Density-driven instabilities between miscible fluids in a vertical Hele-Shaw cell are investigated by means of experimental measurements, as well as two- and three-dimensional numerical simulations. The experiments focus on the early stages of the instability growth, and they provide detailed information regarding the growth rates and most amplified wavenumbers as a function of the governing Rayleigh number Ra. They identify two clearly distinct parameter regimes: a low-Ra, ‘Hele-Shaw’ regime in which the dominant wavelength scales as Ra−1, and a high-Ra ‘gap’ regime in which the length scale of the instability is 5±1 times the gap width. The experiments are compared to a recent linear stability analysis based on the Brinkman equation. The analytical dispersion relationship for a step-like density profile reproduces the experimentally observed trend across the entire Ra range. Nonlinear simulations based on the two- and three-dimensional Stokes equations indicate that the high-Ra regime is characterized by an instability across the gap, wheras in the low-Ra regime a spanwise Hele-Shaw mode dominates.

We analyze the problem of the geometrical aperture between the two faces of a crack described as a self-affine surface. We consider the contact of the two surfaces after a relative rigid-body displacement (translation and/or rotation) of one side of the crack with respect to the opposite one. A number of properties are obtained analytically and illustrated by numerical simulations on generated self-affine surfaces. The results concern the scaling of the average aperture, which is shown to include a very slowly varying correction term (logarithmic with respect to the displacement); this affects severely the dependence of the measured aperture on the displacement. These results are used to analyze experimental data obtained on a granite sample and to estimate the roughness exponent from a global measurement.This estimate is shown to agree with an analysis of the roughness based on profilometry measurements. PACS number(s}: 02.50.r, 47.55.Mh, 47.53.+n, 62.20.Mk

We investigate the use of a digital holographic microscope working in partially coherent illumination to study in three dimensions a micrometer-size particle flow. The phenomenon under investigation rapidly varies in such a way that it is necessary to record, for every camera frame, the complete holographic information for further processing. For this purpose, we implement the Fourier-transform method for optical amplitude extraction. The suspension of particles is flowing in a split-flow lateral-transport thin separation cell that is usually used to separate the species by their sizes. Details of the optical implementation are provided. Examples of reconstructed images of different particle sizes are shown, and a particle-velocity measurement technique that is based on the blurred holographic image is exploited.

Fingering instabilities involving fluids confined between two plates sometimes give rise to a typical wavelength λ proportional to the gap h. This unexplained behavior is investigated for the case of the Rayleigh–Taylor instability between two liquids of the same viscosity. Using qualitative scaling arguments and linear stability analysis for a simplified model of hydrodynamics, we show that, in the miscible case, h becomes a natural cut-off when diffusion is negligible, i.e., when the Péclet number Pe=h3Δρg/(ηD) is large (η viscosity, g gravitational acceleration, D diffusivity, Δρ density difference). The same result holds in the immiscible case for large capillary number Ca=h2Δρg/(12γ) (γ surface tension). In this saturation regime, the dominant wavelength is given by λ≈2.3h, while in the opposite limit (low Pe or low Ca) λ scales, respectively, as h/Pe or h/Ca1/2. These results are in agreement with a recent experimental study.