·490.
999 citations
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TL;DR: In this paper, an apparatus for accelerating small quantities of various liquids vertically downwards at accelerations of the order of 50g ( g being 32.2 ft/sec) is described, and the behavior of small wave-like corrugations initially imposed on the upper liquid surface has been observed by means of high-speed shadow photography.
Abstract: An apparatus for accelerating small quantities of various liquids vertically downwards at accelerations of the order of 50g ( g being 32.2 ft./sec. 2 ) is described, and the behaviour of small wave-like corrugations initially imposed on the upper liquid surface has been observed by means of high-speed shadow photography. The instability observed under a wide variety of experimental conditions has been analyzed, and the initial phases have been found to agree well with the first-order theory given in part I. When the disturbance has attained a considerable amplitude the first-order equations cease to apply and it changes from a wave into a form which has the appearance of large round-ended columns of air extending into the liquid and separated by narrow sheets of liquid. The air columns attain a steady velocity relative to the accelerating liquid and continue to penetrate into the liquid until the lower surface of the liquid is reached. In spite of these very large surface disturbances, the main body of liquid below them is accelerated as though they did not exist.
317 citations
TL;DR: In this paper, the authors considered the flow of an incompressible heavy liquid past a gas bubble in an infinitely long vertical tube, and the gas in the bubble was considered to be at rest, in a state of constant pressure.
Abstract: 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
135 citations