Viscous flows in two dimensions

Bensimon, David; Kadanoff, Leo P.; Liang, Shoudan; Shraiman, Boris I.; Tang, Chao

doi:10.1103/revmodphys.58.977

Cited by 732 publications

(512 citation statements)

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“…My recollection of work that took place nearly 30 years ago is incomplete, but i t is likely that the inclusion of surface tension into the ST instability calculation followed a private communication between Chuoke and Taylor, although I think it was already under consideration by us. Bensimon et al (1985) state that Taylor & Saffman saw instability of fingers at high speed but ignored it. Again, I have no recollection of this observation and cannot find any mention in our papers or correspondence, but the experiments were carried out entirely by Taylor, my contribution being the calculations, and it may be that he saw instabilities and communicated this to others.…”

Section: P G A'affmanmentioning

confidence: 99%

Viscous fingering in Hele-Shaw cells

1986

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The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed. History and statusFlow in a porous medium is a challenging scientific problem of great technological importance. Even though it usually concerns viscous fluids under circumstances where the nonlinear convective terms in the equations of motion are negligible, the mathematical difficulties caused by the randomness and chaotic structure of the medium make fundamental theory as difficult as for turbulence. Moreover, experiments are hard as it is not easy to see into a porous medium or know exactly the position of a measuring probe. The difficulties are compounded when the fluid is not homogeneous. For the case of two immiscible fluids, the added complication of moving microscopic and macroscopic interfaces makes the problem even more intractable.In about 1956, Sir Geoffrey Taylor paid a visit to the Humble Oil Company and became interested in problems of two-phase flow in porous media. He worked out the macroscopic instability which can arise when a less-viscous fluid drives a more-viscous one and which is a t least partly responsible for the coneing in processes of secondary recovery in oil fields. He also realized that two-dimensional flow in a porous medium is modelled by flow in a Hele-Shaw (1898) apparatus consisting of two flat parallel plates separated by a small gap b. Then the average two-dimensional velocity u of a viscous fluid in the space between the plates is related to the pressure p by the formula grad p , div u = 0 ( 1 . 1 ) b2where p is the viscosity of the fluid. This is identical with Darcy's law for motion in a porous medium of permeability b2/12. But it is, of course, an approximation valid when the gap or transverse dimension b is small compared with variations of scale a , say, in the lateral dimension parallel to the plates; see also Lamb (1932, $330).

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Section: P G A'affmanmentioning

confidence: 99%

Viscous fingering in Hele-Shaw cells

1986

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“…Such unstable interfaces are important in applications, significantly for sugar refining, oil recovery, hydrology and carbon sequestration [2][3][4][5][6] and much recent work has focused on how to control the instabilities [7][8][9][10] . Viscous fingering plays a central role in our understanding of pattern formation in part because it is amenable to both theory and experiment 1,6,[11][12][13][14] . Of particular importance is the limit where the characteristic finger width, set by the mostunstable wavelength, l c , approaches zero.…”

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confidence: 99%

Fingering versus stability in the limit of zero interfacial tension

2014

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The invasion of one fluid into another of higher viscosity in a quasi-two dimensional geometry typically produces complex fingering patterns. Because interfacial tension suppresses short-wavelength fluctuations, its elimination by using pairs of miscible fluids would suggest an instability producing highly ramified singular structures. Previous studies focused on wavelength selection at the instability onset and overlooked the striking features appearing more globally. Here we investigate the non-linear growth that occurs after the instability has been fully established. We find a rich variety of patterns that are characterized by the viscosity ratio between the inner and the outer fluid, Z in /Z out , as distinct from the mostunstable wavelength, which determines the onset of the instability. As Z in /Z out increases, a regime dominated by long highly-branched fractal fingers gives way to one dominated by blunt stable structures characteristic of proportionate growth. Simultaneously, a central region of complete outer-fluid displacement grows until it encompasses the entire pattern at Z in /Z out E0.3.

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“…Introduction. A broad class of non-equilibrium growth processes in two dimensions are characterized by a common law: the velocity of the growing interface is determined by the gradient of a harmonic field (often referred to as Laplacian growth) [1].…”

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“…Traditionally air is pumped into one bubble, while oil is extracted from the cell at a constant rate Q > 0 through the edges placed at infinity. Without surface tension, the interface may develop singular cusps at a finite time [1]. At this moment the problem becomes * Also at ITEP, B. Cheremushkinskaya 25, 117259 Moscow, Russia.…”

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Singularities of the Hele-Shaw Flow and Shock Waves in Dispersive Media

et al. 2005

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We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental set-up where the cell is permeable to a non-viscous fluid and study continuation of the flow through singularities. We show that a singular flow in this, non-traditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de-Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.

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Viscous flows in two dimensions

Cited by 732 publications

References 50 publications

Viscous fingering in Hele-Shaw cells

Viscous fingering in Hele-Shaw cells

Fingering versus stability in the limit of zero interfacial tension

Singularities of the Hele-Shaw Flow and Shock Waves in Dispersive Media

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