Radial viscous fingering in miscible Hele-Shaw flows: A numerical study (original) (raw)
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Numerical study of pattern formation in miscible rotating Hele-Shaw flows
Physical Review E, 2006
The dynamics of the diffusing interface separating two miscible fluids in a rotating Hele-Shaw cell is studied by intensive and highly accurate numerical simulations. We perform numerical experiments in a wide range of parameters, focusing on the influence of viscosity contrast and Korteweg stresses on the shape of the interfacial patterns. A great variety of morphological behaviors is systematically introduced, and a wealth of interesting phenomena related to finger competition dynamics, filament stretching, and interface pinch off are reveal. Our simulations exhibit miscible patterns that bear a strong resemblance to their immiscible counterparts for larger Korteweg stresses. The quantitative equivalence between such stresses and the usual immiscible surface tension is studied. The concept of an effective interfacial tension is considered, allowing the direct and precise calculation of the important fingering properties under miscible circumstances. Our results show excellent agreement with existing experiments and simulations for corresponding immiscible displacements. This agreement refers to a striking similarity between miscible and immiscible pattern morphologies, and also to an accurate prediction for the typical number of miscible fingering structures formed. Our findings suggest that the effective interfacial tension is both qualitatively and quantitatively equivalent to its immiscible counterpart.
Viscosity contrast effects on fingering formation in rotating Hele-Shaw flows
Physical Review E, 2005
The different finger morphologies that arise at the interface separating two immiscible fluids in a rotating Hele-Shaw cell are studied numerically. The whole range of viscosity contrast is analyzed and a variety of fingering patterns systematically introduced, including the case in which the inner fluid is less viscous than the outer one. Our numerical results demonstrate that both the magnitude and the sign of the viscosity contrast strongly affect the shape of the emerging fingers, and also their length distribution. We have also found that the occurrence and location of pinch-off singularities are remarkably modified when the inner fluid is less viscous: instead of generating an isolated detaching drop, a full finger is disconnected from the interface. Finally, we have verified that the finger competition phenomena revealed by our simulations are correctly predicted by a weakly nonlinear analysis of the pattern development, showing that such important finger competition dynamics is already set at relatively early stages of interfacial evolution.
Viscous fingering in Hele-Shaw cells
The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.
Stability of viscous fingering in lifted Hele-Shaw cells with a hole
Physical Review Fluids
Lifted Hele-Shaw cells typically display viscous fingering of liquids, which in turn leads to branched fractal patterns in the absence of any anisotropies. Recently, experiments involving parallely lifted Hele-Shaw cells with holes in the cell plates, also termed as "multiport lifted Hele-Shaw cells," have been used to generate more regular meshlike patterns in the liquid film. Although such patterns promise usefulness in several applications, their spatiotemporal evolution needs to be theoretically understood for better synthesis. As a first step, therefore, we examine the stability of fingers evolving from a single hole by focusing on flow of an annular film of liquid placed in a lifted Hele-Shaw cell. We use linear stability analysis to find the growth rate of azimuthally periodic perturbations of the inner and outer interfaces around the evolving base state of the liquid film. To validate the results of our stability analysis, we also perform resolved numerical simulations of the setup via an in-house solver based on lubrication theory, which uses front-tracking method to evolve the interface in time and space. For a wide range of parameters and wave numbers, we find excellent agreement in the growth rates predicted by the linear stability analysis with the numerical simulation. The results of the stability analysis are expressed in terms of the capillary number, initial nondimensional plate separation, and initial ratio of the interface radii. Furthermore, using the results from our linear stability analysis, we generate a phase map to demarcate the flow regimes corresponding to unstable and stable states for the interfaces. Numerical simulations of the interface evolution over finite times are consistent with the results predicted by the proposed analysis. These finite-time simulations successfully capture the presence of shielding of the fingers at both the inner and outer interface. The proposed theoretical analysis and insights obtained through numerical simulations thus provide a framework for accurately predicting and experimentally realizing stable fluid patterns in a multiport Hele-Shaw cell.
Suppression of Viscous Fingers in Miscible Hele-Shaw Flow
2011
The flow of two immiscible fluids between closely-spaced parallel plates can be highly unstable and produce a series of complex fingering patterns when the less viscous injected fluid invades the more viscous one. Air displacing granular material in such a Hele-Shaw geometry shows similar patterns with sharp features consistent with the granular/air surface tension being virtually zero [1]. Here we investigate the flow of two miscible fluids in a radial Hele-Shaw cell, with an inner liquid displacing an outer one of higher viscosity. We use two glycerol-water mixtures so that the viscosity can be tuned by varying the glycerol concentration. We vary the plate spacing and flow rate as well as the fluid viscosites. The nonequilibrium interfacial tension between these two miscible fluids is expected to be nearly zero. However, extrapolating to zero surface tension in the linear theory for Hele-Shaw flow does not describe our results. Specifically, flow becomes stable even when the inner liquid has a much lower viscosity than the outer one. At higher velocity, it is possible to see small amplitude fingering patterns develop.
Coatings, 2020
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 cir...
Dynamics of viscous fingers in rotating Hele-Shaw cells with Coriolis effects
Physical Review E, 2007
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 difference͒ 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.
Radial Hele-Shaw flow with suction: Fully nonlinear pattern formation
Physical Review E, 2014
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.
Controlling radial fingering patterns in miscible confined flows
Physical Review E, 2010
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.