Radial Hele-Shaw Cell Research Articles

A three-layer Hele-Shaw problem driven by a sink

In this paper, we investigate a sink-driven three-layer flow in a radial Hele-Shaw cell. The three fluids are of different viscosities, with one fluid occupying an annulus-like domain, forming two interfaces with the other two fluids. Using a boundary integral method and a semi-implicit time stepping scheme, we alleviate the numerical stiffness in updating the interfaces and achieve spectral accuracy in space. The interaction between the two interfaces introduces novel dynamics leading to rich pattern formation phenomena, manifested by two typical events: either one of the two interfaces reaches the sink faster than the other (forming cusp-like morphology), or they come very close to each other (suggesting a possibility of interface merging). In particular, the inner interface can be wrapped by the other to have both scenarios. We find that multiple parameters contribute to the dynamics, including the width of the annular region, the location of the sink, and the mobilities of the fluids.

Pattern formation in foam displacement in a liquid-filled Hele-Shaw cell

Foam injection in porous media is desirable for displacing less viscous fluids because of its ability to form a stable front and ensure uniform fluid distribution. In our experimental study, we investigated foam invasion in a radial Hele-Shaw cell initially saturated with either water or a surfactant solution at the same concentration as that of the foam. We report on an unexpected phenomenon whereby the typically stable process of foam displacing surfactant solution becomes unstable when the foam displaces water, a less viscous fluid, despite the higher viscosity of foam. Nevertheless, this instability can be moderated by either reducing the gas fraction in the foam or adding a small amount of surfactant solution to the water. Through single-bubble analysis, we observe that surfactant dilution, resulting from the mixture with water during foam injection, can trigger the rupture of thin films between large bubbles and the surfaces of the Hele-Shaw cell. This leads to the immobilization of larger bubbles, around which smaller bubbles accumulate densely. Such disturbances create side branching and a distinct fingering pattern within the foam flow. From the observations, the foam-water contact line growth follows a Lc∼t2/3 relation, deviating from the Lc∼t1/2 growth expected in stable displacement scenarios.

Onset and growth of viscous fingering in miscible annular ring

We investigate the onset and growth of viscous fingering (VF) of miscible annulus in a radial Hele-Shaw cell. Systematic numerical study on a finite annulus domain is performed by employing finite element method solver in COMSOL Multiphysics software. We justify that concentration field analysis is not a good choice for dynamic study in radial flows. Instead, velocity magnitude is a better tool to understand the dynamics. Therefore, we propose velocity field analysis to better differentiate the stable and unstable states and present a new stability criterion using the velocity field method. Most interestingly, using the velocity field analysis and the new stability criterion, we show a restabilization of the VF at a critical time when the system becomes diffusion dominant and able to provide both the onset time, τon (time at which instability develops), and the time at which the interface returns to the stable state, τd. Furthermore, the study successfully suggests the critical values for several dimensionless parameters, the Péclet number (Pe), log-viscosity ratio (R), and volumetric ratio (ra) and time (τ), to induce instability. When Pe is higher than 103, the evolution of VF instability is no longer enhanced by Pe, and Rc converges to a certain value. In particular, for the transiently unstable system of low Pe, the restabilization of VF instability is identified even though R is higher than Rc. The unstable system (τ>τon) returns to the stable state as injection time increases further. Moreover, we obtained a critical value of the volumetric ratio (rc,a).

Exact solutions to interfacial flows with kinetic undercooling in a Hele-Shaw cell of time-dependent gap

Hele-Shaw cells where the top plate is moving uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. This paper focuses on interfacial flows with linear and nonlinear kinetic undercooling regularization in a radial Hele-Shaw cell with a time dependent gap. We obtain some exact solutions of the moving boundary problems when the initial shape is a circle, an ellipse or an annular domain. For the nonlinear case, a linear stability analysis is also presented for the circular solutions. The methodology is to use complex analysis and PDE theory.

Control of interfacial instabilities through variable injection rate in a radial Hele-Shaw cell: A nonlinear approach for late-time analysis.

In this paper, the nonlinear behavior of immiscible viscous fingering in a circular Hele-Shaw cell under the action of different time-dependent injection flow rate schemes is assessed numerically. Unlike previous studies which addressed the infinite viscosity ratio (inviscid-viscous flow), the problem is tackled by paying special attention to flows with finite viscosity ratio (viscous flow) in which the viscosity of the displacing and the displaced fluids can have any arbitrary value. Systematic numerical simulations based on a complex-variable formulation of Cauchy-Green barycentric coordinates are performed at different mobility ratios and capillary numbers with a focus on the late-time fully nonlinear regime. Additionally, numerical optimization is used to obtain the optimal flow rate schedule through a second-order weakly nonlinear stability analysis in contrast to previous studies in which the optimal flow rate was obtained entirely based on linear stability analysis. It is demonstrated that, irrespective of the values of the mobility ratio and/or the capillary number, for patterns whose constant injection counterpart exhibits linear flow regime, the curvature-driven relaxation time is comparable with the operational time of the time-dependent injection flow rate controlling schemes, and most of the controlling schemes work very well and suppress the fingering phenomenon remarkably with the maximum recovery improvement of 15%. As the nonlinearity of the system increases, the schemes may still perform well, but their effectiveness is more pronounced in patterns with less nonlinearity in their constant injection counterpart than those with higher nonlinearity. As the nonlinearity increases, the curvature-driven relaxation time becomes longer than the operational time of the schemes, leading to a reduction in their effectiveness. Additionally, it is shown that employment of the second-order weakly nonlinear stability analysis to formulate the objective function does not result in any remarkable variation in the obtained optimal flow rate schedule.

Flow behavior in a radial Hele-Shaw cell with wettability heterogeneities

Flow behavior in a radial Hele-Shaw cell with wettability heterogeneities

Viscous fingering to fracturing transition in Hele–Shaw flow of shear-thickening fluid

We experimentally investigate the interplay of viscous fingering and fracturing in a radial Hele–Shaw cell displacing a non-Newtonian (shear-thickening) fluid. We have used cornstarch suspension of different compositions (39%–48% w/w dispersed in water–CsCl solution), and the rheology of the suspension exhibits shear thickening behavior beyond a shear rate of 1 s−1. We observe the transition from viscous fingering to dendritic fracturing morphology beyond a critical weight fraction of cornstarch suspension. We analyze the plot of the fraction of injected phase to displaced phase as a function of injected volume for different weight fractions of cornstarch suspension. The injection pressure of the invading fluid (air or oil) used in the present work has no significant effect on the transition from viscous fingering and fracturing patterns. The transition of the pattern is possible if the injection pressure (and consequently the flow rate) is increased. The width of the finger decreases with the injection pressure of the invading fluid and widens with the injected volume due to the reduced local shear rate beyond a critical size. The width of the dendritic fracturing decreases with the injection pressure of invading fluid but increases with the injected volume.

Reactive-infiltration instability in a Hele-Shaw cell influenced by initial aperture and flow rate

We experimentally investigate dissolution morphologies in a radial Hele-Shaw cell. We observe a unique phenomenon where the dissolution front shifts from stable to unstable and uncover the role of buoyancy-driven convection in forming radial stripes (dissolution ``flower'') on the solid surface. We propose a phase diagram to predict the transitions of dissolution patterns.

Emergence of transient reverse fingers during radial displacement of a shear-thickening fluid

A highly sheared dense aqueous suspension of granular cornstarch particles displays rich nonlinear rheology. We had previously demonstrated the growth and onset of interfacial instabilities when shear-thinning cornstarch suspensions were displaced by a Newtonian fluid, and had suggested methods to maximise displacement efficiency [Palak, R. Sathayanath, S. K. Kalpathy and R. Bandyopadhyay, Colloids Surf. A Physicochem. Eng. Asp., 629 (2021) 127405]. In the present work, we explore the miscible displacement of a dense aqueous cornstarch suspension in its discontinuous shear-thickening regime in a quasi-two-dimensional radial Hele-Shaw cell. We systematically study the growth kinetics of the inner interface between water and the cornstarch suspension, and also of the outer interface between the suspension and air. In addition to the growth of interfacial instabilities at the inner interface, we observe a transient withdrawal of the suspension and the formation of fingering instabilities at the outer interface. We demonstrate that these `reverse fingering' instabilities are extremely sensitive to the injection flow rate of water, the gap of the Hele-Shaw cell and the concentration of the displaced cornstarch suspension, {and emerge irrespective of immiscibility between the fluid pair. We believe that as the cornstarch suspension dilates due to the high shear rate imposed by the displacing fluid, the outer suspension-air interface responds with a restoring force, resulting in the penetration of air into the suspension and the formation of reverse fingers. We note that the growth of reverse fingers significantly reduces the displacement efficiency of the suspension. Finally, we demonstrate a correlation in the growth of inner and outer interfacial patterns by computing the velocity with which stresses propagate in the confined dense suspension. Our findings are useful in understanding the flow of granular materials through constrained geometries and can be extended to study stress propagation in shear-thickening materials due to a sudden imposition of high shear rate, such as in impact behaviour.

Control and suppression of viscous fingering displacing non-Newtonian fluid with time-dependent injection strategies

We explore the stabilization mechanism of the fluid–fluid interface in the radial Hele–Shaw cell, displacing a non-Newtonian fluid. It is possible to stabilize the interface following a non-linear injection rate, Q∼t−(2−n)/(2+n), which is related to the displaced fluid rheology (n: power-law index). This suggests the absence of fingering at constant injection when n∼2. We propose a quantitative criterion to control the pattern formation and suppress fingering, through the dimensionless parameter J as a function of the physical and operating parameters, which is applicable for a generalized shear thinning fluid. The parameter J is related to the capillary number in the context of the power-law fluid, relating to the viscous and interfacial forces. The fingering morphology at higher order modes is affected by non-linear effects. The results are non-intuitive, and we have shown a feasible approach toward long term fingering stabilization.

Miscible fluids mixing via alternating injection in a radial Hele-Shaw cell: experimental and numerical studies

Abstract An alternating injection scheme is experimentally employed to study the mixing performance of miscible viscous fluids in a radial Hele-Shaw cell. The concept of the covered area is introduced to quantify the mixing efficiency and various numerical simulations are performed to support and verify the experimental results. It has been observed that the alternating injection can improve the mixing efficiency, depending on the prominence of the channeling interactions. The alternating injection can effectively improve the mixing efficiency under certain conditions, where weaker fingering interactions are dominated by diffusive mixing. However, the results under the high Péclet number conditions are inconsistent, i.e. the mixing efficiency is reduced due to the emergence of orderly viscous fingers prevailing random chaotic interactions. This phenomenon is similar to the channeling interaction in the heterogeneous permeability fields in which less viscous fluid flows through favorable paths due to the local pressure difference between two successively emerging fingers.

Emergent patterns and stable interfaces during radial displacement of a viscoelastic fluid

We explore the interfacial instability that results when a Newtonian fluid (a glycerol-water mixture, inner fluid) displaces a viscoelastic fluid (a dense cornstarch suspension, outer fluid) in a radial Hele-Shaw cell. As the ratio of viscosities of the inner and outer fluids is increased, side branched interfacial patterns are replaced by more stable interfaces that display proportionate growth and finger coalescence. We correlate the average finger spacing with the most dominant wavelength of interfacial instability, computed using a mathematical model that accounts for viscous fingering in miscible Hele-Shaw displacements. The model predictions on the role of viscosity ratio on finger spacing are in close agreement with the experimental observations. Our study lends insight into the significant contribution of the viscoelasticity of the outer fluid on the morphology and growth of interfacial patterns.

Morphological patterns and interface instability during withdrawal of liquid-particle mixtures

HypothesisThe stability of fluid–fluid interface is key to control the displacement efficiency in multiphase flow. The existence of particles can alter the interfacial dynamics and induce various morphological patterns. Moreover, the particle aggregations are expected to have a significant impact on the interface stability and patterns. ExperimentsMonodisperse polyethylene particles of different sizes are uniformly mixed in silicone oil to form the granular mixtures, which are injected into a transparent radial Hele-Shaw cell through different strategies to obtain the homogeneous and inhomogeneous (with particle aggregations) initial states. Subsequently, a systematic study of morphology and interface stability during the withdrawal of granular mixtures is performed. FindingsFor homogeneous mixtures, we observe earlier onset of fingering, more fingers and lower gas saturation at breakthrough than for pure fluid with equivalent viscosity. This effect can be attributed to the particle-induced perturbations. For inhomogeneous mixtures, particle clusters and bands significantly enhance the interface instability. Furthermore, we find that particle deposition due to liquid film entrainment occurs above a critical local flow velocity, and we elucidate the responsible mechanism through force balance analysis and the thin film theory. This work could be of practical significance in geoenergy and industrial applications.

Viscous normal stresses and fingertip tripling in radial Hele-Shaw flows.

Viscous fingering in radial Hele-Shaw cells is markedly characterized by the occurrence of fingertip splitting, where growing fingered structures bifurcate at their tips, via a tip-doubling process. A much less studied pattern-forming phenomenon, which is also detected in experiments, is the development of fingertip tripling, where a finger divides into three. We investigate the problem theoretically, and employ a third-order perturbative mode-coupling scheme seeking to detect the onset of tip-tripling instabilities. Contrary to most existing theoretical studies of the viscous fingering instability, our theoretical description accounts for the effects of viscous normal stresses at the fluid-fluid interface. We show that accounting for such stresses allows one to capture the emergence of tip-tripling events at weakly nonlinear stages of the flow. Sensitivity of fingertip-tripling events to changes in the capillary number and in the viscosity contrast is also examined.

Saffman-Taylor instability in a radial Hele-Shaw cell for a shear-dependent rheological fluid

We present the study of viscous fingering in the radial hele shaw cell, using the shear-dependent rheological fluid which transitions from the Newtonian to shear thinning behaviour, at low shear rates. The experimental observations show that the characteristic features of the fingering phenomena in the case of shear-thinning flow (such as side branching, growth of second generation fingers, inhibition of tip splitting), is observed at late times, from an initial Newtonian regime. On account of the asymmetry of the fingering structure, the local shear rate is non-uniform. This leads to different rheological behaviour in different sections of the network locally, revealing contrasting local fingering patterns, specific to the respective fluid rheological state based on the local shear rate. This is also explained using the numerical simulations for such rheological fluid, in the two-phase porous media flow. The fingering features, such as the length, width are quantified as a function of the flowrate. We have also observed detachment of the fingers beyond a certain critical size of the fingering network. Finally, the stability of the fingering is also investigated at ultra low flow rates, which suggests that the duration of stability exists much beyond what is predicted using the linear stability analysis for classical Newtonian flows.

Entrance effects in a radial Hele-Shaw cell: Numerical and experimental study

Hele-Shaw cells are a frequently used tool in various fields of chemical technology, and in environmental and biomedical engineering. The flow conditions near the inlet of a radial Hele-Shaw cell significantly affect the outcome of its technological applications. The present work combines Computational Fluid Dynamics (CFD) and micro-Particle Image Velocimetry (μPIV) to explain the entrance phenomena, i.e. flow detachment and vortex generation, in radial Hele-Shaw cells. The experiments show that the flow detachment is determined by the inlet flow Reynolds number, Re. Two-dimensional numerical simulations were employed to further investigate the role of the gap width, w to inlet diameter, D aspect ratio, w∕D. The resulting flow regime map is divided by a transitional Re number, Ret, that depends on the aspect ratio. A further parametric study examining how Re and the aspect ratio affect the reattachment length yields an empirical correlation in power-law form. Finally, the impact of the inlet’s geometrical features is briefly examined. The current work can be used as a design guide for future radial HS engineering applications.

Experimental study of liquid-liquid interface oscillating in radial hele-shaw cell

The dynamics of the interface between two immiscible liquids with a high viscosity contrast is studied experimentally under steady displacement of interface and periodic variation of the flow rate of the pumped liquid in radial Hele-Shaw cell. Classic Saffman–Taylor instability, which develops when the viscous fluid is monotonously displaced by the inviscid one, is well known. In the present work, the excitation of Saffman–Taylor instability by means of oscillations of the liquid-liquid interface is demonstrated. The interphase boundary performs axisymmetric radial oscillations at small amplitude of oscillations and in the absence of an average pumping. With the growth of the amplitude of radial oscillations the interface instability is excited, which manifests itself in the development of an azimuthally periodic finger structure during a part of the period. “Finger-like” instability is determined by the relative amplitude of the oscillations of the interphase boundary and under the conditions of the performed experiments depends neither on the oscillation frequency nor on the radial size of the interface.

Explosively driven dynamic compaction of granular media

This paper reports experimental investigations into the dynamic compaction of particle rings subjected to moderate explosions confined in a radial Hele-Shaw cell. The findings reveal marked transitions in the flow regimes corresponding to the evolution of the transient pressure fields inside the granular medium induced by unsteady gas infiltration. As the pressure fields evolve from being localized to diffusive with a substantial reduction in intensity, three sequent flow regimes with distinct rheologies are identified. Specifically, these flow regimes are found to be governed by the localized strong pressure field, then the competition between the diffusive pressure field and wall friction, and finally, solid stresses in the presence of rarefaction waves. A Bingham-type rheology can adequately describe the granular compaction when the pressure gradients remain the dominant driving forces, whereas the frictional nature of the granular flows becomes increasingly significant as the solid stresses set in. As the pressure gradients phase out, rarefaction decompaction commences. However, this only manages to relax the innermost layers of the compacted particles due to a distinctive compressive deformation pattern, giving rise to a discontinuous flow field. These findings shed light on the rheology of dense granular flows subjected to unsteady pressure loadings involving diverse flow–particle and particle–particle interactions.

Pattern formation of the three-layer Saffman-Taylor problem in a radial Hele-Shaw cell

Nonlinear simulations are utilized to investigate the viscous fingering formation in a three-layer radial Hele-Shaw cell. The two interfaces of this system are coupled, and the initial distance between them has a great impact on the interfacial morphologies. This work supplements previous studies, which examined the system via linear and weakly nonlinear analyses.

Control of instability by injection rate oscillations in a radial Hele-Shaw cell

Small spatial perturbations grow into fingers along the unstable interface of a fluid displacing a more viscous fluid in a porous medium or a Hele-Shaw cell. Mitigating this Saffman-Taylor instability increases the efficiency of fluid displacement applications (e.g., oil recovery), whereas amplifying these perturbations is desirable in, e.g., mixing applications. In this work, we investigate the Saffman-Taylor instability through analysis and experiments in which air injected with an oscillatory flow rate outwardly displaces silicone oil in a radial Hele-Shaw cell. A solution for linear instability growth that shows the competing effects of radial growth and surface tension, including wetting effects, is defined given an arbitrary reference condition. We use this solution to define a condition for stability relative to the constant flow rate case and make initial numerical predictions of instability growth by wave number for a variety of oscillations. These solutions are then modified by incorporating reference conditions from experimental data. The morphological evolution of the interface is tracked as the air bubble expands and displaces oil between the plates. Using the resulting images, we analyze and compare the linear growth of perturbations about the mean interfacial radius for constant injection rates with and without superimposed oscillations. Three distinct types of flow rate oscillations are found to modulate experimental linear growth over a constant phase-averaged rate of fluid displacement. In particular, instability growth at the interface is mildly mitigated by adding to the base flow rate provided by a peristaltic pump a second flow with low-frequency oscillations of small magnitude and, to a lesser extent, high-frequency oscillations of large amplitude. In both cases, the increased stability results from the selective suppression of the growth of large wave numbers in the linear regime. Contrarily, intermediate oscillations consistently destabilize the interface and significantly amplify the growth of the most unstable wave numbers of the constant flow rate case. Numerical predictions of low-frequency oscillations of opposite sign (initially decreasing) show promise of even greater mitigation of linear instability growth than that observed in this investigation. Looking forward, proper characterization of the unsteady, wetting, and nonlinear dynamics of instability growth will give further insight into the efficacy of oscillatory injection rates.