Time-dependent gap Hele-Shaw cell with a ferrofluid: evidence for an interfacial singularity inhibition by a magnetic field

Fingering stabilization and adhesion force in the lifting flow with a fluid annulus

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.

Role of interfacial rheology on fingering instabilities in lifting Hele-Shaw flows

The lifting Hele-Shaw cell setup is a popular modification of the classic, fixed-gap, radial viscous fingering problem. In the lifting cell configuration, the upper cell plate is lifted such that a more viscous inner fluid is invaded by an inward-moving outer fluid. As the fluid-fluid interface contracts, one observes the rising of distinctive patterns in which penetrating fingers having rounded tips compete among themselves, reaching different lengths. Despite the scholarly and practical relevance of this confined lifting flow problem, the impact of interfacial rheology effects on its pattern-forming dynamics has been overlooked. Authors of recent studies on the traditional injection-induced radial Hele-Shaw flow and its centrifugally driven variant have shown that, if the fluid-fluid interface is structured (i.e., laden with surfactants, particles, proteins, or other surface-active entities), surface rheological stresses start to act, influencing the development of the viscous fingering patterns. In this paper, we investigate how interfacial rheology affects the stability as well as the shape of the emerging fingered structures in lifting Hele-Shaw flows, at linear and early nonlinear dynamic stages. We tackle the problem by utilizing the Boussinesq-Scriven model to describe the interface and by employing a perturbative mode-coupling scheme. Our linear stability results show that interfacial rheology effects destabilize the interface. Furthermore, our second-order findings indicate that interfacial rheology significantly alters intrinsically nonlinear morphological features of the shrinking interface, inducing the formation of narrow sharp-tip penetrating fingers and favoring enhanced competition among them.

Delayed Hopf bifurcation and control of a ferrofluid interface via a time-dependent magnetic field

A ferrofluid droplet confined in a Hele-Shaw cell can be deformed into a stably spinning "gear," using crossed magnetic fields. Previously, fully nonlinear simulation revealed that the spinning gear emerges as a stable traveling wave along the droplet's interface bifurcates from the trivial (equilibrium) shape. In this work, a center manifold reduction is applied to show the geometrical equivalence between a two-harmonic-mode coupled system of ordinary differential equations arising from a weakly nonlinear analysis of the interface shape and a Hopf bifurcation. The rotating complex amplitude of the fundamental mode saturates to a limit cycle as the periodic traveling wave solution is obtained. An amplitude equation is derived from a multiple-time-scale expansion as a reduced model of the dynamics. Then, inspired by the well-known delay behavior of time-dependent Hopf bifurcations, we design a slowly time-varying magnetic field such that the timing and emergence of the interfacial traveling wave can be controlled. The proposed theory allows us to determine the time-dependent saturated state resulting from the dynamic bifurcation and delayed onset of instability. The amplitude equation also reveals hysteresislike behavior upon time reversal of the magnetic field. The state obtained upon time reversal differs from the state obtained during the initial (forward-time) period, yet it can still be predicted by the proposed reduced-order theory.

Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons

We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted non-uniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports travelling waves, governed by a novel modified Kuramoto–Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equation allows for the existence of dissipative solitons. These permanent travelling waves’ propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The travelling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the travelling waves. Interestingly, multi-periodic waves, which are a non-integrable analogue of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multi-periodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified.

Tuning a magnetic field to generate spinning ferrofluid droplets with controllable speed via nonlinear periodic interfacial waves

Two-dimensional free surface flows in Hele-Shaw configurations are a fertile ground for exploring nonlinear physics. Since Saffman and Taylor's work on linear instability of fluid-fluid interfaces, significant effort has been expended to determining the physics and forcing that set the linear growth rate. However, linear stability does not always imply nonlinear stability. We demonstrate how the combination of a radial and an azimuthal external magnetic field can manipulate the interfacial shape of a linearly unstable ferrofluid droplet in a Hele-Shaw configuration. We show that weakly nonlinear theory can be used to tune the initial unstable growth. Then, nonlinearity arrests the instability and leads to a permanent deformed droplet shape. Specifically, we show that the deformed droplet can be set into motion with a predictable rotation speed, demonstrating nonlinear traveling waves on the fluid-fluid interface. The most linearly unstable wave number and the combined strength of the applied external magnetic fields determine the traveling wave shape, which can be asymmetric.

Azimuthal field instability in a confined ferrofluid

We report the development of interfacial ferrohydrodynamic instabilities when an initially circular bubble of a nonmagnetic inviscid fluid is surrounded by a viscous ferrofluid in the confined geometry of a Hele-Shaw cell. The fluid-fluid interface becomes unstable due to the action of magnetic forces induced by an azimuthal field produced by a straight current-carrying wire that is normal to the cell plates. In this framework, a pattern formation process takes place through the interplay between magnetic and surface tension forces. By employing a perturbative mode-coupling approach we investigate analytically both linear and intermediate nonlinear regimes of the interface evolution. As a result, useful analytical information can be extracted regarding the destabilizing role of the azimuthal field at the linear level, as well as its influence on the interfacial pattern morphology at the onset of nonlinear effects. Finally, a vortex sheet formalism is used to access fully nonlinear stationary solutions for the two-fluid interface shapes.

Adhesion force in fluids: effects of fingering, wetting, and viscous normal stresses

Probe-tack measurements evaluate the adhesion strength of viscous fluids confined between parallel plates. This is done by recording the adhesion force that is required to lift the upper plate, while the lower plate is kept at rest. During the lifting process, it is known that the interface separating the confined fluids is deformed, causing the emergence of intricate interfacial fingering structures. Existing meticulous experiments and intensive numerical simulations indicate that fingering formation affects the lifting force, causing a decrease in intensity. In this work, we propose an analytical model that computes the lifting adhesion force by taking into account not only the effect of interfacial fingering, but also the action of wetting and viscous normal stresses. The role played by the system's spatial confinement is also considered. We show that the incorporation of all these physical ingredients is necessary to provide a better agreement between theoretical predictions and experiments.

Determining the number of fingers in the lifting Hele-Shaw problem

The lifting Hele-Shaw cell flow is a variation of the celebrated radial viscous fingering problem for which the upper cell plate is lifted uniformly at a specified rate. This procedure causes the formation of intricate interfacial patterns. Most theoretical studies determine the total number of emerging fingers by maximizing the linear growth rate, but this generates discrepancies between theory and experiments. In this work, we tackle the number of fingers selection problem in the lifting Hele-Shaw cell by employing the recently proposed maximum-amplitude criterion [Dias and Miranda, Phys. Rev. E 88, 013016 (2013)]. Our linear stability analysis accounts for the action of capillary, viscous normal stresses, and wetting effects, as well as the cell confinement. The comparison of our results with very precise laboratory measurements for the total number of fingers shows a significantly improved agreement between theoretical predictions and experimental data.

Taper-induced control of viscous fingering in variable-gap Hele-Shaw flows

Variable-gap Hele-Shaw flows consider viscous fluid displacements resulting from the lifting or squeezing of the upper cell plate, while the lower plate remains at rest. Conventionally, researchers focus on the situation in which the cell plates are perfectly parallel. We study a slightly different version of the problem, where the upper plate is gently inclined so that the plates are no longer parallel. Within this tapered Hele-Shaw cell context we examine how the presence of such a small gap gradient affects the stability properties of the fluid-fluid interface. Linear stability analysis indicates that the existence of the taper offers a simple geometric way to control the development of interfacial fingering instabilities under both lifting and squeeze flow circumstances.

Ferrofluid patterns in a radial magnetic field: linear stability, nonlinear dynamics, and exact solutions

The response of a ferrofluid droplet to a radial magnetic field is investigated, when the droplet is confined in a Hele-Shaw cell. We study how the stability properties of the interface and the shape of the emerging patterns react to the action of the magnetic field. At early linear stages, it is found that the radial field is destabilizing and determines the growth of fingering structures at the interface. In the weakly nonlinear regime, we have verified that the magnetic field favors the formation of peaked patterned structures that tend to become sharper and sharper as the magnitude of the magnetic effects is increased. A more detailed account of the pattern morphology is provided by the determination of nontrivial exact stationary solutions for the problem with finite surface tension. These solutions are obtained analytically and reveal the development of interesting polygon-shaped and starfishlike patterns. For sufficiently large applied fields or magnetic susceptibilities, pinch-off phenomena are detected, tending to occur near the fingertips. We have found that the morphological features obtained from the exact solutions are consistent with our linear and weakly nonlinear predictions. By contrasting the exact solutions for ferrofluids under radial field with those obtained for rotating Hele-Shaw flows with ordinary nonmagnetic fluids, we deduce that they coincide in the limit of very small susceptibilities.

Confined ferrofluid droplet in crossed magnetic fields

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.

Fingering patterns in the lifting flow of a confined miscible ferrofluid

Miscible flow displacements of a ferrofluid droplet subjected to various magnetic field configurations and confined in a time-dependent gap Hele-Shaw cell are examined through highly accurate numerical simulations. The interplay between lifting, miscibility, and applied magnetic fields resulted in complex interfacial pattern formation. By varying the symmetry properties of the applied magnetic fields and by considering the action of Korteweg stresses, a number of interesting droplet morphologies are identified and characterized. The possibility of controlling the degree of fluid mixing and the ultimate shape of the emerging patterns by appropriately adjusting the strength of the applied magnetic fields is also discussed.

Stretching of a confined ferrofluid: influence of viscous stresses and magnetic field

An analytical investigation is presented for the stretch flow of a viscous Newtonian ferrofluid highly confined between parallel plates. We focus on the development of interfacial instabilities when the upper plate is lifted at a described rate, under the action of an applied magnetic field. We derive the mode-coupling differential equation for the interface perturbation amplitudes and study both linear and nonlinear flow regimes. In contrast to the great majority of works in stretch flow we take into account stresses originated from velocity gradients normal to the ferrofluid interface. The impact of such normal stresses is accounted for through a modified Young-Laplace pressure jump interfacial boundary condition, which also includes the contribution from magnetic normal traction. We study how the stability properties of the interface and the shape of the emerging patterns respond to the combined action of normal stresses and magnetic field, both in the presence and absence of surface tension. We show that the inclusion of normal viscous stresses introduces a pertinent dependence on the initial aspect ratio, indicating that the number of fingers formed would be overestimated if such stresses are not taken into account. At early linear stages it is found that such stresses regularize the system, acting as an effective interfacial tension. At weakly nonlinear stages we verified that normal stresses reduce finger competition, which can be completely suppressed with the assistance of an azimuthal magnetic field. We have also found that the magnetic normal traction introduces a purely nonlinear contribution to the problem, revealing the key role played by the magnetic susceptibility in the control of finger competition.

Viscosity contrast effects on fingering formation in rotating Hele-Shaw flows

The different finger morphologies that arise at the interface separating two immiscible fluids in a rotating Hele-Shaw cell are studied numerically. The whole range of viscosity contrast is analyzed and a variety of fingering patterns systematically introduced, including the case in which the inner fluid is less viscous than the outer one. Our numerical results demonstrate that both the magnitude and the sign of the viscosity contrast strongly affect the shape of the emerging fingers, and also their length distribution. We have also found that the occurrence and location of pinch-off singularities are remarkably modified when the inner fluid is less viscous: instead of generating an isolated detaching drop, a full finger is disconnected from the interface. Finally, we have verified that the finger competition phenomena revealed by our simulations are correctly predicted by a weakly nonlinear analysis of the pattern development, showing that such important finger competition dynamics is already set at relatively early stages of interfacial evolution.

Numerical study of miscible fingering in a time-dependent gap Hele-Shaw cell

We perform a detailed numerical study of the evolution of a miscible fluid droplet in a time-dependent gap Hele-Shaw cell. The development of the emerging fingering instabilities is systematically analyzed by intensive and highly accurate numerical simulations. We focus on the influence of three relevant physical parameters on the interface dynamics: the Pélclet number Pe, the viscosity contrast A, and the Korteweg stress parameter delta. Consistently with conventional miscible Saffman-Taylor studies in constant-gap Hele-Shaw cells, our results demonstrate that more vigorous fingering is observed at higher Pe and larger A. Concerning the specific role of Pe and A, we deduce two general results: higher Péclet number favors branching around a nearly circular region (which leads to longer interfacial lengths); while larger viscosity contrast results in more significant finger penetrations (which is quantitatively expressed by larger diameter of gyration). We have also verified that the Korteweg stress parameter delta does act as an effective interfacial tension: it stabilizes the miscible interface, leading to fingering patterns that present a greater resemblance with the structures obtained in similar immiscible situations. Finally, we have identified the development of a visually striking phenomenon in the limit of high Pe, large A , and relatively small delta: some outward fingers pinch, and subsequent droplet detachment is observed. We show that such a droplet detachment process can be prevented by the action of stronger interfacial stresses. This last finding provides additional evidence for the claim that the Korteweg stresses can be treated as an ersatz interfacial tension in diffusing fluids.

Finger competition dynamics in rotating Hele-Shaw cells

We report analytical results for the development of interfacial instabilities in rotating Hele-Shaw cells. We execute a mode-coupling approach to the problem and examine the morphological features of the fluid-fluid interface at the onset of nonlinear effects. The impact of normal stresses is accounted for through a modified pressure jump boundary condition. A differential equation describing the early nonlinear evolution of the interface is derived, being conveniently written in terms of three relevant dimensionless parameters: viscosity contrast A , surface tension B , and gap spacing b . We focus our study on the influence of these parameters on finger competition dynamics. It is deduced that the link between finger competition and A , B , and b can be revealed by a mechanism based on the enhanced growth of subharmonic perturbations. Our results show good agreement with existing experimental and numerical investigations of the problem both in low and high A<0 limits. In particular, it is found that the condition of vanishing A suppresses the dynamic competition between fingers, regardless of the value of B and b . Moreover, our study enables one to extract analytical information about the problem by exploring the whole range of allowed values for A , B , and b . Specifically, it is verified that pattern morphology is significantly modified when the viscosity contrast -1< or =A< or =1 varies: increasingly larger values of A>0 (A<0) lead to enhanced competition of outward (inward) fingers. Within this context the role of B and b in determining different finger competition behaviors is also discussed.