Growth Of Fingers Research Articles

Experimental study of the growth of cholesteric fingers subjected to an AC electric field and a temperature gradient

We study the behaviour of cholesteric fingers of the first and second types (CF1 and CF2, respectively) which form in cholesteric samples treated for homeotropic anchoring when they are subjected to the combined action of an AC electric field and a temperature gradient. We investigate under which conditions each type of finger can drift and form a spiral. We then show that the ends of the growing CF1 follow circular trajectories, the curvatures of which depend linearly on the temperature gradient. This observation opens the possibility of measuring the thermal Lehmann coefficient in any cholesteric liquid crystal at all temperatures.

Differences in miscible viscous fingering of finite width slices with positive or negative log-mobility ratio

When a sample of fluid of finite size is displaced in a porous medium by another miscible fluid, viscous fingering may occur when the two fluids have different viscosities. Depending whether the sample is more or less viscous than the carrier fluid, the log-mobility ratio R [defined as R=ln(mu_{2}mu_{1}) where mu_{2} and mu_{1} are the viscosities of the sample and of the carrier] is respectively positive or negative. In the case of a linear displacement of a finite slice of fluid, R>0 leads to fingering of the rear interface of the sample where the less viscous carrier invades the more viscous sample. If R<0 , it is on the contrary the frontal interface of the sample that develops fingers where the less viscous sample displaces the more viscous bulk solution. We investigate here numerically the differences in fingering dynamics between the positive and negative log-mobility ratio cases leading to the growth of fingers against or along the direction of the flow, respectively. To do so, we integrate Darcy's law coupled to a convection-diffusion equation for the concentration of a solute ruling the viscosity of the finite-size sample. The statistical moments of the solute's concentration distribution are analyzed as a function of dimensionless parameters of the problem such as the length of the slice l , the log-mobility ratio R , and the ratio between transverse and axial dispersion coefficients . We find that, on average, the mixing zones and the width of the sample broadening due to fingering are larger for negative R than for positive R . This is due to the fact that fingers travel quicker in the flow direction than against the flow. Relevance of our results are discussed for interpretation of experimental results obtained in chromatographic separation and for understanding conditions of enhanced spreading of contaminants in aquifers.

The influence of wettability on NAPL dissolution fingering

Laboratory experiments and numerical simulations in homogeneous porous media were used to investigate the influence of porous medium wettability on the formation and growth of preferential dissolution pathways, dissolution fingers, during nonaqueous phase liquid (NAPL) dissolution. As the porous medium became increasingly NAPL-wet, dissolution fingers grew wider and slower. This result was observed in physical experiments with 0% and 100% NAPL-wet conditions and confirmed with numerical simulations at these and intermediate wettabilities. A previously derived expression for an upscaled mass transfer rate coefficient that accounts for the growth of dissolution fingers was used to quantify the effect of fingering on overall NAPL removal rates. For the test cases evaluated, NAPL dissolution fingering controlled the overall rate of NAPL dissolution after the dissolution front moved 4 cm in 0% NAPL-wet conditions and 18 cm in 100% NAPL-wet conditions. Thus, even in completely NAPL-wet media dissolution fingering may control the overall rate of NAPL dissolution after relatively short travel distances. The importance of NAPL dissolution fingering in heterogeneous systems with spatially varying NAPL saturations, though, remains an important question for future work.

The asymmetric self-assembly mechanism of adherens junctions: a cellular push–pull unit

To form adherens junctions (AJ), cells first establish contact by sending out lamellipodia onto neighboring cells. We investigated the role of contacting cells in AJ assembly by studying an asymmetric AJ motif: finger-like AJ extending across the cell–cell interface. Using a cytoskeleton replica and immunofluorescence, we observed that actin bundles embedded in the lamellipodia are co-localized with stress fibers in the neighboring cell at the AJ. This suggests that donor lamellipodia present actin fingers, which are stabilized by acceptor lamellae via acto-myosin contractility. Indeed, we show that changes in actin network geometry promoted by Rac overexpression lead to corresponding changes in AJ morphology. Moreover, contractility inhibition and enhancement (via drugs or local traction) lead respectively to the disappearance and further growth of AJ fingers. Thus, we propose that receiving lamellae exert a local pull on AJ, promoting further polymerization of the donor actin bundles. In spite of different compositions, AJ and focal contacts both act as cellular mechanosensors.

The Magnetic Rayleigh‐Taylor Instability in Three Dimensions

We study the magnetic Rayleigh-Taylor instability in three dimensions, with focus on the nonlinear structure and evolution that results from different initial field configurations. We study strong fields in the sense that the critical wavelength λc, at which perturbations along the field are stable, is a large fraction of the size of the computational domain. We consider magnetic fields that are initially parallel to the interface, but have a variety of configurations, including uniform everywhere, uniform in the light fluid only, and fields that change direction at the interface. Strong magnetic fields do not suppress instability; in fact, by inhibiting secondary shear instabilities they reduce mixing between the heavy and light fluid and cause the rate of growth of bubbles and fingers to increase in comparison to hydrodynamics. Fields parallel to, but whose direction changes at, the interface produce long, isolated fingers separated by the critical wavelength λc, which may be relevant to the morphology of the optical filaments in the Crab nebula.

An experimental study of miscible displacement with gravity-override and viscosity-contrast in a Hele Shaw cell

An experimental investigation of miscible displacements at constant volume flow-rate under the coupled effects of mobility contrast and gravitational segregation has been performed in a Hele Shaw cell having an aspect ratio, width to length, of 1:2. While the viscosity ratio was large (M > 180), the experiments covered both the neutrally buoyant case through to gravity-override-dominated unstable displacements. Dependence of the global displacement properties on the Gravity number (G) and the Peclet number (Pe) were quantified using a flow visualization technique. Within the experiment’s parameter range, no matter how complex the finger patterns became, and independent of G, the area grew linearly in time. As a result, the thickness of the injected less dense and less viscous fluid was almost constant at a value of 0.5–0.58 of the cell thickness with a weak dependence on Peclet number. Based on transversely averaged concentration profiles, the dependence of the average finger length was investigated and it also grew linearly in time. The displacement efficiency and breakthrough time decreased with increase of G, while the longitudinal finger growth rate increased with G. The averaged finger width followed the opposite trend and decreased as G increased. Velocity of the leading fingertip grew linearly with G at fixed Pe. The larger the value of Pe, the faster fingertips spread. As was to be expected, the larger the gravity number, the larger the global tilting of the whole finger pattern. The fractal dimension of the distorted interface at breakthrough was investigated, and it varied from 1.54 for the neutrally buoyant case to 1.08 for the gravity override dominated case.

Density-dependent dispersion in heterogeneous porous media Part I: A numerical study

In this paper, we describe carefully conducted numerical experiments, in which a dense salt solution vertically displaces fresh water in a stable manner. The two-dimensional porous media are weakly heterogeneous at a small scale. The purpose of these simulations, conducted for a range of density differences, is to obtain accurate concentration profiles that can be used to validate nonlinear models for high-concentration-gradient dispersion. In this part we focus on convergence of the computations, in numerical and statistical sense, to ensure that the uncertainty in the results is small enough. Concentration variances are computed, which give estimates of the uncertainty in local concentration values. These local variations decrease with increasing density contrast. For tracer transport, obtained longitudinal dispersivities are in accordance with analytical findings. In the case of high-density contrasts, stabilizing gravity forces counteract the growth of dispersive fingers, decreasing the effective width of the transition zone. For small log-permeability variances, the decrease of the apparent dispersivity that is found is in agreement with laboratory results for homogeneous columns.

Fabrication of Nano‐Scaled α‐Al2O3 Crystallites Through Heterogeneous Precipitation of Boehmite in a Well‐Dispersed θ‐Al2O3‐Suspension

The possibility of eliminating finger or vermicular growth of α‐Al2O3 particles obtained by calcination of boehmite was examined. Heterogeneous precipitation of boehmite in a well‐dispersed θ‐Al2O3 suspension was first prepared, in which the mass ratio of boehmite to θ‐crystallite was evaluated to form agglomerates of similar sizes that will form α‐Al2O3 crystallites of &lt;100 nm in diameter. θ‐ to α‐phase transformation of alumina experiences a nucleation and growth mechanism, with the critical size of nucleation being ∼25 nm for θ‐Al2O3 and the size for accomplishment of transformation followed by finger growth being ∼100 nm. Hence, fabricating agglomerates that would form α‐Al2O3 crystallites with sizes &lt;100 nm accompanied with appropriate thermal treatments can be a method for obtaining α‐Al2O3 crystallites free of finger growth. It is found that proper preparation of the agglomerate with appropriate size may initiate a simultaneous and lower temperature θ‐ to α‐Al2O3 phase transformation for such powder systems, substantially limiting the mass transfer among the newly formed α‐Al2O3 particles. Moreover, α‐Al2O3 crystallites free of finger growth can be obtained.

Pressure Head Profile within Growing Fingers in Initially Dry Glass Beads

Wetting front instability is one of the mechanisms producing preferential flow in soils. In this study, we measured the pressure head within growing fingers in three-dimensional soil columns and examined the applicability of the prefingered flow theory and the similarity solution. The pressure head within a growing finger was measured during infiltration into a porous medium consisting of fine over coarse layered glass beads in a transparent, acrylic column. To ensure the measurement of pressure heads within growing fingers whose pathways cannot be known a priori, we made 10 insertion ports for every measurement position in the column. The fingertip velocity was essentially constant with time, while the infiltration rate increased. Fingers merged as they grew, which complicated the finger flow physics in three dimensions. The profiles of the pressure head were nearly linear during early finger growth. As the finger lengthened, the pressure head at all measured points decreased to an almost constant value. The prefingered flow model showed good applicability not only to induction zone formation, but also to the early finger growth when it remained saturated. Conversely, the similarity solution was not as applicable to growing fingers that were not in a steady state.

Surface tension driven fingering of a viscoplastic film

We investigate surface-tension-driven fingering of a thin layer of viscoplastic fluid. Using standard lubrication theory we derive an equation for the flow of the film which includes effects of surface tension and yield stress. We obtain traveling-wave solutions to model the advance of a steadily propagating front and then apply a linear stability analysis to model finger growth and to determine the effect of the yield stress. Qualitative agreement is demonstrated between the numerical results and experiments.

Anomalous Richtmyer-Meshkov Fingering in Dissipative Particle Systems

We study the fingering instability induced by a shock that propagates across a perturbed interface that separates two types of discrete particles. If collisions between particles conserve energy, then the relative sizes and growth rates of the fingers are similar to those in the analogous shock-induced fingering instability in fluids. However, we show that energy loss during particle collisions, even when very small, causes the qualitative features of the finger growth to be completely opposite to the fluid case. The fingers formed by light particles grow faster and become longer and narrower than the fingers formed by heavy particles. In addition, the finger composed of light particles collapses into an extremely compact, tortuous filament, and diffusive mixing between particle types at the particle scale is heavily suppressed.

Dynamics of driven liquid films on heterogeneous surfaces

A computational study is reported of the instability and growth of fingers for liquid films driven over heterogeneous surfaces. Computations are performed using a variation of the precursor-film model, in which a disjoining pressure term is used to introduce variation in the static contact angle, which in turn models surface heterogeneity. The formulation is shown to yield results consistent with the Tanner–Hoffman–Voinov dynamic contact angle formula for sufficiently small values of the precursor film thickness. A modification of the disjoining pressure coefficient is introduced which yields correct variation of dynamic contact angle for finite values of the precursor film thickness. The fingering instability is examined both for cases with ordered strips of different static contact angle and for cases with random variation in static contact angle. Surface heterogeneity is characterized by strip width and amplitude of static contact angle variation for the case with streamwise strips and by correlation length and variance of the static contact angle variation from its mean value for the random distribution case.

Fingering Instability in Water-Oil Displacement

Following the classical Buckley–Leverett theory for the two-phase immiscible flows in porous media a non-linear evolution equation for the water-oil displacement front is formulated and studied numerically. The numerical simulations yield a physically plausible picture of the fingering instability known to develop in water-oil systems. A way to control the unrestricted growth of fingers is discussed. Distinctions and similarities with dynamically related Saffman–Taylor and Darrieus–Landau problems are outlined.

Anomalous motion of viscous fingers in surfactant solutions in a Hele–Shaw cell

Viscous fingering in surfactant solutions in a rectangular Hele–Shaw cell was investigated. Test fluids were aqueous solutions of cetyltrimethylammonium bromide (CTAB) with sodium salicylate (NaSal) as a counter ion, and the ratio of mole concentration of CTAB and that of NaSal was 1–7.7. Two fluids that had a mole concentration different from that of CTAB were used. Air was injected into the cell and the growth of the interface between air and a CTAB/NaSal solution was observed. The fingertip grew similar to the finger growth in shear-thinning fluids at low pressure gradients. It took a cuspidate shape at the intermediate pressure gradient, and a sudden protrusion at a critical shear rate occurred. In high shear rate regions, the finger behaved as in a less shear-thinning fluid. These phenomena relate to rheological properties of the test fluids. Comparison with flow curves for CTAB/NaSal systems showed that the critical shear rate related to the shear rate at which a bending point appeared in the flow curve.

Stretch flow of thin layers of Newtonian liquids: Fingering patterns and lifting forces

We study the stretch flow of a thin layer of Newtonian liquid constrained between two circular plates. The evolution of the interface of the originally circular bubble is studied when lifting one of the plates at a constant velocity and the observed pattern is related to the measured lifting force. By comparing experimental results to numerical simulations using a Darcy’s law model we can account for the fully nonlinear evolution of the observed fingering pattern. One observes an initial destabilization of the interface by growth of air fingers due to a Saffman–Taylor-like instability and then a coarsening of the pattern toward a circular interface until complete debonding of the two plates occurs. Numerical simulations reveal that when relating the observed patterns to the lifting force not only the number of fingers but also the amplitude of the fingering growth has to be taken into account. This is consistent with the experimental observations.

Fingering instability in particle systems

In many multiphase systems, material interfaces can be destabilized by shocks. Small disturbances at these interfaces can grow in size to form large-scale fingers. We consider a shock propagating through a system that consists of two types of particles, of different mass, that are initially separated by an interface, but are free to mix. In the classical case of immiscible fluids, the finger of heavy fluid propagating into the light fluid grows faster and becomes much thinner than the finger of light fluid propagating into the heavy fluid. We show that collisions between particles of different types lead to shock focusing that causes a secondary flow that is initially similar to the fluid case. However, the particle system can exhibit completely different qualitative behavior in the nonlinear-growth phase and can give rise to the situation where the finger of heavy material is actually wider than the finger of the light material. We show that this qualitative change is due to a strong decompression that occurs in the heavy material. We also show that microscopic mixing can have an important impact on finger growth.

Non-Normal Effects on Salt Finger Growth

Abstract Salt fingers, which occur because of the difference in diffusivities of salt and heat in water, may play an important role in ocean mixing and circulation. Previous studies have suggested the long-time dominance of initially fastest growing finger perturbations. Finger growth has been theoretically derived in terms of the normal modes of the idealized system, which include a growing mode and a pair of decaying internal wave modes. Because these normal modes are not orthogonal, however, transient effects can occur related to the interaction between the modes, as explained by the generalized stability theory of non-normal growth. Initial growth of a perturbation that is not along a normal mode can be faster than the leading normal mode. In this study, the effects of non-normal growth on salt finger formation are investigated. It is shown that some salt finger perturbations that are a superposition of the growing mode and the decaying modes initially grow faster than pure growing normal mode perturbations. These non-normal effects are found to be significant for up to 10 or more e-folding times of the growing normal mode. The generalization of the standard idealized salt finger growth dynamics to include non-normal effects is found to lead to fastest-growing fingers that agree less well with observed fully developed salt fingers than the fastest-growing normal mode previously investigated.

Tip Velocity of Viscous Fingers in Shear-Thinning Fluids in a Hele-Shaw Cell

Viscous fingering in non-Newtonian fluids in a rectangular Hele-Shaw cell was investigated. The cell was filled with a 0.5 or 1.0wt% aqueous solution of carboxymethylcellulose (CMC), a shear-thinning fluid. Air was injected into the cell and the growth of viscous fingers was observed. The velocity of finger tip was characterized by the pressure gradient. A modified Darcy law was able to describe the characteristics of the tip velocity that the growth rate of the tip velocity increased with increasing pressure gradient in the CMC solutions. The prediction of tip velocity with the modified Darcy law indicated that an effective pressure gradient near the tip was larger than the average pressure gradient between the finger tip and the cell exit and that the rate of increase depended on the cell gap width.

Study of the long-time dynamics of a viscous vortex sheet with a fully adaptive nonstiff method

A numerical investigation of the long-time dynamics of two immiscible two-dimensional fluids shearing past one another is presented. The fluids are incompressible and the interface between the bulk phases is subjected to surface tension. The simple case of density and viscosity matched fluids is considered. The two-dimensional Navier–Stokes equations are solved numerically with a fully adaptive nonstiff strategy based on the immersed boundary method. Dynamically adaptive mesh refinements are used to cover at all times the separately tracked fluid interface at the finest grid level. In addition, by combining adaptive front tracking, in the form of continuous interface marker equidistribution, with a predictor–corrector discretization an efficient method is introduced to successfully treat the well-known numerical difficulties associated with surface tension. The resulting numerical method can be used to compute stably and with high resolution the flow for wide-ranging Weber numbers but this study focuses on the computationally challenging cases for which elongated fingering and interface roll-up are observed. To assess the importance of the viscous and vortical effects in the interfacial dynamics the full viscous flow simulations are compared with inviscid counterparts computed with a state-of-the-art boundary integral method. In the examined cases of roll-up, it is found that in contrast to the inviscid flow in which the interface undergoes a topological reconfiguration, the viscous interface remarkably escapes self-intersection and rich long-time dynamics due to separation, transport, and diffusion of vorticity is observed. An even more striking motion occurs at an intermediate Weber number for which elongated interpenetrating fingers of fluid develop. In this case, it is found that the Kelvin–Helmholtz instability weakens due to shedding of vorticity and unlike the inviscid counterpart in which there is indefinite finger growth the viscous interface is pulled back by surface tension. As the interface recedes, thin necks connecting pockets of fluid with the rest of the fingers form. Narrow jets are observed at the necking regions but the vorticity there ultimately appears to be insufficient to drain all the fluid and cause reconnection. However, at another point, two disparate portions of the interface come in close proximity as the interface continues to contract. Large curvature points and an intense concentration of vorticity are observed in this region and then the motion is abruptly terminated by the collapse of the interface.

Impact of molten metal droplets on the tip of a pin projecting from a flat surface

We photographed impact of molten zinc and tin droplets on a flat steel surface from which the tip of a small pin was projecting. The height of the pin and its distance from the droplet center was varied. A three-dimensional model of droplet impact and solidification was used to predict splat shapes formed by droplets after they had spread and frozen. The model calculated velocity and pressure distributions in droplets and the growth of solid layers. When the offset distance of the pin was less than the droplet radius liquid flowed over the pin so that it had relatively little impact on the final splat shape. At larger offset distances a liquid sheet jetted from under the droplet after impact and impinged on the vertical surface of the pin. If the pin height was sufficiently large, approximately the same as the final splat thickness, it obstructed flow of liquid so that the solidified splat had a V-shaped notch in it. If pin height was made less than the average splat thickness liquid flowed over it and the final splat was circular, but the pin reduced flow velocity and suppressed growth of fingers along the edges of the splat that passed over the pin. Reduced velocities also resulted in faster growth of the solidified layer around the pin.