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

Simulation of Nonlinear Viscous Fingering in a Reactive Flow Displacement: A Multifractal Approach

fractal analysis of viscous fingering of a reactive miscible flow displacement in homogeneous porous media is investigated and multifractal spectrum, and fractal dimension are introduced as two essential features to characterize the irregularity of finger patterns. The Reaction of the two reactant fluids generates a miscible chemical product C in the contact zone. Considering the similarity between chemical products and coastline, monofractal and multifractal analyzes are performed. In monofractal analysis, the box-counting method is implemented on binary images and in multifractal analysis, due to the image processing; the fractal characteristics of viscous fingering instability are analyzed by means of fractal quantities such as Holder exponent, multifractal spectrum, f (α)-image and fractal dimension dynamics. Fractal analysis shows that the fractal dimension increases with time. Also, by considering five different nonlinear simulations, the results show that in the case both sides of the chemical product C are unstable, the multifractal spectrum curve has the highest peak, which means the more complex finger patterns lead to more values of fractal dimension. In addition, a comparison between different values of Ar is conducted and the results show similar behavior. However, small value of aspect ratio leads to a broader width of the multifractal spectrum curve. Furthermore, f (α)-images of concentration contour were investigated for different precisions and some undetectable finger patterns were observed in these images. It can be concluded that the use of f (α)-image represents more detailed image than concentration contours.

Effect of Hele–Shaw cell gap on radial viscous fingering

The flow through a Hele–Shaw cell is an experimental prototype to study the flow through a porous medium as well as the flow in microfluidic devices. In context with porous medium flows, it is used to visualize and understand hydrodynamic instabilities like viscous fingering (VF). The gap between the plates of the cell is an important parameter affecting the flow dynamics. However, the effect of the gap on the Hele–Shaw cell flows has been minimally explored. We perform experiments to understand the effect of the gap on VF dynamics. It is observed that a minimum gap is required to observe rigorous fingering instability. The onset time of instability, as well as the width of the fingers, increases with an increment in the gap due to a decrease in the convection. The instability increases with an increase in Péclet number, but the effect of gap width on fingering patterns is evident with broader fingers observed for larger b. The results are validated by performing numerical simulations. It is further shown that the gap-averaged three-dimensional simulations using the Stokes law approach and the two-dimensional Darcy’s law result in a small gap Hele–Shaw cell.

The effect of a crosslinking chemical reaction on pattern formation in viscous fingering of miscible fluids in a Hele-Shaw cell

Viscous fingering can occur in fluid motion whenever a high mobility fluid displaces a low mobility fluid in a Darcy type flow. When the mobility difference is primarily attributable to viscosity (e.g., flow between the two horizontal plates of a Hele-Shaw cell), viscous fingering (VF) occurs, which is sometimes termed the Saffman-Taylor instability. Alternatively, in the presence of differences in density in a gravity field, buoyancy-driven convection can occur. These instabilities have been studied for decades, in part because of their many applications in pollutant dispersal, ocean currents, enhanced petroleum recovery, and so on. More recent interest has emerged regarding the effects of chemical reactions on fingering instabilities. As chemical reactions change the key flow parameters (densities, viscosities, and concentrations), they may have either a destabilizing or stabilizing effect on the flow. Hence, new flow patterns can emerge; moreover, one can then hope to gain some control over flow instabilities through reaction rates, flow rates, and reaction products. We report effects of chemical reactions on VF in a Hele-Shaw cell for a reactive step-growth cross-linking polymerization system. The cross-linked reaction product results in a non-monotonic viscosity profile at the interface, which affects flow stability. Furthermore, three-dimensional internal flows influence the long-term pattern that results.

Off-equilibrium surface tension in miscible fluids

The interfacial tension between immiscible fluids is responsible for a wealth of every-day phenomena, from the spherical shape of small drops and bubbles to the ability to walk on water of many insects. More than a century ago, physicist and mathematician D. Korteweg postulated the existence of an effective interface tension for miscible fluids, whenever a composition gradient exists, as encountered, e.g., in many flow geometries. In this mini-review, we discuss experimental work performed in the last decades that demonstrates the existence of a positive effective interface tension in a variety of systems, from molecular, near-critical liquids to complex fluids such as polymer solutions and colloidal suspensions. The various experimental strategies that have been deployed are discussed, together with their advantages and limitations. Finally, some of the key theoretical questions still open are outlined.

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.

Interface evolution during radial miscible viscous fingering

We study experimentally the miscible radial displacement of a more viscous fluid by a less viscous one in a horizontal Hele-Shaw cell. For the range of tested injection rates and viscosity ratios we observe two regimes for the evolution of the fluid-fluid interface. At early times the interface length increases linearly with time, which is typical of the Saffman-Taylor instability for this radial configuration. However, as time increases, the interface growth slows down and scales as ∼t(1/2), as one expects in a stable displacement, indicating that the overall flow instability has shut down. Surprisingly, the crossover time between these two regimes decreases with increasing injection rate. We propose a theoretical model that is consistent with our experimental results, explains the origin of this second regime, and predicts the scaling of the crossover time with injection rate and the mobility ratio. The key determinant of the observed scalings is the competition between advection and diffusion time scales at the displacement front, suggesting that our analysis can be applied to other interfacial-evolution problems such as the Rayleigh-Bénard-Darcy instability.

Crystal growth from a supersaturated melt: relaxation of the solid-liquid dynamic stiffness

We discuss the growth process of a crystalline phase out of a metastable over-compressed liquid that is brought into contact with a crystalline substrate. The process is modeled by means of molecular dynamics. The particles interact via the Lennard-Jones potential and their motion is locally thermalized by Langevin dynamics. We characterize the relaxation process of the solid-liquid interface, showing that the growth speed is maximal for liquid densities above the solid coexistence density, and that the structural properties of the interface rapidly converge to equilibrium-like properties. In particular, we show that the off-equilibrium dynamic stiffness can be extracted using capillary wave theory arguments, even if the growth front moves fast compared to the typical diffusion time of the compressed liquid, and that the dynamic stiffness converges to the equilibrium stiffness in times much shorter than the diffusion time.

Radial Hele-Shaw flow with suction: fully nonlinear pattern formation

We study the development of intricate, fully nonlinear immiscible interfacial patterns in the suction-driven radial Hele-Shaw problem. The complex-shaped, contracting fluid-fluid interface arises when an initially circular blob of more viscous fluid, surrounded by less viscous one, is drawn into an eccentric point sink. We present sophisticated numerical simulations, based on a diffuse interface model, that capture the most prominent interfacial features revealed by existing experimental studies of the problem. The response of the system to changes in the capillary number is investigated, accurately revealing the occurrence of finger competition phenomena, and correctly describing the velocity behavior of both inward- and outward-pointing fingers. For the large-capillary-number regime, a set of complex interfacial features (finger merging, shielding, and pinch-off) whose experimental realization is still not available, are predicted.

Off-equilibrium surface tension in colloidal suspensions

We study the fingering instability of the interface between two miscible fluids, a colloidal suspension and its own solvent. The temporal evolution of the interface in a Hele-Shaw cell is found to be governed by the competition between the nonlinear viscosity of the suspension and an off-equilibrium, effective surface tension Γe. By studying suspensions in a wide range of volume fractions, ΦC, we show that Γe∼ΦC2, in agreement with Korteweg's theory for miscible fluids. The surface tension exhibits an anomalous increase with particle size, which we account for using entropy arguments.

Radial viscous fingering in yield stress fluids: onset of pattern formation

We report analytical results for the development of interfacial instabilities in a radial Hele-Shaw cell in which a yield stress fluid is pushed by a Newtonian fluid of negligible viscosity. By dealing with a gap averaging of the Navier-Stokes equation, we derive a Darcy-law-like equation for the problem, valid in the regime of high viscosity compared to yield stress effects. A mode-coupling approach is executed to examine the morphological features of the fluid-fluid interface at the onset of nonlinearity. Within this context, mechanisms for explaining the rising of tip-splitting and side-branching events are proposed.

Quantifying mixing in viscously unstable porous media flows

Viscous fingering is a well-known hydrodynamic instability that sets in when a less viscous fluid displaces a more viscous fluid. When the two fluids are miscible, viscous fingering introduces disorder in the velocity field and exerts a fundamental control on the rate at which the fluids mix. Here we analyze the characteristic signature of the mixing process in viscously unstable flows, by means of high-resolution numerical simulations using a computational strategy that is stable for arbitrary viscosity ratios. We propose a reduced-order model of mixing, which, in the spirit of turbulence modeling and in contrast with previous approaches, recognizes the fundamental role played by the mechanical dissipation rate. The proposed model captures the nontrivial interplay between channeling and creation of interfacial area as a result of viscous fingering.

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.

Coriolis effects on rotating Hele-Shaw flows: a conformal-mapping approach

The zero surface tension fluid-fluid interface dynamics in a radial Hele-Shaw cell driven by both injection and rotation is studied by a conformal-mapping approach. The situation in which one of the fluids is inviscid and has negligible density is analyzed. When Coriolis force effects are ignored, exact solutions of the zero surface tension rotating Hele-Shaw problem with injection reveal suppression of cusp singularities for sufficiently high rotation rates. We study how the Coriolis force affects the time-dependent solutions of the problem, and the development of finite time singularities. By employing Richardson's harmonic moments approach we obtain conformal maps which describe the time evolution of the fluid boundary. Our results demonstrate that the inertial Coriolis contribution plays an important role in determining the time for cusp formation. Moreover, it introduces a phase drift that makes the evolving patterns rotate. The Coriolis force acts against centrifugal effects, promoting (inhibiting) cusp breakdown if the more viscous and dense fluid lies outside (inside) the interface. Despite the presence of Coriolis effects, the occurrence of finger bending events has not been detected in the exact solutions.

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.

Transient surface tension in miscible liquids

Evidence of the existence of a transient surface tension between two miscible fluid phases is given. This is done by making use of a density matched free of gravity perturbations, binary liquid of isobutyric acid and water, which presents a miscibility gap and is studied by light scattering. The experiment is performed very near the critical point of the binary liquid, where the diffusion of phases is extremely slow. The surface tension is deduced from the evolution of the structure factor obtained from low angle light scattering. The latter evolution is successfully analyzed in terms of a local equilibrium diffusive approach that makes explicit how the surface tension decreases with time.

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.