Rectangular Hele-Shaw Cell Research Articles

Experimental study of solute diffusion in oscillating flow in a rectangular cell

Abstract The mass transfer of a passive substance dissolved in a fluid in a rectangular Hele–Shaw cell is experimentally studied. We consider mass transfer when the fluid is (i) at rest and (ii) oscillates. In the first case, mass transfer is carried out due to molecular diffusion. The results of measuring the molecular diffusion coefficient of the fluorescent dye Rhodamine B in water are in good agreement with the data obtained by other methods. In the presence of oscillations, the mass transfer rate increases. The comparison of the obtained experimental results with theoretical predictions reveals that the mass transfer is enhanced due to the Taylor dispersion.

Mass transfer of solute in an oscillating flow in a two-dimensional channel

The effective diffusion of a solute in a rectangular two-dimensional channel is experimentally studied. We experimentally examine the effective diffusion of Rhodamine B dissolved in water oscillating in a rectangular Hele–Shaw cell. The concentration of Rhodamine B in water is measured by the intensity of its fluorescence emission. In particular, we consider two problems: (i) effective diffusion of solute in water oscillating in a two-dimensional rectangular channel (Hele–Shaw cell) and (ii) effective diffusion of solute in pores between monosized hard spheres randomly packed in a rectangular Hele–Shaw cell. It is revealed that the rate of solute mass transfer exceeds the molecular diffusion rate in both cases. It has been demonstrated that when water oscillates between parallel walls, diffusion is accelerated by Taylor dispersion with the effective diffusion coefficient Deff exceeding the molecular diffusion coefficient Dm by 1–2 orders of magnitude. The effective diffusion coefficient Deff depends only on the relative amplitude but not on the frequency of the fluid oscillations in the studied range of frequencies and amplitudes. When the fluid oscillates in the pores of the porous medium, solute transport is faster than in the case of Taylor dispersion. Here, the effective diffusion coefficient depends on both the frequency and amplitude of oscillations. The analysis shows that the experimental data obtained at various frequencies and amplitudes of oscillations are consistent with the relation Deff/Dm∼Pe2 (Pe is the Peclet number). We suggest that the enhanced solute mass transfer is associated with the time-averaged fluid flows that arise due to spatial heterogeneity of the amplitude of water oscillations in the pores between randomly packed hard spheres.

CFD analysis of viscous fingering in Hele-Shaw cell for air-glycerin system

In this research paper, a numerical study has been done on the viscous fingering instability for the air-glycerin system in the 2D horizontal Hele-Shaw cell. The air and glycerin are immiscible fluids and the viscosity of glycerin is too much higher than air and viscous fingering strongly depends upon the viscosity differences of fluids pairs so that the air-glycerin system is taken. In this work, an immiscible fluid–fluid displacement flow process occurs and a low viscous fluid air displaces another high viscous fluid glycerin in the rectangular Hele-Shaw cell then due to unstable displacement of glycerin by air a single finger shape instability appears at the air-glycerin interface. This phenomenon is called viscous fingering. The viscous fingering has got importance in many natural and industrial applications such as geologic carbon dioxide sequestration, oil recovery, water infiltration into the soil, mixing of fluids at low Reynolds numbers in microfluidic devices, and many more. This numerical simulation is based on the transient solver. Volume of fluid (VOF) model has been accounted for tracking of the air- glycerin interface. The Geo-reconstruct scheme has been selected for precise interface tracking. The PISO (Pressure Implicit with Splitting of Operators) scheme has been adopted because it has improved the calculation efficiency. The objectives of this paper are to track the growth of a single finger at the air-glycerin interface and calculate the displacement efficiency by image processing. We found that air displaces only 74.15% of glycerin from the entire cell. We have also shown the distribution of pressure and velocity in the entire cell domain.

Development of magnetoelastic fingering patterns in a rectangular Hele-Shaw cell

A ferrofluid and a nonmagnetic fluid flow in a rectangular Hele-Shaw cell subjected to an in-plane magnetic field. A chemical reaction occurs, and the two-fluid interface becomes elastic. The produced patterns range from straight front to sharp tip ones, and they differ from usual Saffman-Taylor fingers.

Peak instability in an elastic interface ferrofluid

The instability of an elastic interface separating a ferrofluid and a nonmagnetic fluid subjected to an applied magnetic field perpendicular to the initially undisturbed interface is investigated in the effectively two-dimensional environment of a vertical, rectangular Hele-Shaw cell. By using a third-order mode-coupling perturbative scheme, and considering that the elastic interface has a curvature-dependent bending rigidity, the emergence of elastic ferrofluid peaks is detected at the onset of nonlinearities. In this context, the approximate profile of the interface is obtained. It is also shown that the morphology of the resulting peaks is sensitive to changes in a dimensionless magnetoelastic number (relative measure of magnetic and elastic forces), as well as in a bending rigidity fraction parameter, which expresses variations in the bending rigidity with the local interface curvature.

A paradox in Hele-Shaw displacements

The Saffman–Taylor instability occurs when a less viscous Stokes fluid is displacing a more viscous one, in a rectangular Hele-Shaw cell. We study the case when all surface tensions on the interfaces are zero. Consider N liquid-layers with constant viscosities, inserted between the two initial fluids. The corresponding growth constants become infinite with increasing wave numbers, even if N is very large. On the contrary, a single intermediate liquid with a suitable variable viscosity gives us an almost stable flow. We compute the total amount of intermediate liquid used in both models.

Immiscible liquid-liquid displacement flows in a Hele-Shaw cell including shear thinning effects

We experimentally studied the viscous fingering instability considering a Newtonian oil displacing viscoelastic shear-thinning liquids and vice versa. The non-Newtonian liquids are aqueous solutions of polyacrylamide and xanthan gum, i.e., flexible and rigid polymers, respectively. A rectangular Hele-Shaw cell, connected by two plenum chambers, was developed to evaluate the displacement of a fixed volume. The experiment consists of analyzing the interface time evolution through a digital camera as a function of the geometric, dynamic, and rheological parameters. The displacement efficiency was determined through image processing in order to identify the formation of fingers or plugs. Unlike the Newtonian case, the transition does not occur when the viscosity ratio is roughly equal to one, but nevertheless, it was observed that the stability of the interface depends on the viscosity ratio. Specifically, more branches are observed at low viscosity ratios. Furthermore, a higher stability is observed when the Newtonian liquid displaces the shear thinning liquid, especially when the polymer is more rigid. When the Newtonian liquid is being displaced, elastic effects favor the displacing efficiency.

Viscous fingering of yield stress fluids: The effects of wettability

We consider experimentally the viscous fingering problem wherein a silicone oil displaces a Carbopol gel in a rectangular Hele-Shaw cell. Unlike traditional cases, the displacing fluid perfectly wets the walls of the Hele-Shaw cell. While the silicone oil used is a Newtonian fluid (with a small viscosity), the Carbopol gel employed is a non-Newtonian fluid (with a large viscosity), which exhibits an interesting range of complex rheology such as yield stress, shear-thinning and weakly-elastic behaviours. We demonstrate that non-Newtonian effects together with wettability conditions exert a significant impact on fingering patterns, which we describe through a detailed comparison between wetting displacements (oil-Carbopol) and non-wetting displacements (air-Carbopol). We analyze the flow from various perspectives, using the relevant dimensionless groups, including the capillary number (Ca), the Bingham number (Bn), the Weissenberg number (Wi), Weber number (We), the channel aspect ratio (δ), and the shear-thinning power-law index (n). We succeed in classifying four main flow regimes versus these dimensionless groups: a capillary regime, a yield stress regime, a viscous regime and an elasto-inertial regime. The capillary regime, observed at very low velocities, is a unique regime which appears due to an interplay between yield stress and interfacial tension effects. In this regime, the finger width is nearly equal to the cell width. The other flow regimes are observed in both wetting and non-wetting displacements, but with certain critical differences that are made clear in the paper.

Viscous fingering regimes in elasto-visco-plastic fluids

We experimentally study the Saffman–Taylor instability of air invasion into a non-Newtonian fluid (i.e., Carbopol solution) in a rectangular Hele-Shaw cell. In addition to viscous features, the non-Newtonian fluid used exhibits yield stress, shear-thinning as well as elastic behaviors. The key dimensionless parameters that govern the various flow regimes are the Bingham number (Bn), the capillary number (Ca), the Weber number (We), the Weissenberg number (Wi), the channel aspect ratio (δ ≫ 1), and the shear-thinning power-law index (n). Three main flow regimes are observed, i.e., a yield stress regime, a viscous regime and an elasto-inertial regime. We present a detailed description for each regime and quantify their transition boundaries versus dimensionless groups. Some of the secondary flow aspects, e.g., the wall residual layer thickness and a network structure regime, have been also studied.

Hydrodynamics of fingering instability in the presence of a magnetic field

The hydrodynamics of two immiscible fluids in a rectangular Hele-Shaw cell under the influence of a magnetic field is studied, both theoretically and numerically. A linear stability analysis is conducted to determine the effect of magnetic fields on the formation of viscous fingers. As a result, an analytical solution is found to calculate the growth rate of perturbations. For numerical simulation of the two-phase flow, the interfacial tension is treated as a body force using the continuum surface force model and the interface tracking is performed by the volume of fluid method. The variations of the width and growth rate of fingers in an unstable displacement versus Hartmann number, a dimensionless number characterizing the strength of the applied magnetic field, are investigated. By varying the value of Hartmann number systematically, a suppressing effect on the formation of viscous fingers is observed. Consequently, it is detected that there exists a minimum Hartmann number preventing the formation of viscous fingers and ensuring a stable displacement. Our numerical simulations are in agreement with the results of the linear stability analysis and quantify the effect of magnetic fields in mitigating viscous fingering effects and improving the efficiency of the fluid displacement.

Bridging aero-fracture evolution with the characteristics of the acoustic emissions in a porous medium

The characterization and understanding of rock deformation processes due to fluid flow is a challenging problem with numerous applications. The signature of this problem can be found in Earth Science and Physics, notably with applications in natural hazard understanding, mitigation or forecast (e.g. earthquakes, landslides with hydrological control, volcanic eruptions), or in industrial applications such as hydraulic-fracturing, steam-assisted gravity drainage, CO₂ sequestration operations or soil remediation. Here we investigate the link between the visual deformation and the mechanical wave signals generated due to fluid injection into porous media. In a rectangular Hele-Shaw Cell, side air injection causes burst movement and compaction of grains along with channeling (creation of high permeability channels empty of grains). During the initial compaction and emergence of the main channel, the hydraulic fracturing in the medium generates a large non-impulsive low frequency signal in the frequency range 100 Hz - 10 kHz. When the channel network is established, the relaxation of the surrounding medium causes impulsive aftershock-like events, with high frequency (above 10 kHz) acoustic emissions, the rate of which follows an Omori Law. These signals and observations are comparable to seismicity induced by fluid injection. Compared to the data obtained during hydraulic fracturing operations, low frequency seismicity with evolving spectral characteristics have also been observed. An Omori-like decay of microearthquake rates is also often observed after injection shut-in, with a similar exponent p≃0.5 as observed here, where the decay rate of aftershock follows a scaling law dN/dt ∝(t-t₀ )-p . The physical basis for this modified Omori law is explained by pore pressure diffusion affecting the stress relaxation.

Viscous fingering in yield stress fluids: a numerical study

The effect of yield stress is numerically investigated on the viscous fingering phenomenon in a rectangular Hele–Shaw cell. It is assumed that the displacing fluid is Newtonian, while the displaced fluid is assumed to obey the bi-viscous Bingham model. The lubrication approximation together with the creeping-flow assumption is used to simplify the governing equations. The equations so obtained are made two-dimensional using the gap-averaged variables. The initially flat interface between the two (immiscible) fluids is perturbed by a waveform perturbation of arbitrary amplitude/wavelength to see how it grows in the course of time. Having treated the interfacial tension like a body force, the governing equations are solved using the finite-volume method to obtain the pressure and velocity fields. The volume-of-fluid method is then used for interface tracking. Separate effects of the Bingham number, the aspect ratio, the perturbation parameters (amplitude/wavelength), and the inlet velocity are examined on the steady finger width and the morphology of the fingers (i.e., tip-splitting and/or side-branching). It is shown that the shape of the fingers is dramatically affected by the fluid’s yield stress. It is also shown that a partial slip has a stabilizing effect on the viscous fingering phenomenon for yield-stress fluids.

Two-phase viscous fingering of immiscible thixotropic fluids: A numerical study

The effects of a fluid’s thixotropic behavior is investigated on the viscous fingering phenomenon in a rectangular Hele-Shaw cell assuming that the displacing fluid is Newtonian while the displaced fluid obeys the Moore model for thixotropic fluids. Lubrication theory is used to simplify the gap-averaged governing equations in which the interfacial tension is treated as a body force. It is shown that the shapes of the fingers are dramatically affected by the displaced fluid’s thixotropic behavior. For highly thixotropic fluids, a chaotic behavior, accompanied by a blowup at prolonged times, is predicted to occur for certain set of parameter values. The viscosity ratio of the Moore fluid is also predicted to influence the shapes of the fingers provided that the zero-shear viscosity of the displaced fluid is higher than the viscosity of the displacing fluid. It is shown that the amplitude and wavenumber of the initial perturbation plays a crucial role on its time evolution. Also, a partial slip of the displaced and/or the displacing fluid is predicted to have a stabilizing effect on the viscous fingering phenomenon.

Viscous Fingering in LAPONITE® and Mud

Abstract We have studied the Saffman–Taylor instability for LAPONITE® and mud in a rectangular Hele–Shaw cell. The experiments showed several fingering patterns that did not obey the classical theory of Saffman–Taylor instability. LAPONITE® fingers at high and low shear rates showed shear thinning and shear thickening behavior, respectively. This result is in agreement with rheological analyses. The results of both rheology and Hele–Shaw experiments show a diversion point in the shear viscosity diagram at a specific shear rate. The generalized Darcy’s law for shear thinning fluid in which the viscosity is a function of shear rate can be used to measure viscosity. The validity of this approach was tested using three Newtonian fluids (detergent, oil, and clothing conditioner) and one non-Newtonian fluid (LAPONITE®). These analyses on LAPONITE® as a non-Newtonian fluid verified that the generalized Darcy’s law is in good agreement with rheological analysis in the shear thinning region. In contrast, the mud suspension showed Newtonian behavior at all pressure gradients.

Discrete families of Saffman–Taylor fingers with exotic shapes

The mathematical problem of determining the shape of a steadily propagating Saffman–Taylor finger in a rectangular Hele-Shaw cell is known to have a countably infinite number of solutions for each fixed surface tension value. For sufficiently large surface tension values, we find that fingers on higher solution branches are non-convex. The tips of the fingers have increasingly exotic shapes as the branch number increases.

Fingering dynamics on the adsorbed solute with influence of less viscous and strong sample solvent.

Viscous fingering is a hydrodynamic instability that sets in when a low viscous fluid displaces a high viscous fluid and creates complex patterns in porous media flows. Fundamental facets of the displacement process, such as the solute concentration distribution, spreading length, and the solute mixing, depend strongly on the type of pattern created by the unstable interface of the underlying fluids. In the present study, the frontal interface of the sample shows viscous fingering and the strong solvent causes the retention of the solute to depend on the solvent concentration. This work presents a computational investigation to explore the effect of the underlying physico-chemical phenomena, (i.e., the combined effects of solvent strength, retention, and viscous fingering) on the dynamics of the adsorbed solute. A linear adsorption isotherm has been assumed between the mobile and stationary phases of the solute. We carried out the numerical simulations by considering a rectangular Hele-Shaw cell as an analog to 2D-porous media containing a three component system (displacing fluid, sample solvent, solute) to map out the evolution of the solute concentration. We observed that viscous fingering at the frontal interface of the strong sample solvent intensifies the band broadening of the solute zone. Also notable increase in the spreading dynamics of the solute has been observed for less viscous and strong sample solvent as compared to the high viscous sample slices or in the pure dispersive case. On the contrary, the solute gets intensively mixed at early times for more viscous sample in comparison to less viscous one. The results of the simulations are in qualitative agreement with the behavior observed in the liquid chromatography column experiments.

Dynamic aerofracture of dense granular packings

A transition in hydraulically induced granular displacement patterns is studied by means of discrete numerical molecular dynamics simulations. During this transition the patterns change from fractures and fingers to finely dispersed bubbles. The dynamics of the displacement patterns are studied in a rectangular Hele-Shaw cell filled with a dense but permeable two-dimensional granular layer. At one side of the cell the pressure of the compressible interstitial gas is increased. At the opposite side from the inlet of the cell a semipermeable boundary is located. This boundary is only permeable towards the gas phase while preventing grains from leaving the cell. The imposed pressure gradient compacts the grains. In the process we can identify and describe a mechanism that controls the transition of the emerging displacement patterns from fractures and fingers to finely dispersed bubbles as a function of the interstitial gas's properties and the characteristics of the granular phase.

From viscous fingering to elastic instabilities

An analytical and numerical study of the linear Saffman–Taylor instability for a Maxwell viscoelastic fluid is presented. Results obtained in a rectangular Hele–Shaw cell are complemented by experiments in a circular cell corroborating the universality of our main result: The base flow becomes unstable and the propagating disturbances develop into crack-like features. The full hydrodynamics equations in a regime where viscoelasticity dominates show that perturbations to the pressure remain Laplacian. Darcy’s law is expressed as an infinite series in the cell thickness. An unique dimensionless parameter λ¯, equivalent to a relaxation time, controls the growth rate of the perturbation. λ¯ depends on the applied gradient of pressure, the surface tension, the cell thickness, and the elastic modulus of the fluid. For small values of λ¯, Newtonian behavior dominates whereas for higher values of λ¯ viscoelastic effects appear. For the critical value λ¯=λc¯≃10 a blowup is predicted and fracture-like patterns are observed.

Sedimentation instabilities: Impact of the fluid compressibility and viscosity

The effect of an interstitial fluid on the mixing of sedimenting grains is studied numerically in a closed rectangular Hele-Shaw cell. We investigate the impact of the fluid compressibility and fluid viscosity on the dynamics and structures of the granular Rayleigh-Taylor instability. First we discuss the effect of the fluid compressibility on the initial fluid pressure evolution and on the dynamics of the particles. Here, the emerging patterns do not seem highly affected by the compressibility change studied. To characterize the patterns and motion the combined length of the particle trajectories in relation to the movement of the center of mass is analyzed, and the separation of particle pairs is measured as a function of the fluid viscosity.

Natural convection of magnetic fluid in a rectangular Hele–Shaw cell of different aspect ratios

The nature convection of a magnetic fluid in a rectangular Hele-Shaw cell with an imposition of an even vertical magnetic field is studied experimentally. Heat transfer measurements with thermocouples and flow visualization with liquid crystal thermography are conducted in Hele-Shaw cells of aspect ratios 1.0, 1.5, and 2.0. Both results show that the vertically imposed magnetic field has destabilizing influence. The flow instability modes become different from that in two-dimensional cavity cases, with and without the magnetic field. A pair of symmetric counter-rotating vortices is observed for the first instability mode. Increasing the aspect ratio destabilizes the flow. Second instability mode is observed in the cases of aspect ratios 1.0 and 1.5, but not in 2.0.