Rafael M. Oliveira

Viscous fingering instabilities reduce the efficiency of immiscible displacement processes in radial Hele-Shaw flows. Time-dependent injection protocols provide a practical means of mitigating these instabilities, and previous studies have shown that linearly increasing injection rates can strongly suppress finger growth when both the total injected volume and the total injection time are fixed. Here we reformulate this optimization problem by prescribing a fixed radial distance and terminating the injection when the most advanced finger reaches it, a condition inspired by breakthrough in displacement processes. We then introduce a linear-plateau injection protocol, in which the injection rate increases linearly up to a switching time t_{m} and subsequently remains constant at a maximum value Q_{max}, reflecting the finite capacity of pumping systems. Using linear stability analysis and fully nonlinear boundary-integral simulations, we show that this linear-plateau protocol stabilizes the interface and increases the swept area, even though the interface evolves for a longer time than in a reference protocol with constant injection Q_{max} throughout the process. Finally, we introduce a profitlike function that balances the gain in swept area against the cost associated with longer injection times. This formulation is motivated by potential applications of the proposed control strategy to displacement processes in porous media, such as enhanced oil recovery and CO_{2} injection for geological storage. Optimization of this function identifies the switching time that provides the best trade-off between displacement efficiency and injection duration. These results establish a Hele-Shaw framework for optimizing injection protocols under a swept-area-injection-time trade-off while accounting for a maximum allowable injection rate.

We investigate the dynamic and morphological behaviors of a ferrofluid drop initially located at the center of a Hele–Shaw cell, under the influence of an off-centered radial magnetic field. Previous studies have demonstrated that when the center of the cell coincides with the center of the radial magnetic field, rotationally symmetric, stationary, starlike ferrofluid shapes may arise. In this work, we employ a perturbative, second-order mode-coupling approach to examine the changes in the emergent ferrofluid structures when the radial magnetic field is shifted off-center. Our results show that the symmetry and location of the applied radial magnetic field play a key role in determining the shape and dynamical response of the ferrofluid droplet, with the off-centered radial field leading to nonrotationally symmetric interfacial patterns that drift away from the magnetic field's center. Our early nonlinear findings indicate that stationary shapes only develop when the field's center is aligned with the center of the Hele–Shaw cell (i.e., with the center of the originally circular ferrofluid drop). We have also found that small misalignments between the center of the droplet and the center of the radial field can induce important changes in the pattern formation process, including modifications in the number of fingered protrusions formed. The influence of the ferrofluid's magnetic susceptibility on the interfacial shapes and drift velocity and the impact of random perturbations in the initial conditions are also scrutinized.

The lifting Hele-Shaw cell flow commonly involves the stretching of a viscous oil droplet surrounded by air, in the confined space between two parallel plates. As the upper plate is lifted, viscous fingering instabilities emerge at the air-oil interface. Such an interfacial instability phenomenon is widely observed in numerous technological and industrial applications, being quite difficult to control. Motivated by the recent interest in controlling and stabilizing the Saffman-Taylor instability in lifting Hele-Shaw flows, we propose an alternative way to restrain the development of interfacial disturbances in this gap-variable system. Our method modifies the traditional plate-lifting flow arrangement by introducing a finite fluid annulus layer encircling the central oil droplet, and separating it from the air. A second-order, perturbative mode-coupling approach is employed to analyze morphological and stability behaviors in this three-fluid, two-interface, doubly connected system. Our findings indicate that the intermediate fluid ring can significantly stabilize the interface of the central oil droplet. We show that the effectiveness of this stabilization protocol relies on the appropriate choice of the ring's viscosity and thickness. Furthermore, we calculate the adhesion force required to detach the plates, and find that it does not change significantly with the addition of the fluid envelope as long as it is sufficiently thin. Finally, we detect no distinction in the adhesion force computed for stable or unstable annular interfaces, indicating that the presence of fingering at the ring's boundaries has a negligible effect on the adhesion force.

The displacement of a fluid by another less viscous one in isotropic quasi-two-dimensional Hele–Shaw cells typically leads to viscous fingering patterns characterized by repeated tip-splitting and side-branching. When anisotropy is present, the viscous fingering patterns are replaced by dendritic instability. In isotropic cells, Couder et al. [“Dendritic growth in the Saffman-Taylor experiment,” Europhys. Lett. 2, 437 (1986)] induced the growth of a dendritic finger by placing a small isolated bubble in contact with the tip of a finger. Moreover, in rectangular displacements, Kopf-Sill and Homsy [“Narrow fingers in a Hele–Shaw cell,” Phys. Fluids 30, 2607–2609 (1987)] observed for specific ranges of modified capillary numbers, Ca, that narrow fingers develop into dendritic patterns after carefully cleaning the Hele–Shaw cell with a soap solution. In the current work, by injecting air to radially displace a lubricant oil with strong adsorption properties, we observe for the first time the growth of dendritic instability in displacements confined in isotropic Hele–Shaw cells in the absence of air bubbles driving the fingertip. These structures are driven by the formation of capillary waves inside the finger adjacent to the Hele–Shaw surfaces. Furthermore, we compare displacements of different materials at similar high Ca values and obtain significantly different morphological results, indicating the pattern formation may not scale with this parameter in this range. Discrepancies are explained based on both wetting and inertia effects.

We conduct nonlinear simulations to investigate the radial growth of viscous fingers in a Hele–Shaw cell by performing a wide parametric study on the two dimensionless parameters that govern the dynamics. These are the viscosity contrast, A, and the effective surface tension, B, which compares the influence of surface tension with the injection rate that drives instability. It is well known that the surface tension between the fluids has a stabilizing influence on pattern formation leading to slower fingering growth and a reduced number of fingers. Moreover, the current results show that, for fixed A values, larger values of the effective surface tension lead to the formation of fingering patterns that are similar to each other. As such, the calculated interfacial shapes are larger and take longer to develop. Furthermore, the calculated shapes offer a visually striking portrait of instability evolution, and comparison to our experimental measurements highlights the close resemblance between the two with a few discrepancies pointed out. Finally, we observe how changes to the initial nearly circular interface affect both the morphology of viscous fingers and the timescale for pattern formation.

The growth of viscous fingers in the radial displacement of a yield strength material confined between the plates of a Hele–Shaw cell is investigated. The apparatus is filled with an aqueous solution of Carbopol® before air is injected to start the displacement process. In addition to striking fingering patterns, we identify unyielded residuals of the Carbopol solution arrested on the plates' surfaces with the assistance of digital mobile microscopes placed above the top plate. These unyielded residuals are subjected to slip conditions on the surface walls and appear in different forms. The experimental observations are correlated with the wall slip behavior detected in rheometric measurements, i.e., observed in the flow curve for shear rates below a critical value. This correlation provides an estimate of a critical propagating radius beyond which shear rates drop to values lower than the critical one, and the influence of wall slip becomes significant. We observe that these residuals are uniformly distributed and appear as thin films where the radii are smaller than the critical value and the wall slip is minimum. However, in locations where the radii are larger than the critical one, the residuals turn into isolated blobs of different sizes, which may propagate in a stick-slip motion radially downstream inside the air fingers. In addition, we observe that the morphology of residuals depends on the gap width between the plates, the injection rate of the invading air, the yield strength of the Carpobol solution, and the wettability conditions of the surface walls.

We analyze the morphology and dynamic behavior of the interface separating a ferrofluid and a nonmagnetic fluid in a Hele-Shaw cell, when crossed radial and azimuthal magnetic fields are applied. In addition to inducing the formation of a variety of eye-catching, complex interfacial structures, the action of the crossed fields makes the deformed ferrofluid droplet to rotate. Numerical simulations and perturbative mode-coupling theory are employed to look into early linear, intermediate weakly nonlinear, and fully nonlinear dynamic regimes of the pattern-forming process. We investigate how the system responds to variations in the viscosity difference between the fluids, the magnetic susceptibility of the ferrofluid, the effects of surface tension, and in the relative strength between radial and azimuthal applied magnetic fields. The role played by random perturbations at the initial conditions in determining the ultimate shape and dynamic stability of the spinning ferrofluid patterns is also studied.

We investigate the onset of instability in a stably stratified two-component fluid in a vertical Hele-Shaw cell when the unstably stratified scalar has a settling velocity. This linear stability problem is analysed on the basis of Darcy’s law, for constant-gradient base states. The settling velocity is found to trigger a novel instability mode characterized by pairs of inclined waves. For unequal diffusivities, this new settling-driven mode competes with the traditional double-diffusive mode. Below a critical value of the settling velocity, the double-diffusive elevator mode dominates, while, above this threshold, the inclined waves associated with the settling-driven instability exhibit faster growth. The analysis yields neutral stability curves and allows for the discussion of various asymptotic limits.