Interfacial Fingering Research Articles

Experimental and analytical approaches toward controlling viscous fingering at the interface of two immiscible fluids using the optimal constant piecewise injection rate

Viscous fingering (VF) at the interface of two immiscible fluids can have disruptive effects in many technological applications. Therefore finding methods for reducing it with an effective and simple approach plays a significant role. A proposed solution that restrains the expansion of such complex morphologies, is the time-dependent injection rate. This proposed solution is often done either analytically or by numerical simulations which need to be experimentally proven. In this study, we perform multiple experiments of low viscosity fluid injection into high viscosity fluid in order to study the formation of VF and find injection rates or protocols that reduce VF at the onset of the nonlinear regime and in the fully nonlinear regime by applying several constant piecewise injection rates. Moreover, analytically and numerically studies are done, in which we derive the optimal parameters in piecewise injection profiles to limit the VF and the tip-splitting effect. This optimal constant piecewise injection rate includes specified number of stages as well as the injection time and its rate in each stage. The obtained results revealed that depending on being at the beginning of the nonlinear regime or in the fully nonlinear regime, the optimal constant piecewise injection rate can considerably reduce the length of fingers and the interfacial fingering perturbations. Empirical results show similar trend in terms of instability reduction by applying the same optimal number of piecewise injection rate stages. In addition, evaluating the difference between this piecewise results and the results of linear injection rates, which we have done before, reveals that employing linear injection rates is more effective in controlling the instability in lower injection rates, while for higher injection rates, it is required to utilize a piecewise injection rate.

Suppression of droplet pinch-off by early onset of interfacial instability

HypothesisInterfacial instabilities cause undesirable droplet breakage during impact. Such breakage affects many applications, such as printing, spraying, etc. Particle coating over a droplet can significantly change the impact process and stabilize it against breakage. This work investigates the impact dynamics of particle-coated droplets, which mostly remains unexplored. ExperimentsParticle-coated droplets of different mass loading were formed using volume addition. The prepared droplets were impacted on superhydrophobic surfaces, and their dynamics were recorded using a high-speed camera. FindingsWe report an intriguing phenomenon where an interfacial fingering instability helps suppress pinch-off in particle-coated droplets. This island of breakage suppression, where the droplet maintains its intactness upon impact, appears in a regime of Weber numbers where bare droplet breakage is inevitable. The onset of fingering instability in particle-coated droplets is observed at much lower impact energy, around two times less than the bare droplet. The instability is characterized and explained using the rim Bond number. The instability suppresses pinch-off because of the higher losses associated with the formation of stable fingers. Such instability can also be seen in dust/pollen-covered surfaces, making it useful in many applications related to cooling, self-cleaning, anti-icing etc.

A unified theoretical model for spatiotemporal development of Rayleigh–Taylor and Richtmyer–Meshkov fingers

An initially perturbed interface between two fluids of different densities is usually unstable when driven by an acceleration or a shock wave; it is known as a Rayleigh–Taylor instability or a Richtmyer–Meshkov instability. One of the most significant issues in these instabilities is the spatiotemporal development of fingers generated at the interface, which plays an important role in both scientific research (e.g. supernova explosion) and engineering applications (e.g. inertial confinement fusion). Accurate theoretical solution of these interfacial fingers remains as an unsolved and challenging problem since Taylor's seminal work more than seven decades ago. This paper reports a unified theory established for such phenomena by combining the classical potential-flow theory and a dual-source model to address the long-standing difficulty highlighted by the initial-value sensitivity and strong nonlinearity. It is the first time for a theory to accurately predict the long-time developments in both growth rate and shape curvature of interfacial fingers at all density ratios in two and three dimensions. Moreover, the new theory clearly reveals the nonlinear coupling mechanism for interfacial evolution, and especially explains the origin of overshot in the growth rate curve.

Relative role of short interfacial fingers and long internally driven streamers in convective flows below growing sea ice

As sea ice grows from sea water, salt initially dissolved in the liquid is rejected from the solid. Buoyancy-driven flows develop under the ice layer in two ways: 1) interfacial boundary layer convection resulting in small-scale fingers and 2) internal convection originating from brine drainage channels inside the ice, flushing out longer-scale convective streamers. We study these dynamics experimentally by freezing salt water from above in a quasi-2D Hele-Shaw cell, observing with Schlieren and direct imaging systems. Interfacial fingers turn out more significant as a salt-transport pathway than previously thought, persisting throughout ice growth, whereas streamers show on-off behavior.

Emergent patterns and stable interfaces during radial displacement of a viscoelastic fluid

We explore the interfacial instability that results when a Newtonian fluid (a glycerol-water mixture, inner fluid) displaces a viscoelastic fluid (a dense cornstarch suspension, outer fluid) in a radial Hele-Shaw cell. As the ratio of viscosities of the inner and outer fluids is increased, side branched interfacial patterns are replaced by more stable interfaces that display proportionate growth and finger coalescence. We correlate the average finger spacing with the most dominant wavelength of interfacial instability, computed using a mathematical model that accounts for viscous fingering in miscible Hele-Shaw displacements. The model predictions on the role of viscosity ratio on finger spacing are in close agreement with the experimental observations. Our study lends insight into the significant contribution of the viscoelasticity of the outer fluid on the morphology and growth of interfacial patterns.

Fluid Morphologies Governed by the Competition of Viscous Dissipation and Phase Separation in a Radial Hele-Shaw Flow

The displacement of a less viscous fluid by a more viscous fluid in a radial Hele-Shaw cell makes a circular pattern because the interface is hydrodynamically stable in this condition. Very recently, it has been experimentally reported that the hydrodynamically stable displacement in a partially miscible system induces fingering patterns while stable circular patterns are made at fully miscible and immiscible systems. The fingering instability in the partially miscible system results from complex and entangled elements involving viscous dissipation, molecular diffusion, and phase separation. The analyzing mechanism requires a quantitative relationship between the hydrodynamic interfacial fingering patterns and underlying physicochemical properties. Here, we experimentally investigated the change in fluid patterns formed by the progression of phase separation in the partially miscible systems and categorized them into three patterns: finger-like pattern, annular-like pattern, and circular pattern. Moreover, we propose the mechanism of the pattern formation by an interfacial tension measurement and evaluate the patterns by modified capillary number and newly defined body force ratio, Bf. Our analysis revealed that the deformation index of the pattern can be expressed as a function of Bf on a single curve regardless of the miscibility.

Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.

Self-similar and disordered front propagation in a radial Hele-Shaw channel with time-varying cell depth

An air bubble expanding into a viscous liquid in the small gap between parallel plates deforms into highly branched, continuously evolving interfacial fingering patterns. By separating the plates following a power law in time, it is found that the interface can also form self-similar fingering patterns.

Short, flat-tipped, viscous fingers: novel interfacial patterns in a Hele-Shaw channel with an elastic boundary

Injecting a less viscous fluid into a more viscous fluid in a Hele-Shaw cell triggers two-dimensional viscous fingering patterns which are characterised by increasingly long fingers undergoing tip splitting and branching events. These complex structures are considered to be a paradigm for interfacial pattern formation in porous media flow and other related phenomena. Over the past five years, there has been a flurry of interest in manipulating these interfacial fingering patterns by altering the physical components of the Hele-Shaw apparatus. In this Focus on Fluids article, we summarise some of this work, concentrating on a very recent study in which the alterations include replacing one of the two bounding plates with an elastic membrane (Ducloué et al., J. Fluid Mech., vol. 826, 2017, R2). The resulting experimental set-up gives rise to a wide variety of novel interfacial patterns including periodic sideways fingers, dendritic-like patterns and short, flat-tipped viscous fingers that appear to resemble molar teeth. These latter fingers are similar to those observed in the printer’s instability and when peeling off a layer of adhesive tape. This delightful work brings together a number of well-studied themes in interfacial fluid mechanics, including how viscous and surface tension forces compete to drive fingering patterns, how interfaces are affected by fluid–solid interactions and, finally, how novel strategies can be implemented to control interfacial instabilities.

Droplet Formation by Rupture of Vibration-Induced Interfacial Fingers.

By imposing vibration to a core-annular flow of an aqueous two-phase system (ATPS) with ultralow interfacial tension, we observe a liquid finger protruding from the interface of an expanding jet. We find that the protruded finger breaks up only when its length-to-width ratio exceeds a threshold value. The breakup follows a constant wavelength-to-width ratio that is consistent with that of breakup under Rayleigh-Plateau instability. The mechanism is applicable to aqueous two-phase systems with a large range of viscosity ratios. The protruded finger can break up into small droplets that are monodisperse in size, controllable in generation frequency under a wide range of flow rates. This work suggests a way to generate small water-water droplets with high monodispersity and production rate from a single nozzle.

Viscous fingering and dendritic growth under an elastic membrane

We present an experimental investigation of interfacial fingering instabilities in a compliant channel, where the interface can adopt a planar front orthogonal to the direction of propagation over most of the channel width. Finite-length fingers develop on that front, similarly to the previously studied radial configuration with injection of air at constant flow rate (Pihler-Puzović et al., Phys. Rev. Lett., vol. 108 (7), 2012, 074502), but, unlike the radial case, the interface propagates steadily. This allows us to present the first quantification of the nonlinearly saturated fingering pattern and to demonstrate that the morphological features of the fingers are selected in a simple way by the local geometry of the compliant cell. In contrast, the local geometry itself is determined from a complex fluid–solid interaction. Furthermore, we find that changes to the geometry of the channel cross-section lead to a rich variety of possible interfacial patterns.

Rotationally induced fingering patterns in a two-dimensional heterogeneous porous medium.

Rotating fluid flows under two-dimensional homogeneous porous media conditions (or in a rotating Hele-Shaw cell) reveal the development of complex interfacial fingering patterns. These pattern-forming structures are characterized by the occurrence of finger competition events, finger pinch-off episodes, as well as by the production of satellite droplets. In this work, we use intensive numerical simulations to investigate howthesefully nonlinear pattern growth phenomena are altered by the presence of permeability heterogeneities in the rotating porous medium. This is done by employing a diffuse-interface Darcy-Cahn-Hilliard description of the problem and considering a permeability field presenting a log-Gaussian distribution, characterized by a variance s and a correlation length l. We study how the heterogeneity measures s and l couple to the governing hydrodynamic dimensionless parameters of the problem and introduce important changes on the pattern formation dynamics of the system.

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.

Stationary patterns in centrifugally driven interfacial elastic fingering.

A vortex sheet formalism is used to search for equilibrium shapes in the centrifugally driven interfacial elastic fingering problem. We study the development of interfacial instabilities when a viscous fluid surrounded by another of smaller density flows in the confined environment of a rotating Hele-Shaw cell. The peculiarity of the situation is associated to the fact that, due to a chemical reaction, the two-fluid boundary becomes an elastic layer. The interplay between centrifugal and elastic forces leads to the formation of a rich variety of stationary shapes. Visually striking equilibrium morphologies are obtained from the numerical solution of a nonlinear differential equation for the interface curvature (the shape equation), determined by a zero vorticity condition. Classification of the various families of shapes is made via two dimensionless parameters: an effective bending rigidity (ratio of elastic to centrifugal effects) and a geometrical radius of gyration.

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.

Stretch flow of confined non-Newtonian fluids: Nonlinear fingering dynamics

We employ a weakly nonlinear perturbative scheme to investigate the stretch flow of a non-Newtonian fluid confined in Hele-Shaw cell for which the upper plate is lifted. A generalized Darcy's law is utilized to model interfacial fingering formation in both the weak shear-thinning and weak shear-thickening limits. Within this context, we analyze how the interfacial finger shapes and the nonlinear competition dynamics among fingers are affected by the non-Newtonian nature of the stretched fluid.

Interfacial elastic fingering in Hele-Shaw cells: A weakly nonlinear study

We study a variant of the classic viscous fingering instability in Hele-Shaw cells where the interface separating the fluids is elastic, and presents a curvature-dependent bending rigidity. By employing a second-order mode-coupling approach we investigate how the elastic nature of the interface influences the morphology of emerging interfacial patterns. This is done by focusing our attention on a conventionally stable situation in which the fluids involved have the same viscosity. In this framework, we show that the inclusion of nonlinear effects plays a crucial role in inducing sizable interfacial instabilities, as well as in determining the ultimate shape of the pattern-forming structures. Particularly, we have found that the emergence of either narrow or wide fingers can be regulated by tuning a rigidity fraction parameter. Our weakly nonlinear findings reinforce the importance of the so-called curvature weakening effect, which favors the development of fingers in regions of lower rigidity.

Scaling Normal Adhesion Force Capacity with a Generalized Parameter

The adhesive response of a rigid flat cylindrical indenter in contact with a compliant elastic layer of varying confinement is investigated experimentally and described analytically. Using a soft elastic gel with substrate thickness, t, and indenter radius, a, 28 unique combinations of the confinement parameter, a/t, are examined over a range of 0.016 < a/t < 7.2. Continuous force capacity predictions as a function of a/t and material properties are provided through a scaling theory and are found to agree well with the experimental data. We further collapse all of the data over orders of magnitude in adhesive force capacity onto a single line described by a generalized reversible adhesion scaling parameter, A/C, where A is the contact area and C is the compliance. As the scaling analysis does not assume a specific separation mechanism the adhesive force capacity is well described during both axisymmetric edge separation and during interfacial fingering and cavitation instabilities. We discuss how the geometry of the contact, specifically increasing the degree of confinement, allows reversible adhesive materials to be designed that are not "sticky" or "tacky", yet can be very strong and provide high performance.

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.

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.