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.

Controlling fluid adhesion force with electric fields.

Developing adhesives whose bond strength can be externally manipulated is a topic of considerable interest for practical and scientific purposes. In this work, we propose a method of controlling the adhesion force of a regular fluid, such as water and/or glycerol, confined between two parallel plates by applying an external electric field. Our results show the possibility of enhancing or diminishing the bond strength of the liquid sample by appropriately tuning the intensity and direction of the electric current generated by the applied electric field. Furthermore, we verify that, for a given direction of the electric current, the adhesion force can be reduced enough for the fluid to lose its adhesive properties and begin exerting a force to move apart the confining plates. In these circumstances, we obtain an analytical expression for the minimum electric current required to detach the plates without requiring the action of an external force.

Elastic fingering in three dimensions.

Recent studies on quasi-two-dimensional (2D) fluid flows in Hele-Shaw cells revealed the emergence of the so-called elastic fingering phenomenon. This pattern-forming process takes place when a reaction occurs at the fluid-fluid interface, transforming it into an elastic gel-like boundary. The interplay of viscous and elastic forces leads to the development of pattern morphologies significantly different from those seen in the conventional, purely hydrodynamic viscous fingering problem. In this work, we investigate the occurrence of elastic fingering for radial fluid displacements in a 3D uniform porous medium. A perturbative third-order mode-coupling approach is employed to examine how the combined action of viscous and elastic effects influences the linear stability of the interface, and the weakly nonlinear pattern formation in such a 3D environment. In addition, a variational method is used to determine how to minimize the growth of interfacial perturbation amplitudes via a time-dependent injection rate scheme.

Hamiltonian formulation towards minimization of viscous fluid fingering.

A variational approach has been recently employed to determine the ideal time-dependent injection rate Q(t) that minimizes fingering formation when a fluid is injected in a Hele-Shaw cell filled with another fluid of much greater viscosity. However, such a calculation is approximate in nature, since it has been performed by assuming a high capillary number regime. In this work, we go one step further, and utilize a Hamiltonian formulation to obtain an analytical exact solution for Q(t), now valid for arbitrary values of the capillary number. Moreover, this Hamiltonian scheme is applied to calculate the corresponding injection rate that minimizes fingering formation in a uniform three-dimensional porous media. An analysis of the improvement offered by these exact injection rate expressions in comparison with previous approximate results is also provided.

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.

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.

Controlling and minimizing fingering instabilities in non-Newtonian fluids.

The development of the viscous fingering instability in Hele-Shaw cells has great practical and scientific importance. Recently, researchers have proposed different strategies to control the number of interfacial fingering structures, or to minimize as much as possible the amplitude of interfacial disturbances. Most existing studies address the situation in which an inviscid fluid displaces a viscous Newtonian fluid. In this work, we report on controlling and minimizing protocols considering the situation in which the displaced fluid is a non-Newtonian, power-law fluid. The necessary changes on the controlling schemes due to the shear-thinning and shear thickening nature of the displaced fluid are calculated analytically and discussed.

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.

Wavelength selection in Hele-Shaw flows: a maximum-amplitude criterion.

As in most interfacial flow problems, the standard theoretical procedure to establish wavelength selection in the viscous fingering instability is to maximize the linear growth rate. However, there are important discrepancies between previous theoretical predictions and existing experimental data. In this work we perform a linear stability analysis of the radial Hele-Shaw flow system that takes into account the combined action of viscous normal stresses and wetting effects. Most importantly, we introduce an alternative selection criterion for which the selected wavelength is determined by the maximum of the interfacial perturbation amplitude. The effectiveness of such a criterion is substantiated by the significantly improved agreement between theory and experiments.

Control of centrifugally driven fingering in a tapered Hele-Shaw cell.

Conventional studies of the centrifugally driven fingering instability are performed in rotating Hele-Shaw cells presenting perfectly parallel plates. In this setup, the fluid-fluid interface can become unstable due to the density difference between the fluids, forming a variety of complex patterns. We study a modified, tapered version of this rotating flow problem where the cell plates are not exactly parallel, but present a constant gap gradient in the radial direction. Our analytical results indicate that the stability of the interface is significantly sensitive to the presence of the depth gradient. This allows a proper monitoring of pattern-forming features just by slightly changing the cell's taper angle. In this context, we have verified that by manipulating the sign and magnitude of the gap gradient one can favor, restrain, or even suppress the development of centrifugally driven instabilities. A taper-induced mechanism for the selection of the number of emerging fingers is also discussed.

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.

Minimization of instabilities in growing interfaces: a variational approach.

The Mullins-Sekerka and the electric breakdown instabilities are well known to lead to the spontaneous formation of a variety of complex spatial structures, among them dendritic crystal shapes, and treelike electric discharge patterns. Controlling such systems by suppressing predominantly excited solutions offers the opportunity to manipulate and stabilize these patterns in a defined way for a wide range of technological applications. In this work, we employ a variational approach which enables one to systematically search for the ideal conditions under which the patterns grow, but where interfacial deformations are efficiently minimized. The effectiveness of our variational control method is illustrated via linear stability calculations on both two-dimensional and three-dimensional contour-dynamics models for crystal growth and electric discharge phenomena.

Viscous-fingering minimization in uniform three-dimensional porous media.

In this paper, we consider a radial displacement of a viscous fluid by another one of much lower viscosity through a three-dimensional uniform porous medium. It is well known that when a less viscous fluid is pumped at a constant injection rate, very complex interfacial patterns are formed. The control and eventual suppression of these instabilities are relevant to a large number of areas in science and technology. Here, we use a variational approach to search for an analytical form of an optimal flow rate so that the interface between two almost neutrally buoyant fluids grows, but the emergence of interfacial disturbances is minimized. We find a closed analytical solution for the ideal flow rate which surprisingly does not depend on either the properties of the fluids or the permeability of the porous medium.

Variational scheme towards an optimal lifting drive in fluid adhesion.

One way of determining the adhesive strength of liquids is provided by a probe-tack test, which measures the force or energy required to pull apart two parallel flat plates separated by a thin fluid film. The vast majority of the existing theoretical and experimental works in fluid adhesion use very viscous fluids, and consider a linear drive L(t)∼Vt with constant lifting plate velocity V. This implies a given energy cost and large lifting force magnitude. One challenging question in this field pertains to what would be the optimal time-dependent drive Lopt(t) for which the adhesion energy would be minimized. We use a variational scheme to systematically search for such Lopt(t). By employing an optimal lifting drive, in addition to saving energy, we verify a significant decrease in the adhesion force peak. The effectiveness of the proposed lifting procedure is checked for both Newtonian and power-law fluids.

Minimization of viscous fluid fingering: a variational scheme for optimal flow rates.

Conventional viscous fingering flow in radial Hele-Shaw cells employs a constant injection rate, resulting in the emergence of branched interfacial shapes. The search for mechanisms to prevent the development of these bifurcated morphologies is relevant to a number of areas in science and technology. A challenging problem is how best to choose the pumping rate in order to restrain the growth of interfacial amplitudes. We use an analytical variational scheme to look for the precise functional form of such an optimal flow rate. We find it increases linearly with time in a specific manner so that interface disturbances are minimized. Experiments and nonlinear numerical simulations support the effectiveness of this particularly simple, but nontrivial, pattern controlling process.

Effect of fluid inertia on probe-tack adhesion.

One way of determining the adhesive strength of liquids is provided by a probe-tack test, which involves measuring the force required to pull apart two parallel flat plates separated by a thin fluid film. The large majority of existing theoretical and experimental work on probe-tack adhesion use very viscous fluids and considers relatively low lifting plate velocities, so that effects due to fluid inertia can be neglected. However, the employment of low-viscosity fluids and the increase in operating speeds of modern lifting apparatus can modify this scenario. By dealing with a proper gap averaging of the Navier-Stokes equation, we obtain a modified Darcy-law-like description of the problem and derive an adhesion force which incorporates the effects of fluid inertia, fluid viscosity (for Newtonian and power law fluids), and the contribution of the compliance and inertia of the probe-tack apparatus. Our results indicate that fluid inertia may have a significant influence on the adhesion force profiles, inducing a considerable increase in the force peaks and producing oscillations in the force-displacement curves as the plate-plate separation is increased. The combined role of inertial and non-Newtonian fluid behaviors on the adhesion force response is also investigated.

Influence of inertia on viscous fingering patterns: rectangular and radial flows.

Recently, there has been a growing interest in the impact of inertial effects on the development of the Saffman-Taylor instability. Experiments and theory indicate that inertia may have a significant influence on the system's behavior. We employ a perturbative-mode-coupling method to examine how the stability and morphology of the viscosity-driven fingering patterns are affected by inertia. Both rectangular and radial Hele-Shaw flow geometries are considered. In the rectangular configuration useful results can be deduced analytically, and in closed form. In particular, we have found that inertia has a stabilizing role at the linear stage, and tends to widen the fingers at the weakly nonlinear regime. These analytical results are consistent with existing experimental findings. The analysis of the system is not as simple in radial flow geometry, but it still allows the capture of inertially induced, enhanced finger tip splitting events at the onset of nonlinearities.