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We analyze the displacement of a viscous fluid by a miscible more viscous one in heterogeneous porous media. We performed lattice Bhatnagar–Gross–Krook simulations, which were previously successfully applied to the study of the dispersion of a passive tracer in a stochastic heterogeneous porous medium. In the present situation, the flow is stable (no viscous fingering) and leads to an overall Gaussian dispersion, the coefficient of which decreases as the viscosity ratio increases. The results are in reasonable agreement with the stochastic approach of Welty and Gelhar.

Buoyant mixing of two fluids in tubes is studied experimentally as a function of the tilt angle θ from vertical, the density contrast and the common viscosity μ. At high contrasts and low θ, longitudinal mixing is macroscopically diffusive, with a diffusivity D increasing strongly with θ and μ. At lower contrasts and higher θ, a counterflow of the two fluids with little transverse mixing sets in. The transition occurs at an angle increasing with density contrast and decreasing with μ. These results are discussed in terms of the dependence of transerse mixing on θ and an analogy with the Boycott effect.

We study the Brinkman correction to Darcy’s equation for unidirectional flows in a Hele-Shaw cell. Three examples, describing gravity-driven flow with variable density, pressure-driven flow with variable viscosity, and pressure-driven flow in a cell with a specific variation in aperture are discussed. In general, the Brinkman correction involves nonlocal terms, and it is not simply equal to an effective viscous shear stress involving the gap-averaged velocity. The latter is applicable at long wavelengths, however, provided that the viscosity is augmented by a prefactor equal to 12/π2.

We revisit the Rayleigh–Taylor instability when the two fluids are miscible and in the geometry of a Hele-Shaw cell. We provide analytical dispersion relations for the particular cases of either a sharp front between the two fluids or of a uniform density gradient stratification and for various fluid flow models, including an unbounded geometry, a two-dimensional gap-averaged Navier–Stokes–Darcy equation, and an effective porous medium. The results are compared to three-dimensional lattice BGK simulations, based on which the relevance of the various models in different wavelength regimes is discussed.

Non-Brownian sedimenting suspensions exhibit density and velocity fluctuations. We have performed experiments on a quasi-two-dimensional counter-flow stabilized suspension of 2000 spherical particles, namely a liquid–solid fluidized bed in a Hele–Shaw cell. This two-dimensional suspension displays a uniform concentration but the particle radial distribution function and the fluctuations of the particle number in a subvolume of the suspension suggest that the microstructure is far from being random. We have also measured the velocity fluctuations of a test particle and the fluctuations of the mean particle velocity in a subvolume. It happens that the relation between velocity and concentration fluctuations in a subvolume can be deduced from a balance between buoyancy and parietal friction forces.

This paper uses an acoustic technique to determine the concentration profile developing during the sedimentation of noncolloidal bidisperse suspensions of glass beads in a Newtonian fluid. Various bead diameter ratios have been used and a wide range of relative concentrations is covered. From the shock front velocities and the concentrations in different zones, the sedimentation velocities of small and large particles in a homogeneous suspension of respective concentrations cs0 and cl0 have been determined. The semidilute regime (c0=ss0+cl0

For sufficiently large viscosity ratios and injection rates, miscible displacements in a vertical Hele–Shaw cell at high rates become unstable, leading to three-dimensional (3D) fingering patterns. Below the instability threshold, the base state is 2D in the form of a “tongue” of constant thickness. We apply the long wave Saffman–Taylor stability analysis to find an expression for the threshold of instability as a function of the viscosity ratio and the injection rate. The results are in agreement with the experimental data.

The onset of viscous instability and the subsequent fingering have been mostly studied under conditions of a sharp mobility contrast. In stable miscible flows in porous media, however, mobility gradients of variable extent can develop due to hydrodynamic dispersion. Graded mobility and density profiles will affect instabilities, either by mitigating (in the case of a monotonic variation) or by enhancing (nonmonotonic variation) the rates of growth. The mitigating effect was exploited by Claridge and Gorrel and Homsy for the optimal design of graded mobility banks. Hickernell and Yortsos showed that in the linear stability limit, the growth rate is controlled by the maximum of the logarithmic derivative of the mobility profile. We present an experimental study of miscible displacements in porous media to study effects of mobility gradients in viscous instability. We take advantage of molecular diffusion in miscible displacements to create a mobility gradient zone and subsequently initiate the instability b...

We address the question of using a lattice gas method to study flows of complex fluids, such as non‐Newtonian or miscible fluids. A Bhatnagar, Gross, and Krook lattice technique provides a tool to simulate the flow of one fluid and the diffusion of a tracer in that fluid. We extend the technique to flows in which the viscosity is space and time dependent. This approach is suitable for non‐Newtonian fluids (shear dependent viscosity) and miscible fluids (concentration dependent viscosity). The modified scheme is tested on physical flow situations, analytically tractable for the sake of comparison.

Hydrodynamic dispersion is responsible for the spreading of the sedimentation front even in a noncolloidal monodisperse suspension. Measurements of the broadening of the top front observed during sedimentation have been used in determining the hydrodynamic dispersion coefficient. Hindered settling has an opposed effect and leads to the self‐sharpening of the front. Both effects have to be taken into account simultaneously. This Letter provides a simple, but complete determination of the space and time concentration profile and shows that the final front should consist of a steady‐shape profile propagating at constant velocity. With such a solution, the data of Davis et al. [AIChE J. 34, 123 (1988); J. Fluid Mech. 196, 107 (1988)] give hydrodynamic dispersion coefficient five times larger than their former analysis, in agreement with Lee et al. [Phys. Fluids A 4, 2601 (1992)].

International audience; Buoyancy-driven turbulent mixing of fluids of slightly different densities [At = Δρ/(2〈ρ〉) = 1.15×10−2] in a long circular tube tilted at an angle θ = 15° from the vertical is studied at the local scale, both experimentally from particle image velocimetry and laser induced fluorescence measurements in the vertical diametrical plane and numerically throughout the tube using direct numerical simulation. In a given cross section of the tube, the axial mean velocity and the mean concentration both vary linearly with the crosswise distance z from the tube axis in the central 70% of the diameter. A small crosswise velocity component is detected in the measurement plane and is found to result from a four-cell mean secondary flow associated with a nonzero streamwise component of the vorticity. In the central region of the tube cross section, the intensities of the three turbulent velocity fluctuations are found to be strongly different, that of the streamwise fluctuation being more than twice larger than that of the spanwise fluctuation which itself is about 50% larger than that of the crosswise fluctuation. This marked anisotropy indicates that the turbulent structure is close to that observed in homogeneous turbulent shear flows. Still in the central region, the turbulent shear stress dominates over the viscous stress and reaches a maximum on the tube axis. Its crosswise variation is approximately accounted for by a mixing length whose value is about one-tenth of the tube diameter. The momentum exchange in the core of the cross section takes place between its lower and higher density parts and there is no net momentum exchange between the core and the near-wall regions. A sizable part of this transfer is due both to the mean secondary flow and to the spanwise turbulent shear stress. Near-wall regions located beyond the location of the extrema of the axial velocity (|z|≳0.36 d) are dominated by viscous stresses which transfer momentum toward (from) the wall near the top (bottom) of the tube.

To address the issue of miscible viscous fingering instability in buoyancy free conditions, experiments have been performed under microgravity conditions in parabolic flights. A Hele-Shaw cell, two parallel plates separated by a small gap, has been used with two miscible fluids of viscosity ratio 100 (the injected fluid is the less viscous). The influence of the initial thickness of the pseudointerface between the two fluids has been studied, using flow rates large enough to prevent further mixing during displacement. The selected wavelength, measured on the observed fingering pattern, does not depend on the initial front thickness: It is around three times the gap of the cell, i.e., significantly lower than the value of five, observed on earth. However, the initial thickness does control the displacement length required for the instability to occur. Our results are in reasonable agreement with existing and new numerical simulations.

Autocatalytic reaction fronts between reacted and unreacted species may propagate as solitary waves, that is, at a constant front velocity and with a stationary concentration profile, which result from a balance between molecular diffusion and chemical reaction. A velocity field in the supporting medium may affect the propagation of such fronts through different phenomena: advection, diffusion enhancement, front shape changes, etc. Here, we report on an experimental study and lattice Bhatnagar–Gross–Krook numerical simulations of the effect of an oscillating flow on the autocatalytic reaction between iodate and arsenous acid in a Hele–Shaw cell. In the low frequency range covered by the experiments, the front behavior is controlled by the flow across the gap and is reproduced with two-dimensional numerical simulations. It is shown that the front velocity oscillates at the frequency of the flow, whereas the front width oscillates at twice that frequency. Moreover, the Taylor regime in the presence of a Poi...

We report on experiments with two miscible fluids of equal density but different viscosities. The fluids were injected co-currently and concentrically into a cylindrical pipe. The resulting base state is an axisymmetric parallel flow. The ratio of the two fluid flow rates determines the relative amount of the fluids, thus the radius of the inner core fluid. Depending on this radius and the total flow rate, two different and unstable axisymmetric patterns, denoted by mushrooms and pearls, were observed. We delineate the diagram of occurrence of the two patterns as a function of the various parameters.

The buoyancy driven interpenetration of two fluids of different densities has been studied in a long tilted tube in the strong mixing regime for which the mean concentration profile along the tube length satisfies a macroscopic diffusion equation. Variations of the corresponding macroscopic diffusion coefficient D and of the front velocity Vf are studied as a function of the Atwood number At, the viscosity ν, the tube diameter d, and the tilt angle θ. Introducing the characteristic inertial velocity Vt and the Reynolds number Ret, the normalized front velocity Vf∕Vt and dispersion coefficient D∕(Vtd) are observed to scale, respectively, as Ret−3∕4 and Ret−3∕2 for Ret≲1000. Also, Vf increases linearly with tanθ and the ratio (D∕Vf2) remains of the order of (35±10)d∕Vt in a wide range of values of the tilt angle and of the other control parameters. This close relation observed between the variations of D and Vf2 is discussed in terms of the characteristic time for transverse mixing across the flow channel.

The interpenetration of light and heavy liquids has been studied in a long tube inclined at small angles α to the horizontal. For angles greater than a critical angle αc (whose value decreases when the density contrast measured by the Atwood number At increases), the velocity of the interpenetration front is controlled by inertia and takes the steady value Vf=ki(Atgd)1∕2, with ki≃0.7. At lower angles, the front is initially controlled by inertia, but later limited by viscous effects. The transition occurs at a distance Xfc, which increases indefinitely as α increases to αc. Once the viscous effects act, the velocity of the front decreases in time to a steady value Vf∞ which is proportional to sinα. For a horizontal tube in the viscous regime, the velocity of the front decreases to zero as t−1∕2. At the same time, the profile of the interface h(x,t) only depends on the reduced variable x∕t1∕2. A quasi-unidirectional model reproduces well the variation of the velocity of the front and the profiles of the in...

The buoyancy driven motion of a lighter fluid rising spontaneously into a heavier miscible fluid in vertical or tilted tubes give rise to flows as different as diffusive turbulent mixing and counterflow without mixing Fig. 1 . Two miscible fluids of different densities are initially in an unstable configuration, each of them occupying one halflength of a long transparent tube d=20 mm, L=4 m . The FIG. 1. The distribution of fluorescent dye in a vertical diametral plane of the tube for different tilt angles of the tube from vertical see Fig. 3 . The field of view is located between 200 and 500 mm above the gate valve Fig. 2 . Red and purple-blue shades correspond, respectively, to the pure heavy and light fluids and intermediate colors to mixtures of various compositions.

The front of a light fluid rising into a miscible heavier fluid inside a long tilted tube has been analyzed by laser-induced fluorescence. Both the local concentration field and the front velocity Vf have been studied in the inertial flow regime as a function of the tilt angle θ for a constant density contrast (At=4×10−3). We demonstrate experimentally that the velocity Vf is directly related to the local density contrast Cf by the relation Vf∝(Cf)0.5. This relation reflects a local instantaneous equilibrium between inertia and buoyancy; it is valid in the transient relaxation phase as well as in the quasistationary regime reached thereafter.

The velocity Vf of the fronts of light and heavy fluids in a tilted tube, interpenetrating many diameters, is studied as a function of the fluid viscosity μ, Atwood number At⪡1 and tilt angle θ from vertical. Three flow regimes are observed: starting from vertical, Vf first increases with θ, reaches a plateau and then decreases again. In the first regime, Vf is controlled by segregation and mixing effects, respectively, increasing and decreasing with θ. On the plateau, Vf is independent of the fluid viscosity and proportional to (Atgd)1∕2, indicating a balance between inertia and buoyancy. In the third regime close to horizontal, the fluids separate into two parallel countercurrents controlled by viscosity. The variations of Vf with θ, At, and μ in the second and third regimes and the crossover from one to the other are described by scaling laws based on characteristic viscous and inertial velocities.