Philippe Petitjeans

Miscible displacements in capillary tubes. Part 1. Experiments

Experiments have been performed, in capillary tubes, on the displacement of a viscous fluid (glycerine) by a less viscous one (a glycerine-water mixture) with which it is miscible in all proportions. A diagnostic measure of the amount of viscous fluid left behind on the tube wall has been found, for both vertical and horizontal tubes, as a function of the Peclet (Pe) and Atwood (At) numbers, as well as a parameter that is a measure of the relative importance of viscous and gravitational effects. At values of the average Pe greater than 1000 a sharp interface existed so that it was possible to make direct comparisons between the present results and a prior experiment with immiscible fluids, in particular an effective surface tension at the diffusing interface could be evaluated. The effect of gravity on the amount of viscous fluid left on the tube wall has been investigated also.

Density-driven unstable flows of miscible fluids in a Hele-Shaw cell

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.

Comparison between an experimental turbulent vortex and the Lundgren vortex

In a recent letter (Cuypers Y et al 2003 Phys. Rev. Lett. 91 194502), the authors presented experimental results on a structure resulting from a vortex burst. The temporal evolution of this structure results in the k  − 5/3 Kolmogorov spectrum and some common features with the Lundgren theoretical vortex have been shown. The purpose of the present paper is to go further in the comparison with the Lundgren model by a parallel analysis of the experimental structure and of a Lundgren single spiral vortex, whose evolution is numerically obtained based on the calculations of Pullin et al (1993 Phys. Fluids A 5 126; 1994 Phys. Fluids 6 3010).

Experimental and theoretical inspection of the phase-to-height relation in Fourier transform profilometry

The measurement of an objects shape using projected fringe patterns needs a relation between the measured phase and the objects height. Among various methods, the Fourier transform profilometry proposed by Takeda and Mutoh [Appl. Opt.22, 3977-3982 (1983)] is widely used in the literature. Rajoub et al. have shown that the reference relation given by Takeda is erroneous [J. Opt. A. Pure Appl. Opt.9, 66-75 (2007)]. This paper follows from Rajoubs study. Our results for the phase agree with Rajoubs results for both parallel- and crossed-optical-axes geometries and for either collimated or noncollimated projection. Our two main results are: (i) we show experimental evidence of the error in Takedas formula and (ii) we explain the error in Takedas derivation and we show that Rajoubs argument concerning Takedas error is not correct.

Miscible quarter five-spot displacements in a Hele-Shaw cell and the role of flow-induced dispersion

Miscible quarter five-spot displacements in a Hele-Shaw cell are investigated by means of experimental measurements and numerical simulations. The experiments record both the volumetric as well as the surface efficiency at breakthrough as a function of the dimensionless flow rate in the form of a Peclet number and the viscosity contrast. For small flow rates, both of these efficiency measures decrease uniformly with increasing Peclet numbers. At large flow rates, an asymptotic state is reached where the efficiencies no longer depend on the Peclet number. Up to Atwood numbers of approximately 0.5, the less viscous fluid occupies close to 23 of the gap width, which indicates a near-parabolic velocity profile across the gap. Consequently, in this parameter range a Taylor dispersion approach should be well suited to account for flow-induced dispersion effects. For larger viscosity contrasts, accompanying two-dimensional numerical simulations based on Taylor dispersion predict an increased stabilization for hi...

Wavelength selection of fingering instability inside Hele–Shaw cells

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 Peclet 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.

Penetration of a negatively buoyant jet in a miscible liquid

We report experimental results on the evolution of a laminar liquid jet injected with negatively buoyant condition in a miscible surrounding liquid. Since molecular diffusion is negligible, the only significant miscible effect is the absence of any surface tension. After an initial intrusion phase, the jet reaches a steady-state characterized by a constant penetration depth. A simple theoretical model is derived which successfully predicts the transient phase as well as the subsequent steady state in terms of stationary penetration depth and jet’s profile. All the experimental points collapse on a master curve involving two dimensionless numbers: the densimetric Froude number Fr and S, a number comparing viscous friction to buoyancy. Finally, this curve obtained for laminar flows is compared to classical results on turbulent fountains.

Measurement of the Diffusion Coefficient of Miscible Fluids Using Both Interferometry and Wiener's Method

Measurements of the diffusion coefficient of two miscible liquids are reported. The liquids are various combinations of pure silicone oils and those to which small amounts of solvents are added to control the difference in density between the fluids. The liquids were placed in a quartz cell such that the interface is initially horizontal. As the fluids diffuse, the profile of the index of refraction near the interface is time dependent and is related to the local concentration of the diffusing fluids. The concentration gradient profile was measured by a shearing interferometer incorporating a Wollaston prism, as well as Wieners method. In the latter technique, a 45° light sheet was passed through the test cell, and the local deflection of the light beam was measured. The average diffusion coefficient was obtained by analysis of the measured concentration gradient profile, assuming that the diffusion process is one-dimensional and is characterized by a constant value of the diffusion coefficient.

Experimental study on water-wave trapped modes

We present an experimental study on the trapped modes occurring around a vertical surface-piercing circular cylinder of radius a placed symmetrically between the parallel walls of a long but finite water waveguide of width 2 d . A wavemaker placed near the entrance of the waveguide is used to force an asymmetric perturbation into the guide, and the free-surface deformation field is measured using a global single-shot optical profilometric technique. In this configuration, several values of the aspect ratio a / d were explored for a range of driving frequencies below the waveguides cutoff. Decomposition of the obtained fields in harmonics of the driving frequency allowed for the isolation of the linear contribution, which was subsequently separated according to the symmetries of the problem. For each of the aspect ratios considered, the spatial structure of the trapped mode was obtained and compared to the theoretical predictions given by a multipole expansion method. The waveguide–obstacle system was further characterized in terms of reflection and transmission coefficients, which led to the construction of resonance curves showing the presence of one or two trapped modes (depending on the value of a / d ), a result that is consistent with the theoretical predictions available in the literature. The frequency dependency of the trapped modes with the geometrical parameter a / d was determined from these curves and successfully compared to the theoretical predictions available within the frame of linear wave theory.

Velocity fields and streamline patterns of miscible displacements in cylindrical tubes

AbstractDisplacements of a viscous fluid by a miscible fluid of a lesser viscosity and density in cylindrical tubes were investigated experimentally. Details of velocity and Stokes streamline fields in vertical tubes were measured using a DPIV (digital particle image velocimetry) technique. In a reference frame moving with the fingertip, the streamline patterns around the fingertip obtained from the present measurements confirm the hypothesis of Taylor (1961) for the external patterns, and that of Petitjeans and Maxworthy (1996) for the internal patterns. As discussed in these papers, the dependent variable, m, a measure of the volume of viscous fluid left on the tube wall after the passage of the displacing finger, is a parameter that determines the flow pattern. When m>0.5 there is one stagnation point at the tip of the finger; when m<0.5 there are two stagnation points on the centerline, one at the tip and the other inside the fingertip, and a stagnation ring on the finger surface with a toroidal recirculation in the fingertip between the two stagnation points. The finger profile is obtained from the zero streamline of the streamline pattern.