Shear-induced force and dispersion due to buoyancy in a horizontal Hele-Shaw cell

If a less viscous fluid displaces a more viscous one in a porous medium or Hele-Shaw cell, the unfavorable mobility gradient is known to cause the classical ’viscous fingering instability’. In the past, this instability has been investigated mostly on the basis of Darcy’s law, which involves certain averaging procedures that are usually not valid for the case of a Hele-Shaw displacement. In this diploma thesis, miscible fingering in a horizontal Hele-Shaw cell is studied by means of Stokes simulations and linear stability analysis. The difference from former studies is that a nonmonotonic concentration viscosity profile is used which can exist in nature for specific fluid pairs. Here, particular emphasis is placed on identifying parameter regimes in terms of the Peclet-number Pe, the viscosity contrast R, the value and position of the viscosity maximum μm and cm. Λ is an additional parameter depending on the end-point derivatives. The existing nonlinear Stokes code for monotonic exponential viscosity profiles developed by N. Goyal was modified to incorporate any general viscosity concentration functional relationship. These twodimensional simulations lead to a moving finger with a quasisteady state near the tip of the displacement front for sufficiently large Peclet number and high viscosity ratios. The front thickness d0 and the tip velocity vtip of the quasisteady state is primarily affected by the Peclet number Pe and the viscosity contrast R. The scale of d0 with Pe−1/2 which is found for the monotonic profile is deviating for the nonmonotonic profile. A higher value of the maximum of the nonmonotonic viscosity profile leads to a slower tip and thicker front, while a shift of the maximum from the less viscous fluid to the more viscous fluid leads to a increase of the tip velocity and a decrease of the front thickness. In a second step the results of the direct numerical Stokes flow simulation are used to examine the stability of the quasisteady front to spanwise perturbations. In the linear stability analysis the viscosity contrast R is the dominant factor that affects the stability of the flow. Manickam&Homsy [10] investigated that for the Darcy flow a negative Λ leads to a more stable configuration than a positive Λ, while we see a reversal of this behavior. Only higher values of Λ lead to strong changes for the growth rate σmax of the most dangerous mode and the corresponding wavenumber βmax. Because the shape of the profile is also changing a lot with high values of Λ it is really hard to figure out what leads to this change. The exact reason could not be identified and would need further investigation.

When a given fluid displaces another less viscous miscible one in a horizontal Hele-Shaw cell, the displacement is stable from the viscous point of view. Nevertheless, thin stripes perpendicular to the moving interface can be observed in the mixing zone between the fluids both in rectilinear and radial displacements. This instability is due to buoyancy effects within the gap of the cell which develop because of an unstable density stratification associated with the underlying concentration profile. To characterize this buoyancy-driven instability and the related striped pattern, we perform a parametric experimental study of viscously stable miscible displacements in a horizontal Hele-Shaw cell with radial injection. We analyze the influence of the flow rate, the thickness of the gap, and the relative physical fluid properties on the development and characteristics of the instability.

We examine the flow in a horizontal Hele-Shaw cell in which the undisturbed unidirectional flow at infinity is required to stream around a vertical cylinder spanning the gap between the two (horizontal) plates of the cell. A combination of matched asymptotic expansions and numerical methods is employed to elucidate the structure of the boundary layer near the surface of the cylinder. The two length scales of the problem are the gap, h, and the length of the body, l; it is assumed that h/l<<1. The characteristic Reynolds number based on l is O(1). The length scales associated with the boundary layer and the classical Hele-Shaw flow pattern are O(h) and O(l), respectively.It is found that the boundary layer contains streamwise vorticity. This vorticity is generated at the three no-slip surfaces (the two plates and the cylinder wall) as a result of the cross-flow induced by the streamwise acceleration/deceleration of the flow around the curved cylinder. The strength of the secondary flow, hence the associated streamwise vorticity, is proportional to changes in body curvature. The validity of the classical Hele-Shaw flow is examined systematically, and higher-order corrections are worked out. This results in a displacement thickness that is roughly 30% of the gap. In other words, the lowest-order correction to the classical Hele-Shaw flow may be obtained by requiring the outer flow (on the scale O(l)) to satisfy the no-penetration boundary condition on a displaced cylinder surface. The boundary layer contains ‘corner’ vortices at the intersections of the horizontal plates and the vertical cylinder surface.

When air is injected into silicone oil contained in a horizontal Hele-Shaw cell, a single air bubble forms and grows showing various interesting phenomena. In this study the effects of the bubble front velocity, air injection velocity at a nozzle, fluid properties and cell depth on the stability of the growing bubble were investigated experimentally. By using the modified capillary number involving the aspect ratio, we obtained the onset conditions of the unstable bubble. Also, the bubble width was analyzed both quantitatively and qualitatively. Before the bubble experiences splitting, the bubble front velocity is almost proportional to the air injection velocity. Therefore the latter velocity may be used in a practical sense.

The motion and shape of a liquid drop through another continuous liquid phase (conveying phase) in a vertical Hele- Shaw cell with different two different apertures were investigated experimentally. Two different liquid-liquid systems were tested. In all cases, the continuous phase was more viscous and wetted the bounding walls. In the capillarity-dominated region, the irregular shape of the discontinuous phase changed with time and distance, with much lower velocity than that of the conveying phase. In contrast to gas-liquid systems, the velocity of these stabilized elongated drops was 2.5 to almost 5 times higher than that of conveying liquid. Despite the similarities between flow in vertical and horizontal Hele-Shaw cell, the velocity of droplets in vertical fracture is different from that of horizontal fracture. A new correlation is derived from dimensionless analysis and the experimental data to predict the elongated drop velocity as a function of the dimensionless parameters governing the system. Introduction Two-phase flow in micro-fractures is fundamental to many different fields of advanced science and technology, such as chemical process engineering, bioengineering, medical and genetic engineering as well as petroleum engineering. For instance understanding the flow of two-phase fluids in near parallel gaps through fractured rocks has a significant effect on design of different recovery methods for naturally fractured reservoir. The flow pattern of two-phase immiscible flow in a fracture depends on the flow rates of the phases, the geometry, aperture, and roughness of the fracture, the flow properties of the phases, and interfacial tension between the phases. The flow patterns in a fracture are different from that in macro size rectangular ducts or pipes due to the small aperture, which can enhance capillary effects. The flow structure in the fracture affects the flow and transport through the surrounding porous matrix blocks. The slug flow pattern in a fracture, which occurs over a wide range of parameters, is frequently encountered in oil-wet fractured reservoirs during the immiscible displacement of viscous oil. It also occurs in natural gas reservoirs during displacement of water during gas production. In the case of a smooth-walled fracture, the slug shape can be regular elongated drops (or bubbles) flowing through the conveying phase. The elongated drops (or bubbles) can have a rounded leading edge and occupy a considerable portion of the gap with length of several times to several hundred times that of the gap. The behavior of single elongated drops (or bubbles) through a continuous liquid phase problem is as rich and complex as the viscous fingering problem6. The conditions corresponding to a swarm of the elongated drops can be inferred by considering the behavior of a single drop (bubble) and their conglomeration. The interaction of many drops or bubbles across the flow cross section can also be inferred from single drop or bubble dynamics. A firm understanding of the behavior of drop or bubble flow in a single fracture is prerequisite to build predictive models for flow in complex fracture systems.

We investigate the effect of horizontal quasi-periodic oscillation on the stability of two superimposed immiscible fluid layers confined in a horizontal Hele-Shaw cell. To approximate real oscillations, a quasi-periodic oscillation with two incommensurate frequencies is considered. Thus, the linear stability analysis leads to a quasi-periodic oscillator, with damping, which describes the evolution of the amplitude of the interface. Two types of quasi-periodic instabilities occur: the low-wavenumber Kelvin-Helmholtz instability and the large-wavenumber resonances. We mainly show that, for equal amplitudes of the superimposed accelerations, and for a low irrational frequency ratio, there is competition between several resonance modes allowing a very large selection of the wavenumber from lower to higher values. This is a way to control the sizes of the waves. Furthermore, increasing the frequency ratio has a stabilizing effect for both types of instability whose thresholds are found to correspond to quasi-periodic solutions using the frequency spectrum. For a ratio of the two superimposed displacement amplitudes equal to unity and less than unity, the number of resonances and competition between their modes also become significant for the intermediate values of the ratio of frequencies. The effects of other physical and geometrical parameters, such as the damping coefficient, density ratio, and heights of the two fluid layers, are also examined.

The instabilities in vibrating granular materials may be interpreted as segregation processes of vacuum and powder particles. Through an attempt of mesocale modeling of the instabilities within the cell-dynamics scheme, a notable distinction between the ordinary segregation processes such as binary alloy spinodal decomposition and the granular processes is highlighted. ‘Vibrating particles in a horizontal Hele-Shaw cell’ experiment is proposed.

Scalar image velocimetry (SIV) is the technique to extract velocity vectors from scalar field measurements. The usual technique involves minimising a cost functional, that penalises the deviation from the scalar conservation equation. This approach requires the measured scalar field to be sufficiently resolved and relatively noise free, such that space and time derivatives of the measured scalar field can be accurately evaluated. We quantify these requirements for a synthetic two-dimensional (2-D) turbulent flow field by evaluating the velocity reconstruction accuracy as a function of the temporal and spatial resolution and the noise level. We propose an improved SIV scheme, that reconstructs not only the velocity field but also the scalar field, which does not require approximating the space and time derivatives of the measured scalar field. Improved velocity reconstruction is demonstrated for the 2-D synthetic field. We furthermore apply the scheme to interferograms of the thickness field of a falling soap film, where 2-D turbulence is generated by an array of cylindrical obstacles. The statistics of the reconstructed velocity field are within 10 % of laser Doppler velocimetry measurements.

We report the first quantitative measurements of the resonance frequencies of a torus of fluid confined in a horizontal Hele-Shaw cell. By using the unwetting property of a metal liquid, we are able to generate a stable torus of fluid with an arbitrary aspect ratio. When subjected to vibrations, the torus displays azimuthal patterns at its outer periphery. These lobes oscillate radially, and their number n depends on the forcing frequency. We report the instability "tongues" of the patterns up to n=25. These resonance frequencies are well explained by adapting to a fluid torus the usual drop model of Rayleigh. This approach could be applied to the modeling of large-scale structures arisen transiently in vortex rings in various domains.

When a ferrofluid drop is trapped in a horizontal Hele-Shaw cell and subjected to a vertical magnetic field, a fingering instability results in the droplet evolving into a complex branched structure. This fingering instability depends on the magnetic field ramp rate but also depends critically on the initial state of the droplet. Small perturbations in the initial droplet can have a large influence on the resulting final pattern. By simultaneously applying a stabilizing (horizontal) azimuthal magnetic field, we gain more control over the mode selection mechanism. We perform a linear stability analysis that shows that any single mode can be selected by appropriately adjusting the strengths of the applied fields. This offers a unique and accurate mode selection mechanism for this confined magnetic fluid system. We present the results of numerical simulations that demonstrate that this mode selection mechanism is quite robust and "overpowers" any initial perturbations on the droplet. This provides a predictable way to obtain patterns with any desired number of fingers.

We report measurements of the dynamics of an air/water interface moving through a model porous medium made of glass beads packed in a horizontal Hele-Shaw cell. At low flow rates, the interface does not move uniformly. Instead, some small regions move while others stay pinned, at least for a short time. The burst of motion when a pinned region breaks free is called an avalanche. In several theoretical models, such avalanches were found to follow a power-law distribution. In contrast, we find that even for very slow flow (capillary number $\ensuremath{\mu}U/\ensuremath{\gamma}\ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7})$ the number of avalanches of size $s$ decays roughly exponentially with $s.$

The e ect of periodic oscillations on the interfacial instability of two immiscible fluids, confined in a horizontal Hele-Shaw cell, is investigated. A linear stability analysis of the basic state leads to a periodic Mathieu oscillator corresponding to the amplitude of the interface. Then, the threshold of parametric instability of the interface is characterized by harmonic or subharmonic periodic solutions. We show that the relevant parameters that control the interface are the Bond number, density ratio, Weber number and amplitude and frequency of oscillations.

Fingering is a hydrodynamic instability that occurs when a more mobile fluid displaces a fluid of lower mobility. When the primary source of the mobility difference is viscosity, the instability is termed viscous fingering. Viscous fingering is often, though not always, undesirable in industrial processes, particularly secondary petroleum recovery. Linear stability analysis by Hejazi et al. has indicated that the production of a non-monotonic viscosity profile can stabilize the interface. Herein, we use step-growth polymerization at the interface between two miscible monomers as a model system. In particular, a dithiol monomer displaced a diacrylate that reacted to form a linear polymer that behaves as a Newtonian fluid. Viscous fingering was imaged in a horizontal Hele-Shaw cell via Schlieren, which is sensitive to changes in index of refraction, and therefore polymer conversion. By varying reaction rate via initiator concentration along with flow rate via a syringe pump, we were able to demonstrate increasing stabilization of the flow with increasing Damkohler number.

Chemical gardens refer to a class of self-assembling structures of semi-permeable precipitates. They have been attracting significant interest due to their relevance to sub-oceanic hydrothermal vents and the origin of life. We have investigated the growth behaviour of chemical garden walls in a horizontal Hele-Shaw cell. The experiments were conducted with pellets of either solid cobalt(II) or manganese(II) chloride contained in aqueous sodium silicate solutions. It is found that the growth of the chemical garden walls can be well described by a simple, diffusion-controlled dynamical model until their eventual osmotic fracture at a reproducible time. This provides a basis by which wall growth in more complex chemical garden systems can be characterized.This article is part of the theme issue ‘Biological fluid dynamics: emerging directions’.

We study the spatio-temporal evolution of the viscosity field during stable and unstable radial flows of glycerol-water solutions in a horizontal Hele-Shaw cell where a localized temperature gradient is imposed. The viscosity field is reconstructed from the measurement of the fluorescence emitted by a viscosity-sensitive molecular probe (Auramine O). For an immiscible flow, the viscosity and temperature fields are obtained accurately. For miscible displacements, we show how the interplay between the viscosity changes of both fluids and the variation of the fluid thickness in the gap prevents obtaining strict quantitative reconstruction of the viscosity field. We explain how the reconstructed viscosity field can nevertheless be interpreted to obtain information about the fluid thickness and the local viscosity and temperature.