Control of radial fingering patterns: a weakly nonlinear approach

Interface Stabilization in Adhesion Caused by Elastohydrodynamic Deformation

Interfacial instabilities are common phenomena observed during adhesion measurements involving viscoelastic polymers or fluids. Typical probe-tack adhesion measurements with soft adhesives are conducted with rigid probes. However, in many settings, such as for medical applications, adhesives make and break contact from soft surfaces such as skin. Here we study how detachment from soft probes alters the debonding mechanism of a model viscoelastic polymer film. We demonstrate that detachment from a soft probe suppresses Saffman-Taylor instabilities commonly encountered in adhesion. We suggest the mechanism for interface stabilization is elastohydrodynamic deformation of the probe and propose a scaling for the onset of stabilization.

Control of interfacial instabilities through variable injection rate in a radial Hele-Shaw cell: A nonlinear approach for late-time analysis

In this paper, the nonlinear behavior of immiscible viscous fingering in a circular Hele-Shaw cell under the action of different time-dependent injection flow rate schemes is assessed numerically. Unlike previous studies which addressed the infinite viscosity ratio (inviscid-viscous flow), the problem is tackled by paying special attention to flows with finite viscosity ratio (viscous flow) in which the viscosity of the displacing and the displaced fluids can have any arbitrary value. Systematic numerical simulations based on a complex-variable formulation of Cauchy-Green barycentric coordinates are performed at different mobility ratios and capillary numbers with a focus on the late-time fully nonlinear regime. Additionally, numerical optimization is used to obtain the optimal flow rate schedule through a second-order weakly nonlinear stability analysis in contrast to previous studies in which the optimal flow rate was obtained entirely based on linear stability analysis. It is demonstrated that, irrespective of the values of the mobility ratio and/or the capillary number, for patterns whose constant injection counterpart exhibits linear flow regime, the curvature-driven relaxation time is comparable with the operational time of the time-dependent injection flow rate controlling schemes, and most of the controlling schemes work very well and suppress the fingering phenomenon remarkably with the maximum recovery improvement of 15%. As the nonlinearity of the system increases, the schemes may still perform well, but their effectiveness is more pronounced in patterns with less nonlinearity in their constant injection counterpart than those with higher nonlinearity. As the nonlinearity increases, the curvature-driven relaxation time becomes longer than the operational time of the schemes, leading to a reduction in their effectiveness. Additionally, it is shown that employment of the second-order weakly nonlinear stability analysis to formulate the objective function does not result in any remarkable variation in the obtained optimal flow rate schedule.

Controlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects

In this paper, the interfacial motion between two immiscible viscous fluids in the confined geometry of a Hele-Shaw cell is studied. We consider the influence of a thin wetting film trailing behind the displaced fluid, which dynamically affects the pressure drop at the fluid-fluid interface by introducing a nonlinear dependence on the interfacial velocity. In this framework, two cases of interest are analyzed: The injection-driven flow (expanding evolution), and the lifting plate flow (shrinking evolution). In particular, we investigate the possibility of controlling the development of fingering instabilities in these two different Hele-Shaw setups when wetting effects are taken into account. By employing linear stability theory, we find the proper time-dependent injection rate Q(t) and the time-dependent lifting speed b[over ̇](t) required to control the number of emerging fingers during the expanding and shrinking evolution, respectively. Our results indicate that the consideration of wetting leads to an increase in the magnitude of Q(t) [and b[over ̇](t)] in comparison to the nonwetting strategy. Moreover, a spectrally accurate boundary integral approach is utilized to examine the validity and effectiveness of the controlling protocols at the fully nonlinear regime of the dynamics and confirms that the proposed injection and lifting schemes are feasible strategies to prescribe the morphologies of the resulting patterns in the presence of the wetting film.

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.

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.

Capillary and geometrically driven fingering instability in nonflat Hele-Shaw cells

The usual viscous fingering instability arises when a fluid displaces another of higher viscosity in a flat Hele-Shaw cell, under sufficiently large capillary number conditions. In this traditional framing, the reverse flow case (more viscous fluid displacing a less viscous one) and the viscosity-matched situation (fluids of equal viscosities) are stable. We revisit this classical fluid dynamic problem, now considering flow in a nonflat Hele-Shaw cell. For a specific nonflat environment, we show that both the reverse and the viscosity-matched flows can become unstable, even at low capillary number. This peculiar fluid fingering instability is driven by the combined action of capillary effects and geometric properties of the nonflat Hele-Shaw cell. Our theoretical results indicate that the Hele-Shaw cell geometry significantly impacts the linear stability and nonlinear pattern-forming dynamics of the system. This suggests that the geometry of the medium plays an important role in favoring the occurrence of fingering patterns in nonflat, confined fluid flows.

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.

Development of tip-splitting and side-branching patterns in elastic fingering

Elastic fingering supplements the already interesting features of the traditional viscous fingering phenomena in Hele-Shaw cells with the consideration that the two-fluid separating boundary behaves like an elastic membrane. Sophisticated numerical simulations have shown that under maximum viscosity contrast the resulting patterned shapes can exhibit either finger tip-splitting or side-branching events. In this work, we employ a perturbative mode-coupling scheme to get important insights into the onset of these pattern formation processes. This is done at lowest nonlinear order and by considering the interplay of just three specific Fourier modes: a fundamental mode n and its harmonics 2n and 3n. Our approach further allows the construction of a morphology diagram for the system in a wide range of the parameter space without requiring expensive numerical simulations. The emerging interfacial patterns are conveniently described in terms of only two dimensionless controlling quantities: the rigidity fraction C and a parameter Γ that measures the relative strength between elastic and viscous effects. Visualization of the rigidity field for the various pattern-forming structures supports the idea of an elastic weakening mechanism that facilitates finger growth in regions of reduced interfacial bending rigidity.

Enhanced mixing via alternating injection in radial Hele-Shaw flows

Mixing at low Reynolds numbers, especially in the framework of confined flows occurring in Hele-Shaw cells, porous media, and microfluidic devices, has attracted considerable attention lately. Under such circumstances, enhanced mixing is limited due to the lack of turbulence, and absence of sizable inertial effects. Recent studies, performed in rectangular Hele-Shaw cells, have demonstrated that the combined action of viscous fluid fingering and alternating injection can dramatically improve mixing efficiency. In this work, we revisit this important fluid mechanical problem, and analyze it in the context of radial Hele-Shaw flows. The development of radial fingering instabilities under alternating injection conditions is investigated by intensive numerical simulations. We focus on the impact of the relevant physical parameters of the problem (Péclet number Pe, viscosity contrast A, and injection time interval Δt) on fluid mixing performance.

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.

Suppression of viscous fingering in nonflat Hele-Shaw cells

Viscous fingering formation in flat Hele-Shaw cells is a classical and widely studied fluid mechanical problem. Recently, instead of focusing on the development of the fingering instability, researchers have devised different strategies aiming to suppress its appearance. In this work, we study a protocol that intends to inhibit the occurrence of fingering instabilities in nonflat (spherical and conical) Hele-Shaw cell geometries. By using a mode-coupling theory to describe interfacial evolution, plus a variational controlling technique, we show that viscous fingering phenomena can be minimized in such a confined, curved environment by properly manipulating a time-dependent injection flow rate Q(t). Explicit expressions for Q(t) are derived for the specific cases of spherical and conical cells. The suitability of the controlling method is verified for linear and weakly nonlinear stages of the flow.

Interfacial pattern formation in confined power-law fluids

The interfacial pattern formation problem in an injection-driven radial Hele-Shaw flow is studied for the situation in which a Newtonian fluid of negligible viscosity displaces a viscous non-Newtonian power-law fluid. By utilizing a Darcy-law-like formulation, we tackle the fluid-fluid interface evolution problem perturbatively, and we derive second-order mode-coupling equations that describe the time evolution of the perturbation amplitudes. This allows us to investigate analytically how the non-Newtonian nature of the dislocated fluid determines the morphology of the emerging interfacial patterns. If the pushed fluid is shear-thinning, our results indicate the development of side-branching structures. On the other hand, if the displaced fluid is shear-thickening, one detects the formation of petal-like shapes, markedly characterized by strong tip-splitting events. Finally, a time-dependent injection protocol is presented that is able to restrain finger proliferation via side-branching and tip-splitting. This permits the emergence of symmetric n-fold interfacial shapes for which the number of fingers remains fixed as time progresses. This procedure generalizes existing controlling strategies for purely Newtonian flow circumstances to the case of a non-Newtonian, displaced power-law fluid.

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.

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.

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.

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 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.

Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes

Growth of complex dendritic fingers at the interface of air and a viscous fluid in the narrow gap between two parallel plates is an archetypical problem of pattern formation. We find a surprisingly effective means of suppressing this instability by replacing one of the plates with an elastic membrane. The resulting fluid-structure interaction fundamentally alters the interfacial patterns that develop and considerably delays the onset of fingering. We analyze the dependence of the instability on the parameters of the system and present scaling arguments to explain the experimentally observed behavior.

Controlling fingering instabilities in nonflat Hele-Shaw geometries

In analogy to the viscous fingering problem between parallel flat plates, injection-driven flow in nonflat (conical and spherical) Hele-Shaw cells gives rise to ramified interfacial structures. The multiple occurrences of fingertip-splitting events makes the resulting pattern difficult to predict and to control. We calculate expressions for time-dependent injection rates that suppress the development of bifurcated shapes on curved confined environments. Implementation of these injection procedures leads to interfacial shapes with a robust n-fold symmetry. A weakly nonlinear approach is used to extract important analytical information about the influence of the geometric and topological features of the nonflat cells on the efficiency of the proposed fingering control processes.

Suppression of viscous fluid fingering: a piecewise-constant injection process

The injection of a fluid into another of larger viscosity in a Hele-Shaw cell usually results in the formation of highly branched patterns. Despite the richness of these structures, in many practical situations such convoluted shapes are quite undesirable. In this Brief Report, we propose an efficient and easily reproducible way to restrain these instabilities based on a simple piecewise-constant pumping protocol. It results in a reduction in the size of the viscous fingers by one order of magnitude.

Controlling radial fingering patterns in miscible confined flows

Injection-driven immiscible flow in radial Hele-Shaw cells results in highly ramified patterns if the injection rate is constant in time. Likewise, time-dependent gap immiscible flow in lifting Hele-Shaw cells leads to intricate morphologies if the cell's gap width grows exponentially with time. Recent studies show that the rising of these complex fingered structures can be controlled by properly adjusting the injection rate, and the time-dependent gap width. We investigate the effectiveness of these control strategies assuming that the fluids involved are miscible. Despite the absence of surface tension effects, intensive numerical simulations support the stabilizing role of these controlling protocols. Splitting, merging and competition of fingers are all inhibited. The sensitivity of the system to changes in the initial conditions and Péclet numbers is also discussed.

Finger tip behavior in small gap gradient Hele-Shaw flows

Finger tip splitting and finger tip narrowing phenomena are not commonly observed during early flow stages in parallel-plate rectangular Hele-Shaw cells. However, an interesting experimental study performed by Zhao [Phys. Rev. A 45, 2455 (1992)] showed that finger tip behavior can be significantly affected by the addition of a small gradient to the gap of the cell, so that the plates are no longer exactly parallel. It was found that pattern forming events at the finger tip are dictated by the sign of the gap gradient, where positive (negative) sign produces wider (narrower) fingers. We approach the nonparallel-plate situation analytically through a perturbative mode coupling theory and show that these important finger tip features can be predicted at early nonlinear stages of the dynamics.