Systematic weakly nonlinear analysis of interfacial instabilities in Hele-Shaw flows

We develop a systematic method to derive all orders of mode couplings in a weakly nonlinear approach to the dynamics of the interface between two immiscible viscous fluids in a Hele-Shaw cell. The method is completely general: it applies to arbitrary geometry and driving. Here we apply it to the channel geometry driven by gravity and pressure. The finite radius of convergence of the mode-coupling expansion is found. Calculation up to third-order couplings is done, which is necessary to account for the time-dependent Saffman-Taylor finger solution and the case of zero viscosity contrast. The explicit results provide relevant analytical information about the role that the viscosity contrast and the surface tension play in the dynamics of the system. We finally check the quantitative validity of different orders of approximation and a resummation scheme against a physically relevant, exact time-dependent solution. The agreement between the low-order approximations and the exact solution is excellent within the radius of convergence, and is even reasonably good beyond this radius.

Systematic weakly nonlinear analysis of radial viscous fingering

We present a weakly nonlinear analysis of the interface dynamics in a radial Hele-Shaw cell driven by both injection and rotation. We extend the systematic expansion introduced in [E. Alvarez-Lacalle et al., Phys. Rev. E 64, 016302 (2001)] to the radial geometry, and compute explicitly the first nonlinear contributions. We also find the necessary and sufficient condition for the uniform convergence of the nonlinear expansion. Within this region of convergence, the analytical predictions at low orders are compared satisfactorily to exact solutions and numerical integration of the problem. This is particularly remarkable in configurations (with no counterpart in the channel geometry) for which the interplay between injection and rotation allows that condition to be satisfied at all times. In the case of the purely centrifugal forcing we demonstrate that nonlinear couplings make the interface more unstable for lower viscosity contrast between the fluids.

Diffusion in spatially and temporarily inhomogeneous media

In this paper we consider diffusion of a passive substance C in a temporarily and spatially inhomogeneous two-dimensional medium. As a realization for the latter we choose a phase-separating medium consisting of two substances A and B, whose dynamics is determined by the Cahn-Hilliard equation. Assuming different diffusion coefficients of C in A and B, we find that the variance of the distribution function of the said substance grows less than linearly in time. We derive a simple identity for the variance using a probabilistic ansatz and are then able to identify the interface between A and B as the main cause for this nonlinear dependence. We argue that, finally, for very large times the here temporarily dependent diffusion "constant" goes like t-1/3 to a constant asymptotic value D¿. The latter is calculated approximately by employing the effective-medium approximation and by fitting the simulation data to the said time dependence.

Measurements of the bulk and interfacial velocity profiles in oscillating Newtonian and Maxwellian fluids

We present the dynamic velocity profiles of a Newtonian fluid (glycerol) and a viscoelastic Maxwell fluid (CPyCl-NaSal in water) driven by an oscillating pressure gradient in a vertical cylindrical pipe. The frequency range explored has been chosen to include the first three resonance peaks of the dynamic permeability of the viscoelastic-fluid¿pipe system. Three different optical measurement techniques have been employed. Laser Doppler anemometry has been used to measure the magnitude of the velocity at the center of the liquid column. Particle image velocimetry and optical deflectometry are used to determine the velocity profiles at the bulk of the liquid column and at the liquid-air interface respectively. The velocity measurements in the bulk are in good agreement with the theoretical predictions of a linear theory. The results, however, show dramatic differences in the dynamic behavior of Newtonian and viscoelastic fluids, and demonstrate the importance of resonance phenomena in viscoelastic fluid flows, biofluids in particular, in confined geometries.

Relevance of dynamic wetting in viscous fingering patterns

We demonstrate that wetting effects at moving contact lines have a strong impact in viscous fingering patterns. Experiments in a rotating Hele-Shaw (HS) cell, dry or prewetted, show consistent morphological differences. When the wetting fluid invades a dry region, contact angle dynamics yield a kinetic contribution to the interface pressure drop that scales with capillary number as Ca2¿3 but is significantly larger than the Park-Homsy kinetic correction. Numerical results are in very good agreement with experiments and show that standard HS equations work best for prewetted cells.

Growth of unstable interfaces in disordered media

The effects of a disordered medium in the growth of unstable interfaces are studied by means of two local models with multiplicative and additive quenched disorder, respectively. For short times and large pushing the multiplicative quenched disorder is equivalent to a time-dependent noise. In this regime, the linear dispersion relation contains a destabilizing contribution introduced by the noise. For long times, the interface always gets pinned. We model the systematics of the pinned shapes by means of an effective nonlinear model. These results show good agreement with numerical simulations. For the additive noise we find numerically that a depinning transition occurs.

Dynamical scaling of imbibition in columnar geometries

compatible with the usual nonlocal 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 the two 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, the interface is dominated by interfacial tension and can be well modeled by a nonlocal model. We have studied both spontaneous and forced-flow imbibition situations, giving a complete set of scaling exponents in each case, as well as a comparison to the experimental results.

Féminisation du corps médical et dynamiques professionnelles dans le champ de la santé

Si la féminisation du corps médical est désormais un phénomène massif dans tous les pays occidentaux, elle suscite des analyses quelque peu contradictoires. La plupart des études oscillent entre une posture « universaliste », supposant que les femmes médecins adopteront, à terme, les mêmes pratiques professionnelles que les générations précédentes, composées essentiellement d'hommes et une perspective d'inspiration « essentialiste », mettant largement l'accent sur les « spécificités » de l'exercice médical au féminin. Ici, nous montrons que les femmes médecins tendent effectivement à adopter des pratiques spécifiques, notamment en matière de gestion de l'interface vie familiale et vie professionnelle. Toutefois, ces soi-disant « spécificités féminines » sont de plus en plus souvent adoptées par des médecins hommes aussi. En effet, à partir du moment où ces hommes se trouvent en couple avec des femmes diplômées qui tiennent à rentabiliser leurs études sur le marché de l'emploi, il leur est tout aussi difficile de se conformer aux principes de « l'ethos » de la disponibilité permanente à l'égard des patients qui a longtemps prévalu dans cette profession en France. Si les hommes continuent de dominer les échelons supérieurs de la profession, ils n'échappent pas pour autant aux exigences de modification de leurs rapports à l'exercice médical et à la sphère domestique et familiale. L'analyse des dynamiques professionnelles exige donc une attention particulière à la dynamique du genre comme principe d'organisation de l'exercice médical

Pattern formation during mesophase growth in a homologous series

Remarkable differences in the shape of the nematic-smectic-B interface in a quasi-two-dimensional geometry have been experimentally observed in three liquid crystals of very similar molecular structure, i.e., neighboring members of a homologous series. In the thermal equilibrium of the two mesophases a faceted rectanglelike shape was observed with considerably different shape anisotropies for the three homologs. Various morphologies such as dendritic, dendriticlike, and faceted shapes of the rapidly growing smectic-B germ were also observed for the three substances. Experimental results were compared with computer simulations based on the phase field model. The pattern forming behavior of a binary mixture of two homologs was also studied.

Pressure-dependent scaling scenarios in experiments of spontaneous imbibition

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 z¿3 at less negative pressure differences, to finally obeying ordinary Family-Vicsek scaling with z¿2 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.

Influence of disorder strength on phase-field models of interfacial growth

We study the influence of disorder strength on the interface roughening process in a phase-field model with locally conserved dynamics. We consider two cases where the mobility coefficient multiplying the locally conserved current is either constant throughout the system (the two-sided model) or becomes zero in the phase into which the interface advances (one-sided model). In the limit of weak disorder, both models are completely equivalent and can reproduce the physical process of a fluid diffusively invading a porous media, where super-rough scaling of the interface fluctuations occurs. On the other hand, increasing disorder causes the scaling properties to change to intrinsic anomalous scaling. In the limit of strong disorder this behavior prevails for the one-sided model, whereas for the two-sided case, nucleation of domains in front of the invading front are observed.

Experiments in a rotating Hele-Shaw cell

We introduce a modification to Hele-Shaw flows consisting of a rotating cell. A viscous fluid (oil) is injected at the rotation axis of the cell, which is open to air. The morphological instability of the oil-air interface is thus driven by centrifugal force and is controlled by the density (not viscosity) difference. We derive the linear dispersion relation and verify the maximum growth rate selection of initial patterns within experimental uncertainty. The nonlinear growth regime is studied in the case of vanishing injection rate. Several characteristic lengths are studied to quantify the patterns obtained. Experimental data exhibit good collapse for two characteristic lengths, namely, the radius of gyration and the radial finger length, which in the nonlinear regime appear to grow linearly in time.

Phase-field model for Hele-Shaw flows with arabitrary contrast II: numerical approach

We present a phase-field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. With asymptotic matching techniques we check the model to yield the right Hele-Shaw equations in the sharp-interface limit, and compute the corrections to these equations to first order in the interface thickness. We also compute the effect of such corrections on the linear dispersion relation of the planar interface. We discuss in detail the conditions on the interface thickness to control the accuracy and convergence of the phase-field model to the limiting Hele-Shaw dynamics. In particular, the convergence appears to be slower for high viscosity contrasts.

Reaction-diffusion fronts under stochastic advection

We study front propagation in stirred media using a simplified modelization of the turbulent flow. Computer simulations reveal the existence of the two limiting propagation modes observed in recent experiments with liquid phase isothermal reactions. These two modes respectively correspond to a wrinkled although sharp propagating interface and to a broadened one. Specific laws relative to the enhancement of the front velocity in each regime are confirmed by our simulations.

Dynamical Systems approach to Saffman-Taylor fingering. A Dynamical Solvability Scenario

A dynamical systems approach to competition of Saffman-Taylor fingers in a Hele-Shaw channel is developed. This is based on global analysis of the phase space flow of the low-dimensional ordinary-differential-equation sets associated with the classes of exact solutions of the problem without surface tension. Some simple examples are studied in detail. A general proof of the existence of finite-time singularities for broad classes of solutions is given. Solutions leading to finite-time interface pinchoff are also identified. The existence of a continuum of multifinger fixed points and its dynamical implications are discussed. We conclude that exact zero-surface tension solutions taken in a global sense as families of trajectories in phase space are unphysical because the multifinger fixed points are nonhyperbolic, and an unfolding does not exist within the same class of solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed points is argued to be essential to the physically correct qualitative description of finger competition. The restoring of hyperbolicity by surface tension is proposed as the key point to formulate a generic dynamical solvability scenario for interfacial pattern selection.