Fractal dimension and unscreened angles measured for radial viscous fingering

Praud, Olivier; Swinney, Harry L.

doi:10.1103/physreve.72.011406

Cited by 98 publications

(86 citation statements)

References 40 publications

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“…It follows from (55) that this root is selected by the requirement that it has the same poles and residues as the potential A. The function f (z,z), where z andz are treated as two independent complex arguments, defines a Riemann surface with antiholomorphic involution (70). If the involution divides the surface into two disconnected parts, as explained above, the Riemann surface is the Schottky double [48] of one of these parts.…”

Section: Spectral Curvementioning

confidence: 99%

Random matrices in 2D, Laplacian growth and operator theory

Mineev-Weinstein

Putinar

Teodorescu³

2008

J. Phys. A: Math. Theor.

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Abstract. Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own whithin applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.

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Section: Spectral Curvementioning

confidence: 99%

Random matrices in 2D, Laplacian growth and operator theory

Mineev-Weinstein

Putinar

Teodorescu³

2008

J. Phys. A: Math. Theor.

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“…Analytic derivation of this number remains a long-standing challenge in non-equilibrium physics despite numerous efforts [25]. Surprisingly, the same fractal dimension was observed in several Laplacian growth experiments [26,27], where the process is continuous and deterministic.We have unifed LG and DLA as two opposite limits (classical and quantum respectively) of a stochastic Laplacian growth, where instead of one particle the source emits K ≥ 1 uncorrelated particles per time unit. The DLA, when K = 1, can be called a quantum limit of this process, as correlations between particles in this case are maximal.…”

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confidence: 71%

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confidence: 71%

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Stochastic Laplacian growth

Alekseev

Mineev-Weinstein

2016

Phys. Rev. E

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A point source on a plane constantly emits particles which rapidly diffuse and then stick to a growing cluster. The growth probability of a cluster is presented as a sum over all possible scenarios leading to the same final shape. The classical point for the action, defined as a minus logarithm of the growth probability, describes the most probable scenario and reproduces the Laplacian growth equation, which embraces numerous fundamental free boundary dynamics in non-equilibrium physics. For non-classical scenarios we introduce virtual point sources, in which presence the action becomes the Kullback-Leibler entropy. Strikingly, this entropy is shown to be the sum of electrostatic energies of layers grown per elementary time unit. Hence the growth probability of the presented non-equilibrium process obeys the Gibbs-Boltzmann statistics, which, as a rule, is not applied out from equilibrium. Each layer's probability is expressed as a product of simple factors in an auxiliary complex plane after a properly chosen conformal map. The action at this plane is a sum of Robin functions, which solve the Liouville equation. At the end we establish connections of our theory with the tau-function of the integrable Toda hierarchy and with the Liouville theory for non-critical quantum strings. The goal of this work is to unify two fundamental highly non-equilibrium processes, Laplacian growth (LG) [1][2][3][4][5], which is deterministic interface dynamics, and diffusion-limited aggregation (DLA) [6] -a discrete universal stochastic fractal growth. These remarkable processes have a lot in common and were suspected to be deeply related [7][8][9][10][11].Laplacian growth raised enormous interest in physics because of (i) its impressively wide applicability ranging from solidification and oil recovery to biological growth [1], (ii) remarkable universal asymptotic shapes, it exhibits [1,[12][13][14][15][16], and (iii) discoveries of deep intriguing connections of LG to quantum gravity [2] and the quantum Hall effect [17]. In mathematics the Laplacian growth appears so exciting because it possesses beautiful and powerful properties, unusual for most of nonlinear PDEs, such as infinitely many conservation laws [18] and closed form exact solutions [16,[19][20][21][22]. A new splash of intense activity in LG (see [3] for a review) was provoked by the work [2], where strong connections of LG with major integrable hierarchies and the theory of random matrices were established.Mathematical formulation of LG is (deceptively) simple: a droplet of air, D + (t), where t is time, is surrounded by a viscous fluid, D − (t) = C/D + (t), called D(t) for simplicity. Both liquids are sandwiched between two parallel close plates. Fluid velocity in D(t) obeys the Darcy law, v = −∇p (in scaled units), where p(z,z) is pressure and z = x + iy is a complex coordinate on the plane. Because of incompressibility, ∇·v = 0, then ∇ 2 p = 0 in D, except points with sources, which provide growth. Also, p = 0 at the interface, Γ(t) = ∂D(t), between two fluids, if...

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“…Our classification of the displacement pattern is based on visual appearance, as well as the fractal dimension D f (using box counting [Niemeyer et al, 1984]). Visual appearance is an essential consideration in the classification because the estimation of the fractal dimension from the mass vs. distance curves is subject to large fluctuations for finite-size systems [Måløy et al, 1985;Blunt and King, 1990;Løvoll et al, 2004;Praud and Swinney, 2005].…”

Section: Appendixmentioning

confidence: 99%

Mechanisms Leading to Co-Existence of Gas Hydrate in Ocean Sediments [Part 2 of 2]

Bryant¹,

Juanes²

2011

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The experimental boundary conditions do not replicate the situation in gas reservoirs converted to hydrate in the Arctic. The key differences are that in the experiment water can move 117 to the hydrate stability zone only from below, while in nature water can move from below and from above. In experiment gas can move to the hydrate stability zone from below and from above, while in nature it can move only from below. Finally, the fluid phase pressures decrease with time in the experiment (though the conditions remain within the hydrate stability field throughout), while the water phase pressure does not change with time in nature. These differences are minor and do not alter the essential similarities to the natural situation. In the experiment and in nature, the base of the gas hydrate stability zone descends gradually through an existing gas accumulation, and fluid phases are free to move in response to any gradients that arise as hydrate forms. The latter point is important: no fluid flow (neither pressure-driven nor capillarity-driven) is imposed in the experiment. Instead the system evolves a combination of buoyancy-, pressure-and capillarity-driven fluxes of the gas and water phases, and these fluxes balance the various resistances to transport.The model predicts that the contribution of capillarity-driven flux is significant in these experiments. In fact R v~0 .69 when capillary-dominated flow is taken into account while R v~1 when only the viscous dominated portion of the flow was considered. This is consistent with our findings in previous sections of this report, namely, pressure-driven flow cannot supply water from below the BGHSZ fast enough to sustain hydrate formation. Moreover the model predicts that the amount of gas flows to the GHSZ from above quickly diminishes as both the effective permeability, due to hydrate formation within the GHSZ, and gaseous phase relative permeability, due to decrease in gas saturation within the GHSZ, decrease. Therefore, the main contribution for gas flow would be flow from the middle gas inlet. While the gas flux from above and from below could not be independently measured in this experiment, the consistency of the model with other observations suggests that all the fluids needed during conversion to hydrate can be supplied from below. But this is possible only if capillarity-driven flux is accounted for.The consistency of the capillarity-driven flux model with the experimental observations suggests that the finding in the previous section, viz. that the observed hydrate saturation profile in Mt. Elbert arises naturally from the capillarity-driven fluxes, is not a coincidence. That is, these findings suggest that the gas reservoir conversion process is slow, that it introduces modest, even negligible pressure gradients but impose significant gradients in saturation which, along with gas buoyancy, feed the appropriate volumes of fluids to the GHSZ.A complete validation of the model would compare the predicted hydrate saturation profile to the observed profile in F...

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Fractal dimension and unscreened angles measured for radial viscous fingering

Cited by 98 publications

References 40 publications

Random matrices in 2D, Laplacian growth and operator theory

Random matrices in 2D, Laplacian growth and operator theory

Stochastic Laplacian growth

Mechanisms Leading to Co-Existence of Gas Hydrate in Ocean Sediments [Part 2 of 2]

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