Dynamical scaling of imbibition in columnar geometries

M. Pradas, A. Hernández-Machado, M.A. Rodríguez

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Recent experiments of imbibition in columnar geometries show interfacial fluctuations whose dynamic scaling is not compatible with the usual non local model governed by surface tension that results from a macroscopic description. To explore this discrepancy, we exhaustively analyze numerical integrations of a phase-field model with dichotomic columnar disorder. We find that two distinct behaviors are possible depending on the capillary contrast between both values of disorder. In a high contrast case, where interface evolution is mainly dominated by the disorder, an inherent anomalous scaling is always observed. Moreover, in agreement with experimental work, the interface motion has to be described through a local model. On the other hand, in a lower contrast case, interface is dominated by interfacial tension and can be well modeled by a non local model. We have studied both spontaneous and forced-flow imbibition situations, giving a complete set of scaling exponent in each case, as well as, a comparison to the experimental results.

Pressure-dependent scaling scenarios in experiments of spontaneous imbibition

R. Planet, M. Pradas, A. Hernández-Machado, J. Ortín

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The scaling properties of the rough liquid-air interface formed in the spontaneous imbibition of a viscous liquid by a model porous medium are found to be very sensitive to the magnitude of the pressure difference applied at the liquid inlet. Interface fluctuations change from obeying intrinsic anomalous scaling at large negative pressure differences, to being super-rough with the same dynamic exponent z3 at less negative pressure differences, to finally obeying ordinary Family-Vicsek scaling with z2 at large positive pressure differences. This rich scenario reflects the relative importance on different length scales of capillary and permeability disorder, and the role of surface tension and viscous pressure in damping interface fluctuations.

Dynamic instabilities in biological membranes

F Campelo, A. and Hernández-Machado

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Three-dimensional aspects of fluid flows in channels. II. Effects of meniscus and thin film regimes  on viscous fingers

R. Ledesma-Aguilar, I. Pagonabarraga, A. Hernández-Machado

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We perform a three-dimensional study of steady state viscous fingers that develop in linear channels. By means of a three-dimensional lattice-Boltzmann scheme that mimics the full macroscopic equations of motion of the fluid momentum and order parameter, we study the effect of the thickness of the channel in two cases. First, for total displacement of the fluids in the channel thickness direction, we find that the steady state finger is effectively two-dimensional and that previous two-dimensional results can be recovered by taking into account the effect of a curved meniscus across the channel thickness as a contribution to surface stresses. Second, when a thin film develops in the channel thickness direction, the finger narrows with increasing channel aspect ratio in agreement with experimental results. The effect of the thin film renders the problem three-dimensional and results deviate from the two-dimensional prediction.

Three-dimensional aspects of fluid flows in channels. I. Meniscus and thin film regimes

R. Ledesma-Aguilar, A Hernández-Machado, I. Pagonabarraga

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We study the forced displacement of a fluid-fluid interface in a three-dimensional channel formed by two parallel solid plates. Using a lattice-Boltzmann method, we study situations in which a slip velocity arises from diffusion effects near the contact line. The difference between the slip and channel velocities determines whether the interface advances as a meniscus or a thin film of fluid is left adhered to the plates. We find that this effect is controlled by the capillary and Péclet numbers. We estimate the crossover from a meniscus to a thin film and find good agreement with numerical results. The penetration regime is examined in the steady state. We find that the occupation fraction of the advancing finger relative to the channel thickness is controlled by the capillary number and the viscosity contrast between the fluids. For high viscosity contrast, lattice-Boltzmann results agree with previous results. For zero viscosity contrast, we observe remarkably narrow fingers. The shape of the finger is found to be universal.

Model for curvature-driven pearling instability in membranes

F. Campelo, A. Hernéndez-Machado

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A phase-field model for dealing with dynamic instabilities in membranes is presented. We use it to study curvature-driven pearling instability in vesicles induced by the anchorage of amphiphilic polymers on the membrane. Within this model, we obtain the morphological changes reported in recent experiments. The formation of a homogeneous pearled structure is achieved by consequent pearling of an initial cylindrical tube from the tip. For high enough concentration of anchors, we show theoretically that the homogeneous pearled shape is energetically less favorable than an inhomogeneous one, with a large sphere connected to an array of smaller spheres.

Time-dependent couplings and crossover length scales in nonequilibrium surface roughening

M Pradas, López J. M., A. Hernández-Machado

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We show that time-dependent couplings may lead to nontrivial scaling properties of the surface fluctuations of the asymptotic regime in nonequilibrium kinetic roughening models. Three typical situations are studied. In the case of a crossover between two different rough regimes, the time-dependent coupling may result in anomalous scaling for scales above the crossover length. In a different setting, for a crossover from a rough to either a flat or damping regime, the time-dependent crossover length may conspire to produce a rough surface, although the most relevant term tends to flatten the surface. In addition, our analysis sheds light into an existing debate in the problem of spontaneous imbibition, where time-dependent couplings naturally arise in theoretical models and experiments.

Shape instabilities in vesicles: A phase-field model

F. Campelo, A. Hernández-Machado

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A phase field model for dealing with shape instabilities in fluid membrane vesicles is presented. This model takes into account the Canham-Helfrich bending energy with spontaneous curvature. A dynamic equation for the phase-field is also derived. With this model it is possible to see the vesicle shape deformation dynamically, when some external agent instabilizes the membrane, for instance, inducing an inhomogeneous spontaneous curvature. The numerical scheme used is detailed and some stationary shapes are shown together with a shape diagram for vesicles of spherical topology and no spontaneous curvature, in agreement with known results.

Fluctuations in saffman-taylor fingers with quenched disorder

M. Torralba, J. Ortín, A. Hernández-Machado, E. Corvera Poiré

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We make an experimental characterization of the effect that static disorder has on the shape of a normal Saffman-Taylor finger. We find that static noise induces a small amplitude and long wavelength instability on the sides of the finger. Fluctuations on the finger sides have a dominant wavelength, indicating that the system acts as a selective amplifier of static noise. The dominant wavelength does not seem to be very sensitive to the intensity of static noise present in the system. On the other hand, at a given flow rate, rms fluctuations of the finger width, decrease with decreasing intensity of static noise. This might explain why the sides of the fingers are flat for typical Saffman-Taylor experiments. Comparison with previous numerical studies of the effect that temporal noise has on the Saffman-Taylor finger, leads to conclude that the effect of temporal noise and static noise are similar. The behavior of fluctuations of the finger width found in our experiments, is qualitatively similar to one recently reported, in the sense that, the magnitude of the width fluctuations decays as a power law of the capillary number, at low flow rates, and increases with capillary number for larger flow rates.

Intrinsic versus superrough anomalous scaling in spontaneous imbition

M. Pradas, A. Hernández-Machado

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We study spontaneous imbibition using a phase field model in a two-dimensional system with a dichotomic quenched noise. By imposing a constant pressure a0 at the origin, we study the case when the interface advances at low velocities, obtaining the scaling exponents z=3.0±0.1,  =1.50±0.02, and loc=0.95±0.03 within the intrinsic anomalous scaling scenario. These results are in quite good agreement with experimental data recently published. Likewise, when we increase the interface velocity, the resulting scaling exponents are z=4.0±0.1,  =1.25±0.02, and loc=0.95±0.03. Moreover, we observe that the local properties of the interface change from a superrough to an intrinsic anomalous description when the contrast between the two values of the dichotomic noise is increased. From a linearized interface equation we can compute analytically the global scaling exponents which are comparable to the numerical results, introducing some properties of the quenched noise.

Phase-field approach to spatial perturbations in normal Saffman-Taylor fingers

M. Quevedo-Reyes, A. Hernández-Machado, E. Corvera Poiré

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We make a numerical study of the effect that spatial perturbations have in normal Saffman-Taylor fingers driven at constant pressure gradients. We use a phase field model that allows for spatial variations in the Hele-Shaw cell. We find that, regardless of the specific way in which spatial perturbations are introduced, a lateral instability develops on the sides of the propagating Saffman-Taylor finger. Moreover, the instability exists regardless of the intensity of spatial perturbations in the cell as long as the perturbations are felt by the finger tip. If, as the finger propagates, the spatial perturbations felt by the tip change, the instability is nonperiodic. If, as the finger propagates, the spatial perturbations felt by the tip are persistent, the instability developed is periodic. In the later case, the instability is symmetrical or asymmetrical depending on the intensity of the perturbation.

Anomalous roughening of viscous fluid fronts in spontaneous imbition

J. Soriano, A. Mercier, R. Planet, A. Hernández-Machado, M.A. Rodríguez, J. Ortín

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We report experiments on spontaneous imbibition of a viscous fluid by a model porous medium in the absence of gravity. The average position of the interface satisfies Washburn’s law. Scaling of the interface fluctuations suggests a dynamic exponent z ’ 3, indicative of global dynamics driven by capillary forces. The complete set of exponents clearly shows that interfaces are not self-affine, exhibiting distinct local and global scaling, both for time (  0:64  0:02,   0:33  0:03) and space (  1:94  0:20, loc  0:94  0:10). These values are compatible with an intrinsic anomalous scaling scenario.

Side-branch growth in two-dimensional dendrits. Part II: Phase-field model

Y. Couder, R. González-Cinca, A. Hernández-Machado

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The development of side-branching in solidifying dendrites in a regime of large values of the Peclet number is studied by means of a phase-field model. We have compared our numerical results with experiments of the preceding paper and we obtain good qualitative agreement. The growth rate of each side branch shows a power-law behavior from the early stages of its life. From their birth, branches which finally succeed in the competition process of side-branching development have a greater growth exponent than branches which are stopped. Coarsening of branches is entirely defined by their geometrical position relative to their dominant neighbors. The winner branches escape from the diffusive field of the main dendrite and become independent dendrites.

Side-branch growth in two-dimensional dendrits. Part I: Experiments

Y. Couder, J. Maurer, R. González-Cinca, A. Hernández-Machado

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The dynamics of growth of dendrites’ side branches is investigated experimentally during the crystallization of solutions of ammonium bromide in a quasi-two-dimensional cell. Two regimes are observed. At small values of the Peclet number a self-affine fractal forms. In this regime it is known that the mean lateral front grows as t0.5. Here the length of each individual branch is shown to grow sbefore being screened offd with a power-law behavior tan. The value of the exponent ans0.5øanø1d is determined from the start by the strength of the initial disturbance. Coarsening then takes place, when the branches of small an are screened off by their neighbors. The corresponding decay of the growth of a weak branch is exponential and defined by its geometrical position relative to its dominant neighbors. These results show that the branch structure results from a deterministic growth of initially random disturbances. At large values of the Peclet number, the faster of the side branches escape and become independent dendrites. The global structure then covers a finite fraction of the two-dimensional space. The crossover between the two regimes and the spacing of these independent branches are characterized.

Lateral instability in normal viscous fingers

R. Ledesma-Aguilar, M. Quevedo Reyes, E. Corvera Poiré, A. Hernández-Machado

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We study a low-amplitude, long-wavelength lateral instability of the Saffman-Taylor finger by means of a phase-field model. We observe such an instability in two situations in which small dynamic perturbations are overimposed to a constant pressure drop. We first study the case in which the perturbation consists of a single oscillatory mode and then a case in which the perturbation consists of temporal noise. In both cases the instability undergoes a process of selection.

Phase-field models in interfacial pattern formation out of equilibrium

R. González-Cinca, R. Folch, R. Benítez, L. Ramírez-Piscina, J. Casademunt, A. Hernández-Machado

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Anomalous roughening in experiments of interfaces in Hele-Shaw flows with strong quenched disorder

J. Soriano, J. Ortín, A. Hernández-Machado

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Phase-field model of Hele-Shaw flows in the high-viscosity contrast regime

A. Hernández-Machado, A.M. Lacasta, E. Mayoral, Corvera Poiré.

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Experiments of interfacial roughening of Hele-Shaw flows with weak quenched disorder

J. Soriano, J. Ortín, A. Hernández-Machado

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Anomalous roughening of Hele-Shaw flows with quenched disorder

J. Soriano, J.J. Ramasco, M.A. Rodríguez, A. Hernández-Machado, J. Ortín

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