Systematic weakly nonlinear analysis of radial viscous fingering

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.

Effect of interfacial rheology on fingering patterns in rotating Hele-Shaw cells

In rotating Hele-Shaw flows, centrifugal force acts, and the interface separating two viscous fluids becomes unstable, driven by the density difference between them. Complex interfacial structures develop where fingers of various shapes and sizes grow, and compete. These patterns have been well studied over the last few decades, analytically, numerically, and experimentally. However, one feature of the pattern-forming dynamics of much current interest has been underappreciated: the role of surface rheological stresses in the deformation, and time evolution of the fluid-fluid interface. In this paper, we employ a perturbative, second-order mode-coupling analysis to investigate how interfacial rheology effects influence centrifugally driven fingering phenomena, beyond the scope of linear stability theory. Describing the viscous Newtonian interface by using a Boussinesq-Scriven model, we derive a nonlinear differential equation that governs the early linear, and nonlinear time evolution of the system. In this framing, the most prevalent dynamical features of the patterns are described in terms of two dimensionless parameters: the viscosity contrast A (dimensionless viscosity difference between the fluids), and the Boussinesq number Bq which involves a ratio between interfacial and bulk viscosities. At the linear level, our results show that for a given A, surface rheological stresses dictated by Bq have a stabilizing role. Nevertheless, our weakly nonlinear findings reveal a more elaborate scenario in which the shape of the fingers, and their finger competition behavior result from the coupled influence of A and Bq. It is found that, although finger competition phenomena depend on the specific values of A and Bq, the fingers tend to widen as Bq is increased, regardless of the value of A.

Impact of interfacial rheology on finger tip splitting

Fluid-fluid interfaces, laden with polymers, surfactants, lipid bilayers, proteins, solid particles, or other surface-active agents, often exhibit a rheologically complex response to deformations. Despite its academic and practical relevance to fluid dynamics and various other fields of research, the role of interfacial rheology in viscous fingering remains fairly underexplored. A noteworthy exception is the work by Li and Manikantan [Phys. Rev. Fluids 6, 074001 (2021)2469-990X10.1103/PhysRevFluids.6.074001], who used linear stability analysis to show that surface rheological stresses act to stabilize the development of radial viscous fingering at the linear regime. In this paper, we perform a perturbative, second-order mode-coupling analysis of the system and investigate the influence of interfacial rheology on the morphology of the fingering structures at early nonlinear stages of the dynamics. In particular, we focus on understanding how interfacial rheology impacts the emblematic finger tip-widening and finger tip-splitting phenomena that take place in radial viscous fingering in Hele-Shaw cells. We describe the viscous Newtonian fluid-fluid interface by using a Boussinesq-Scriven model, and derive a generalized Young-Laplace pressure jump condition at the fluid-fluid interface. In this framing, we go beyond the purely linear description and use Darcy's law to obtain a perturbative mode-coupling differential equation which describes the time evolution of the perturbation amplitudes, accurate to second order. Our early nonlinear mode-coupling results indicate that regardless of their stabilizing action at the linear regime, interfacial rheology effects favor finger tip widening, leading to the occurrence of enhanced finger tip-splitting events.

Field-controlled flow and shape of a magnetorheological fluid annulus

We investigate the behavior of a magnetorheological (MR) fluid annulus, bounded by a nonmagnetic fluid and confined in a Hele-Shaw cell, under the simultaneous effect of in-plane, external radial and azimuthal magnetic fields. A second-order mode-coupling theory is used to study the early nonlinear stage of the pattern-forming dynamics. We examine changes in the morphology of the MR fluid annular structure as a function of its magnetic-field-tunable rheological properties, as well as the combined magnetic field's intensities, and thickness of the ring. Our weakly nonlinear perturbative results show that, depending on the system control parameters, the MR fluid annulus adopts various stationary shapes. These equilibrium annular structures present slightly bent, asymmetric fingered protrusions which may emerge on the inner, outer, or even on both boundaries of the magnetic fluid ring. On top of these morphological changes, we find that the resulting permanent shape patterns rotate with a well defined angular velocity. We focus on analyzing how the overall shape of the fingered patterns, in particular their sharpness and asymmetric form, as well as the number of resulting fingers are impacted by the magnetic-field-dependent yield stress of the MR fluid annulus. The influence of the magnetically controlled rheological properties of the MR fluid on the angular velocity of the rotating annulus is also scrutinized.

Magnetically induced interfacial instabilities in a ferrofluid annulus

We investigate the flow of a viscous ferrofluid annulus surrounded by two nonmagnetic fluids in a Hele-Shaw cell when subjected to an external radial magnetic field. The interfacial pattern formation dynamics of the system is determined by the interplay of magnetic and surface tension forces acting on the inner and outer boundaries of the annulus, favoring the coupling of the disjoint interfaces. Mode-coupling analysis is employed to examine both linear and weakly nonlinear stages of the flow. Linear stability analysis indicates that the trailing and leading annular boundaries are coupled already at the linear regime, revealing that perturbations arising in the outer interface may induce the emergence of deformed structures in the inner boundary. Moreover, second-order weakly nonlinear analysis is utilized to identify key nonlinear morphological features of the ferrofluid annulus. Our theoretical results show that linear, n-fold symmetric annular patterns having rounded edges are replaced by nonlinear polygonal-like shapes, presenting fairly sharp fingers. It is found that, as opposed to the linear patterns, the nonlinear peaky structures reach a stationary state, characterized by a growth saturation process induced by nonlinear effects. Furthermore, the response of the ferrofluid ring to changes in the thickness of the annulus, in the relative strength of magnetic and surface tensions forces, as well as in the magnetic susceptibility of the ferrofluid material, are also discussed.

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.

Mode-coupling approach to near-cuspidal patterns in planar fluid flows

We investigate the evolution of the interface separating two Newtonian fluids of different viscosities in two-dimensional Stokes flow driven by suction or injection. A second-order, mode-coupling theory is used to explore key morphological aspects of the emerging interfacial patterns in the stage of the flow that bridges the purely linear and fully nonlinear regimes. In the linear regime, our analysis reveals that an injection-driven expanding interface is stable, while a contracting motion driven by suction is unstable. Moreover, we find that the linear growth rate associated with this suction-driven instability is independent of the viscosity contrast between the fluids. However, second-order results tell a different story, and show that the viscosity contrast is crucial in determining the morphology of the interface. Our theoretical description is applicable to the entire range of viscosity contrasts, and provides insights on the formation of near-cusp pattern-forming structures. Reproduction of fully nonlinear, n-fold symmetric near-cuspidal shapes previously obtained through conformal mapping techniques substantiates the validity of our mode-coupling approach.

Control of viscous fingering through variable injection rates and time-dependent viscosity fluids: Beyond the linear regime

During the past few years, researchers have been proposing time-dependent injection strategies for stabilizing or manipulating the development of viscous fingering instabilities in radial Hele-Shaw cells. Most of these studies investigate the displacement of Newtonian fluids and are entirely based on linear stability analyses. In this work, linear stability theory and variational calculus are used to determine closed-form expressions for the proper time-dependent injection rates Q(t) required to either minimize the interface disturbances or to control the number of emerging fingers. However, this is done by considering that the displacing fluid is non-Newtonian and has a time-varying viscosity. Moreover, a perturbative third-order mode-coupling approach is employed to examine the validity and effectiveness of the controlling protocols dictated by these Q(t) beyond the linear regime and at the onset of nonlinearities.

Kinetic undercooling in Hele-Shaw flows

A central topic in Hele-Shaw flow research is the inclusion of physical effects on the interface between fluids. In this context, the addition of surface tension restrains the emergence of high interfacial curvatures, while consideration of kinetic undercooling effects inhibits the occurrence of high interfacial velocities. By connecting kinetic undercooling to the action of the dynamic contact angle, we show in a quantitative manner that the kinetic undercooling contribution varies as a linear function of the normal velocity at the interface. A perturbative weakly nonlinear analysis is employed to extract valuable information about the influence of kinetic undercooling on the shape of the emerging fingered structures. Under radial Hele-Shaw flow, it is found that kinetic undercooling delays, but does not suppress, the development of finger tip-broadening and finger tip-splitting phenomena. In addition, our results indicate that kinetic undercooling plays a key role in determining the appearance of tip splitting in rectangular Hele-Shaw geometry.

Interfacial patterns in magnetorheological fluids: Azimuthal field-induced structures

Despite their practical and academic relevance, studies of interfacial pattern formation in confined magnetorheological (MR) fluids have been largely overlooked in the literature. In this work, we present a contribution to this soft matter research topic and investigate the emergence of interfacial instabilities when an inviscid, initially circular bubble of a Newtonian fluid is surrounded by a MR fluid in a Hele-Shaw cell apparatus. An externally applied, in-plane azimuthal magnetic field produced by a current-carrying wire induces interfacial disturbances at the two-fluid interface, and pattern-forming structures arise. Linear stability analysis, weakly nonlinear theory, and a vortex sheet approach are used to access early linear and intermediate nonlinear time regimes, as well as to determine stationary interfacial shapes at fully nonlinear stages.

Viscous fluid fingering on a negatively curved surface

Viscous fingering formation in flat Hele-Shaw cells is a classical and widely studied fluid mechanical problem. We examine the development of viscous fluid fingering on a two-dimensional surface of constant negative Gaussian curvature, the hyperbolic plane H(2). A perturbative mode-coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system: fingertip splitting and finger competition. We also report on a time-dependent control strategy placed on the injection rate, which is able to minimize viscous fingering growth on H(2).

Field-induced patterns in confined magnetorheological fluids

We study the behavior of a magnetorheological fluid droplet confined to a Hele-Shaw cell in the presence of an applied radial magnetic field. Interfacial pattern formation is investigated by considering the competition among capillary, viscoelastic, and magnetic forces. The contribution of a magnetic field-dependent yield stress is taken into account. Linear stability analysis reveals the stabilizing role played by yield stress. On the other hand, a mode-coupling approach predicts that the resulting fingering structures should become less and less sharp as yield stress effects are increased. By employing a vortex-sheet formalism we have been able to identify a family of exact stationary solutions of the problem, unveiling the development of swollen polygonal patterns. A suggestive magnetically controlled shape transition in which the edges of the patterns change from convex to concave has been also identified.

Experimental investigation of the onset of instability in a radial Hele-Shaw cell

The initial stage of interface instability upon radial displacement of a fluid in a Hele-Shaw cell is investigated. An air-silicone oil system is analyzed. The critical radii of stability relative to long-wave perturbations are determined. It is found that, in the investigated range of parameters, instability most often begins by a translational mechanism. It is ascertained that in the overwhelming majority of cases the critical radii of instability are smaller than the values predicted by the linear stability theory and external effects make this difference even greater. The obtained results are discussed and compared with the existing theories.

Pattern formation and interface pinch-off in rotating Hele-Shaw flows: a phase-field approach

Viscous fingering dynamics driven by centrifugal forcing is studied for arbitrary viscosity contrast. Theoretical methods, including exact solutions, and numerics based on a phase-field approach are used. Both confirm that pinch-off singularities in patterns originated from the centrifugally driven instability may occur spontaneously and be inherent to the two-dimensional Hele-Shaw dynamics. They are systematically more frequent for lower viscosity contrasts consistently with experimental evidence. The analytical insights provide an interpretation of this fact in terms of the asymptotic matching of the different regions of the fingering patterns. The phase-field numerical scheme is shown to be particularly adequate to elucidate the existence of finite-time singularities through the dependence of the singularity time on the interface thickness, in particular for varying viscosity contrast.

Pinch-off singularities in rotating Hele-Shaw flows at high viscosity contrast

We study the evolution of a family of dumbbell-shaped liquid patches surrounded by air inside a rotating Hele-Shaw cell with lubrication methods and numerical simulations. Depending on initial conditions, the dumbbell either stretches to infinity, pinches off at the neck to form a droplet, or collects into a circular drop at the center of rotation. Whether or not pinch-off occurs results from a subtle interplay between centrifugal and capillary forces. In particular, rotation may delay or even prevent pinch-off from occurring owing to stretching and smoothing of the fluid neck. However, frequently rotation may have the opposite effect leading to pinch-off where the relaxation toward a circular drop would be observed in an ordinary Hele-Shaw cell.

Coriolis effects on fingering patterns under rotation

The development of immiscible viscous fingering patterns in a rotating Hele-Shaw cell is investigated. We focus on understanding how the time evolution and the resulting morphologies are affected by the action of the Coriolis force. The problem is approached analytically and numerically by employing a vortex sheet formalism. The vortex sheet strength and a linear dispersion relation are derived analytically, revealing that the most relevant Coriolis force contribution comes from the normal component of the averaged interfacial velocity. It is shown that this normal velocity, uniquely due to the presence of the Coriolis force, is responsible for the complex-valued nature of the linear dispersion relation making the linear phases vary with time. Fully nonlinear stages are studied through intensive numerical simulations. A suggestive interplay between inertial and viscous effects is found, which modifies the dynamics, leading to different pattern-forming structures. The inertial Coriolis contribution plays a characteristic role: it generates a phase drift by deviating the fingers in the sense opposite to the actual rotation of the cell. However, the direction and intensity of finger bending is predominantly determined by viscous effects, being sensitive to changes in the magnitude and sign of the viscosity contrast. The finger competition behavior at advanced time stages is also discussed.

Radial viscous fingering in miscible Hele-Shaw flows: a numerical study

A modified version of the usual viscous fingering problem in a radial Hele-Shaw cell with immiscible fluids is studied by intensive numerical simulations. We consider the situation in which the fluids involved are miscible, so that the diffusing interface separating them can be driven unstable through the injection or suction of the inner fluid. The system is allowed to rotate in such a way that centrifugal and Coriolis forces come into play, imposing important changes on the morphology of the arising patterns. In order to bridge from miscible to immiscible pattern forming structures, we add the surface tensionlike effects due to Korteweg stresses. Our numerical experiments reveal a variety of interesting fingering behaviors, which depend on the interplay between injection (or suction), diffusive, rotational, and Korteweg stress effects. Whenever possible the features of the simulated miscible fronts are contrasted to existing experiments and other theoretical or numerical studies, usually resulting in close agreements. A number of additional complex morphologies, whose experimental realization is still not available, are predicted and discussed.

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.

Dynamics of viscous fingers in rotating Hele-Shaw cells with Coriolis effects

A growing number of experimental and theoretical works have been addressing various aspects of the viscous fingering formation in rotating Hele-Shaw cells. However, only a few of them consider the influence of Coriolis forces. The studies including Coriolis effects are mostly restricted to the high-viscosity-contrast limit and rely on either purely linear stability analyses or intensive numerical simulations. We approach the problem analytically and use a modified Darcy's law including the exact form of the Coriolis effects to execute a mode-coupling analysis of the system. By imposing no restrictions on the viscosity contrast A (dimensionless viscosity difference) we go beyond linear stages and examine the onset of nonlinearities. Our results indicate that when Coriolis effects are taken into account, an interesting interplay between the Reynolds number Re and A arises. This leads to important changes in the stability and morphological features of the emerging interfacial patterns. We contrast our mode-coupling approach with previous theoretical models proposed in the literature.

Numerical study of pattern formation in miscible rotating Hele-Shaw flows

The dynamics of the diffusing interface separating two miscible fluids in a rotating Hele-Shaw cell is studied by intensive and highly accurate numerical simulations. We perform numerical experiments in a wide range of parameters, focusing on the influence of viscosity contrast and Korteweg stresses on the shape of the interfacial patterns. A great variety of morphological behaviors is systematically introduced, and a wealth of interesting phenomena related to finger competition dynamics, filament stretching, and interface pinch off are reveal. Our simulations exhibit miscible patterns that bear a strong resemblance to their immiscible counterparts for larger Korteweg stresses. The quantitative equivalence between such stresses and the usual immiscible surface tension is studied. The concept of an effective interfacial tension is considered, allowing the direct and precise calculation of the important fingering properties under miscible circumstances. Our results show excellent agreement with existing experiments and simulations for corresponding immiscible displacements. This agreement refers to a striking similarity between miscible and immiscible pattern morphologies, and also to an accurate prediction for the typical number of miscible fingering structures formed. Our findings suggest that the effective interfacial tension is both qualitatively and quantitatively equivalent to its immiscible counterpart.

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.

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.

Viscous fingering as a paradigm of interfacial pattern formation: recent results and new challenges

We review recent results on dynamical aspects of viscous fingering. The Saffman-Taylor instability is studied beyond linear stability analysis by means of a weakly nonlinear analysis and the exact determination of the subcritical branch. A series of contributions pursuing the idea of a dynamical solvability scenario associated to surface tension in analogy with the traditional selection theory is put in perspective and discussed in the light of the asymptotic theory of Tanveer and co-workers. The inherently dynamical singular effects of surface tension are clarified. The dynamical role of viscosity contrast is explored numerically. We find that the basin of attraction of the Saffman-Taylor finger depends on viscosity contrast, and that the sensitivity to this parameter is maximal in the usual limit of high viscosity contrast. The competing attractors are identified as closed bubble solutions. We briefly report on recent results and work in progress concerning rotating Hele-Shaw flows, topological singularities and wetting effects, and also discuss future directions in the context of viscous fingering.

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

We consider the flow of a ferrofluid droplet in a Hele-Shaw cell with a time-dependent gap width. When the surface tension and applied magnetic field are zero, interfacial instabilities develop and the droplet breaks. We execute a mode-coupling approach to the problem and focus on understanding how the development of singularities is affected by the action of an external field. Our analytical results indicate that the introduction of an azimuthal magnetic field profoundly modifies pattern formation, allowing the inhibition of interfacial singularities. We suggest the magnetic field can be used as a controllable parameter to discipline singular behavior.