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

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.

Assessment of the mechanical suppression of nonuniform electrodeposition in lithium metal batteries

Dendrite formation is a long-standing issue in lithium metal batteries. Replacing the conventional liquid electrolytes with semi-solid ones, the non-uniform lithium growth can be potentially mitigated by the mechanical deformation in the solid matrix. The underlying dendrite suppression mechanism is investigated in this study using a mechano-electrochemical phase-field method. Two indicators, namely the arithmetic average height and the elongation rate, are proposed to characterize the surface roughness of lithium dendrites. Our simulation results are summarized in two-dimensional design maps as a function of the porosity and the elastic modulus of the semi-solid electrolytes, which could provide us the guidance for the development of dendrite-free lithium metal batteries.

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).

Elastic fingering in rotating Hele-Shaw flows

The centrifugally driven viscous fingering problem arises when two immiscible fluids of different densities flow in a rotating Hele-Shaw cell. In this conventional setting an interplay between capillary and centrifugal forces makes the fluid-fluid interface unstable, leading to the formation of fingered structures that compete dynamically and reach different lengths. In this context, it is known that finger competition is very sensitive to changes in the viscosity contrast between the fluids. We study a variant of such a rotating flow problem where the fluids react and produce a gellike phase at their separating boundary. This interface is assumed to be elastic, presenting a curvature-dependent bending rigidity. A perturbative weakly nonlinear approach is used to investigate how the elastic nature of the interface affects finger competition events. Our results unveil a very different dynamic scenario, in which finger length variability is not regulated by the viscosity contrast, but rather determined by two controlling quantities: a characteristic radius and a rigidity fraction parameter. By properly tuning these quantities one can describe a whole range of finger competition behaviors even if the viscosity contrast is kept unchanged.

Control of centrifugal fingering via a variable interfacial tension approach

We study the centrifugal fingering instabilities that occur in rotating Hele-Shaw cells containing two different fluids. A weakly nonlinear analysis of the problem is performed, considering that the surface tension between the rotating fluids changes with the local curvature of the interface. It is shown that the coupling between the contact angle and the variable interfacial tension permits the control of the interface disturbances. As a result, linear perturbations and nonlinear finger competition phenomena can be properly controlled and suppressed.

Diffuse-interface approach to rotating Hele-Shaw flows

When two fluids of different densities move in a rotating Hele-Shaw cell, the interface between them becomes centrifugally unstable and deforms. Depending on the viscosity contrast of the system, distinct types of complex patterns arise at the fluid-fluid boundary. Deformations can also induce the emergence of interfacial singularities and topological changes such as droplet pinch-off and self-intersection. We present numerical simulations based on a diffuse-interface model for this particular two-phase displacement that capture a variety of pattern-forming behaviors. This is implemented by employing a Boussinesq Hele-Shaw-Cahn-Hilliard approach, considering the whole range of possible values for the viscosity contrast, and by including inertial effects due to the Coriolis force. The role played by these two physical contributions on the development of interface singularities is illustrated and discussed.

Inertial effects on rotating Hele-Shaw flows

We examine the finite surface tension radial viscous fingering problem in a rotating Hele-Shaw cell, and focus on the action of inertial effects on the stability and morphology of the emerging patterns. To study such a flow we use an alternative version of the usual Darcy's law derived by gap-averaging the Navier-Stokes equation, but retaining its inertial terms. The importance of inertial forces is pertinently characterized by a rotational Reynolds number. Linear and weakly nonlinear stages of the dynamics are described analytically through a mode coupling approach. The linear stability results indicate that inertia has a stabilizing role. However, the characteristic number of fingers and the width of the band of linearly unstable modes are not altered by inertia. In the early nonlinear regime we find that inertia acts to favor the development of fingers with narrower tips than those obtained in its absence. We have also verified that finger competition events are affected by inertial effects which tend to restrain finger length variability among inward-moving fingers.

Control of radial fingering patterns: a weakly nonlinear approach

It is well known that the constant injection rate flow in radial Hele-Shaw cells leads to the formation of highly branched patterns, where finger tip-splitting events are plentiful. Different kinds of patterns arise in the lifting Hele-Shaw flow problem, where the cell's gap width grows linearly with time. In this case, the morphology of the emerging structures is characterized by the strong competition among inward moving fingers. By employing a mode-coupling theory we find that both finger tip-splitting and finger competition can be restrained by properly adjusting the injection rate and the time-dependent gap width, respectively. Our theoretical model approaches the problem analytically and is capable of capturing these important controlling mechanisms already at weakly nonlinear stages of the dynamics.

Interfacial instabilities in periodically driven Hele-Shaw flows

We consider flow in a Hele-Shaw cell for which the upper plate moves up and down, making the fluid-fluid interface be driven periodically. To study such a flow we employ a mode-coupling approach, which allows the analytical assessment of important aspects about the stability and morphology of the evolving interface. At the linear level, it is shown that both the amplitude and the frequency of the drive have a significant role in determining the ultimate number of fingers formed. The influence of these factors on the mechanisms of finger competition and finger tip behavior at the onset of nonlinear effects is also studied.

Miscible ferrofluid patterns in a radial magnetic field

Pattern formation in a miscible ferrofluid system is experimentally investigated. The experiment is performed by immersing a thin ferrofluid droplet in a cylindrical container, overfilling it with a nonmagnetic miscible fluid, and applying an in-plane radial magnetic field. Visually striking patterns are obtained whose morphologies change from circular at zero field to complex starburst-like structures at finite field. The evolution of miscible ferrofluid droplets of various initial diameters subjected to different magnetic-field strengths is considered. Proper rescaling of the experimental data indicates that the time evolution of the droplets' area increments obeys a universal 4/3 power-law behavior at long times.

Effects of normal viscous stresses on radial viscous fingering

We revisit the radial viscous fingering problem in a Hele-Shaw cell, and consider the action of viscous stresses originated from velocity gradients normal to the fluid-fluid interface. The evolution of the interface during linear and weakly nonlinear stages is described analytically through a mode-coupling approach. We find that the introduction of normal stresses influences the stability and the ultimate morphology of the emerging patterns. Although at early stages normal stresses tend to stabilize the interface, they act to favor the development of tip-splitting phenomena at the weakly nonlinear regime. We have also verified that finger competition events are only significantly affected by normal stresses for circumstances involving the development of a large number of interfacial fingers.

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.

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.

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.