Trade Off between Hydrodynamic and Thermodynamic Forces at the Liquid-Liquid Interface.

Improving the understanding of mechanisms involved in a low miscible displacement efficiency is significant for a wide spectrum of applications in subsurface from environment such as groundwater remediation and sequestration to energy extraction such as enhanced oil recovery and geothermal recovery. Two key limiting factors to the efficiency are viscous fingering (VF) instability and dead‐end pores in porous media. Previous research on VF simply assumes all pores are well connected and fluids can be mobilized by convection. However, fluids trapped in dead‐end pores, such as nonaqueous phase liquids (NAPLs) in groundwater remediation, are inaccessible to convection, resulting in even less efficient displacements. Instead of the classic convection‐diffusion/dispersion equation, in this work, we use a fundamentally different capacitance model to incorporate the mass transfer between two pore types in miscible displacements. The hybrid pseudospectral and high‐order finite difference methods are employed to solve the governing equations in a fixed reference frame for simulating the flow dynamics. We found that the viscous fingering instabilities in well‐connected pore network led to an unstable, nonuniform distribution of trapped NAPLs in dead‐end pores network. Different with a traditional view that NAPLs are easily cleaned up, our study indicates the existence of dead‐end pores results in an inefficient cleanup of NAPLs in swept area because of the slow mass transfer of trapped NAPLs between two pore types. A simple model is developed to accurately predict the NAPL concentration behind finger trailing front, which has not been previously examined. Six flow regimes, four of which are new, are then identified.

Viscous fingering instability in one-end-lifted Hele-Shaw cells for producing three-dimensional hierarchical structures.

The nonlinear evolution of the interface between miscible fluids in porous media exhibits different spatiotemporal patterns. The understanding of the physical mechanism behind these patterns is relevant in a wide variety of physicochemical processes. The displacement of a high viscous fluid by a less viscous one in uniform porous medium results in classical viscous fingering (VF) instability. We find that the nonlinear Langmuir-type adsorption of the solute, dissolved in the displacing fluid, leading to the formation of a shock layer can alter the fingering dynamics. The influence of the shock layer on the evolving instability is examined by numerical simulations. Of particular interest are the formation of the shock layer and its impact on the onset of viscous fingering. In this paper, we reveal a critical mechanism of Langmuir-type adsorption that plays a vital role in the speed up of instability. We further infer that by controlling the non-linear adsorption parameter and viscosity contrast of the fluids, the shock layer either ceases to exist or it can be suppressed with VF instability. Hence, the Langmuir adsorption is identified as a strategy to manipulate the instability in a system involving porous media flows.

If a less viscous fluid displaces a more viscous one in a porous medium or Hele-Shaw cell, the unfavorable mobility gradient is known to cause the classical ’viscous fingering instability’. In the past, this instability has been investigated mostly on the basis of Darcy’s law, which involves certain averaging procedures that are usually not valid for the case of a Hele-Shaw displacement. In this diploma thesis, miscible fingering in a horizontal Hele-Shaw cell is studied by means of Stokes simulations and linear stability analysis. The difference from former studies is that a nonmonotonic concentration viscosity profile is used which can exist in nature for specific fluid pairs. Here, particular emphasis is placed on identifying parameter regimes in terms of the Peclet-number Pe, the viscosity contrast R, the value and position of the viscosity maximum μm and cm. Λ is an additional parameter depending on the end-point derivatives. The existing nonlinear Stokes code for monotonic exponential viscosity profiles developed by N. Goyal was modified to incorporate any general viscosity concentration functional relationship. These twodimensional simulations lead to a moving finger with a quasisteady state near the tip of the displacement front for sufficiently large Peclet number and high viscosity ratios. The front thickness d0 and the tip velocity vtip of the quasisteady state is primarily affected by the Peclet number Pe and the viscosity contrast R. The scale of d0 with Pe−1/2 which is found for the monotonic profile is deviating for the nonmonotonic profile. A higher value of the maximum of the nonmonotonic viscosity profile leads to a slower tip and thicker front, while a shift of the maximum from the less viscous fluid to the more viscous fluid leads to a increase of the tip velocity and a decrease of the front thickness. In a second step the results of the direct numerical Stokes flow simulation are used to examine the stability of the quasisteady front to spanwise perturbations. In the linear stability analysis the viscosity contrast R is the dominant factor that affects the stability of the flow. Manickam&Homsy [10] investigated that for the Darcy flow a negative Λ leads to a more stable configuration than a positive Λ, while we see a reversal of this behavior. Only higher values of Λ lead to strong changes for the growth rate σmax of the most dangerous mode and the corresponding wavenumber βmax. Because the shape of the profile is also changing a lot with high values of Λ it is really hard to figure out what leads to this change. The exact reason could not be identified and would need further investigation.

The influence of channeling on viscous fingering instability of CO2 miscible displacement is studied in the paper. Due to the viscous fingering not easily observed in porous media, channeling is used to simplify the viscous fingering instability. We adopted nonlinear simulation to investigate the development of viscous fingering instability during the displacement of Newtonian fluids in a channel by miscible fluids, and the influence of different Pe and different viscosity ratio R was studied. Under homogeneous conditions, when R is the same, the larger the Pe is, the more obvious the convection in the process of CO2 displacement is, the earlier the viscous fingering occurs, and the shorter the time for the finger to break through to the right boundary is. When Pe is the same, the larger the R is, the more unstable the contact area of miscible displacement becomes. More finger structures appear, and more complex fingering phenomena occur. The larger the R is, the earlier the fingering phenomenon appears, and the earlier it breaks through to the right boundary. In addition, we studied the change in Relative Mixing Length (RML) during the diffusion process quantitatively. Finally, our investigation delved into the impact of heterogeneity on viscous fingering. We observed that under significant heterogeneity, viscous fingering tends to manifest preferentially in the direction of increasing permeability.

When a low-viscosity fluid displaces into a higher-viscosity fluid, the liquid-liquid interface becomes unstable causing finger-like patterns. This viscous fingering instability has been widely observed in nature and engineering systems with two adjoined fluids. Here, we demonstrate a hitherto-unrealizable viscous fingering in a single fluid-solid interface. In a single polyelectrolyte fluid on a charge selective surface, selective ion rejection through the surface initiates i) stepwise ion concentration and viscosity gradient boundaries in the fluid and ii) electroconvective vortices on the surface. As the vortices grow, the viscosity gradient boundary pushes away from the surface, resulting viscous fingering. Comparable to conventional one with two fluids, i) a viscosity ratio (M) governs the onset of this electroconvective viscous fingering, and ii) the boundary properties (finger velocity and rheological effects) - represented by M, electric Rayleigh ({{Ra}}_{E}), Schmidt ({Sc}), and Deborah ({De}) numbers - determine finger shapes (straight v.s. ramified, the onset length of fingering, and relative finger width). With controllable onset and shape, the mechanism of electroconvective viscous fingering offers new possibilities for manipulating ion transport and dendritic instability in electrochemical systems.

We perform linear stability analyses and direct numerical simulations to investigate the influence of the dynamic viscosity on viscous fingering instability in miscible slices. Selecting the characteristic scales appropriately, the importance of the magnitude of the dynamic viscosity of individual fluids on viscous fingering in miscible slice has been shown in the context of the transient interfacial tension. Further, we have confirmed this result for immiscible fluids and shown the similarities between viscous fingering in immiscible and miscible slices with transient interfacial tension. In a more general setting, the findings of this letter will be very useful for different viscous flows, such as displacement of a buoyant miscible layers in a vertical Hele-Shaw cell, viscous fingering instability with viscosity-dependent dispersion, etc.

Viscous fingering instabilities, common in confined environments such as porous media or Hele-Shaw cells, surprisingly also occur in unconfined, non-porous settings as revealed by recent experiments. These novel instabilities involve free-surface flows of dissimilar viscosity. We demonstrate that such a free-surface flow, involving a thin film of viscous fluid spreading over a substrate that is prewetted with a fluid of higher viscosity, is susceptible to a similar type of novel viscous fingering instability. Such flows are relevant to a range of geophysical, industrial and physiological applications from the small scales of thin-film coating applications and nasal drug delivery to the large scales of lava flows. In developing a theoretical framework, we assume that the intruding layer and the liquid film over which it flows are both long and thin, the effects of inertia and surface tension are negligible, and both layers are driven by gravity and resisted by viscous shear stress so that the principles of lubrication theory hold. We investigate the stability of axisymmetric similarity solutions, describing the base flow, by examining the growth of small-amplitude non-axisymmetric perturbations. We characterise regions of instability across parameter space and find that these instabilities emerge above a critical viscosity ratio. That is, a fluid of low viscosity intruding into another fluid of sufficiently high viscosity is susceptible to instability, akin to traditional viscous fingering in a porous medium. We identify the mechanism of instability, compare with other frontal instabilities and demonstrate that high enough density differences suppress the instability completely.

We demonstrate a novel instability found within unconfined viscous bands/rims, or free-surface flows involving a longitudinal viscosity contrast. Such instabilities may be described as viscous banding instabilities, non-porous viscous fingering instabilities or unconfined viscous fingering instabilities of free-surface flows involving the intrusion of a less viscous fluid into a band of more viscous fluid. A consequence of this work is that viscous fingering instabilities, widely known to occur in porous media following the seminal work of Saffman & Taylor (Proc. R. Soc. Lond. A, vol. 245, 1958, pp. 312–329), also occur in non-porous environments. Although the mechanism of the viscous banding instability is characteristically different from that of the Saffman–Taylor instability, there are important similarities between the two. The main similarity is that a viscosity contrast leads to instability. A distinguishing feature is that confinement, such as the rigid walls of a Hele-Shaw cell, is not necessary for viscous banding instabilities to occur. More precisely, Saffman–Taylor instabilities are driven by a jump in dynamic pressure gradient, whereas viscous banding instabilities, or non-porous viscous fingering instabilities, are driven by a jump in hydrostatic pressure gradient, directly related to a slope discontinuity across the intrusion front. We examine the onset of instability within viscous bands down an inclined plane, determine conditions under which viscous banding instabilities occur and map out a range of behaviours in parameter space in terms of two dimensionless parameters: the viscosity ratio and the volume of fluid ahead of the intrusion front.

Numerical investigations on two-phase fluid flow in a fractured porous medium fully coupled with geomechanics

Viscous fingering instability, as one of the numerous fluid instabilities and mechanisms of pattern formation, is very common in different fields of research. First studied experimentally in an apparatus called a Hele-Shaw cell in 1898, it has been widely used, for example, as a model for porous media in laboratory testing. This paper provides a brief insight into the mathematical model of the Saffman–Taylor instability (pore scale viscous fingering) and the situations in which viscous fingering can be encountered. Since it is a morphogenetic instability, it is usually considered a nuisance and in recent decades many scientists and researchers have focused on the ways to avoid its occurrence. For instance, the petroleum industry has been continuously trying to discover the best solution for its control since viscous fingering limits oil recovery in porous media. The review provides a historical overview of the phenomena and various methods that have been used to control this instability. The final part of the paper gives a summary and the conclusions that have been drawn as the results of diverse studies.

The development of viscous fingers in circular Hele-Shaw cells is a classical and widely studied fluid mechanical problem. The introduction of wall elasticity (via the replacement of one of the bounding plates by an elastic membrane) can weaken or even suppress the fingering instability, but it also makes the system susceptible to additional solid-mechanical instabilities. We show that in elastic-walled Hele-Shaw cells that are bounded by sufficiently thin elastic sheets the (fluid-based) viscous fingering instability can arise concurrently with a (solid-based) wrinkling instability. We study the interaction between these distinct instabilities, using a theoretical model that couples the depth-averaged lubrication equations for the fluid flow to the Föppl-von Kármán equations, which describe the deformation of the thin elastic sheet. We employ a linear stability analysis to determine the growth rate of non-axisymmetric perturbations to the axisymmetrically expanding bubble, and perform direct numerical simulations to study the nonlinear interactions between the instabilities. We show that the system's behaviour may be characterised by a non-dimensional parameter that indicates the strength of the fluid-structure interaction. For small [large] values of this parameter, the system's behaviour is dominated by viscous fingering [wrinkling], with strong interactions between the two instabilities arising in an intermediate regime.

The viscous fingering in the Hele-Shaw cell can be suppressed by replacing the upper-bounding rigid plate with an elastic membrane. Recently, graphene multilayers while polymer-curing-induced blistering showed the dynamical evolution of viscous fingering patterns on a viscoelastic substrate due to their thickness-dependent elasticity. Under certain conditions, the elastic solid-based instability couples with the viscoelastic substrate-based instability. The mechanisms underlying such a coupling in the blisters of 2D materials and the dynamical evolution of the viscous fingering patterns underneath the blisters are yet to be addressed. Herein, we investigate the viscous fingering instabilities in spontaneously formed blisters of MoS2 multilayers, and provide thorough analytical and experimental insights for the elucidation of the dynamical evolution of the viscous fingering patterns and the coupled instabilities in the blisters. We also estimate the interfacial adhesion energy of the MoS2 flakes over a (poly)vinyl alcohol (PVA) substrate and the confinement pressure inside the MoS2 blisters using a conventional blister-test model. It is observed that the presence of instability gives rise to anomalies in the modeling of the blister test. The adhesion mechanical insights would be beneficial for fundamental research as well as practical applications of 2D material blisters in flexible optoelectronics.

When a less viscous fluid pushes a more viscous fluid in a Hele-Shaw cell, the interface between the two fluids develops an instability leading to the formation of fingerlike patterns, called viscous fingers. This is the so-called Saffman-Taylor or viscous fingering instability [1]. The width of these viscous fingers is, for Newtonian fluids, determined by the capillary number Ca = ΔμU/γ which represents the ratio of viscous forces over capillary forces; Δe is the viscosity difference between the two fluids, U the finger velocity and γ the surface tension. The viscous forces tend to narrow the finger, whereas the capillary forces tend to widen it: the width of the finger decreases with increasing finger velocity. Due to its relative simplicity the viscous fingering instability has received much attention as an archetype of pattern forming systems, both theoretically and experimentally [1, 2] and is by now well understood.

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