Saffman-Taylor Fingers Research Articles

Optimal shape of a variable condenser

The shape of a condenser of maximal cross‐sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary‐value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova‐Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor‐Saffman bubble. In the limit of high capacitance, the Polubarinova‐Kochina contour tends to the SaffmanTaylor finger, which in its own turn coincides with the Morse‐Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non‐isoperimetric optimum) and of maximal confined area (isoperimetric extremum).

On the tip-splitting instability of viscous fingers

The instabilities of Saffman–Taylor viscous fingers are revisited experimentally in the standard linear channel as well as in wedges of angle θ0. The local destabilization of a finger occurs by a splitting of its tip and results in the formation of two branches separated by a fjord. It is shown that, in a first approximation, the central line of a fjord follows a curve normal to the successive profiles of stable fingers. These normal curves are computed analytically for the Saffman–Taylor finger in a linear cell and numerically for the wedges. The length of a fjord is critically sensitive to the position of the initial destabilization of the finger. The nearer to the tip it occurs, the longer the fjord will be. Assuming a uniform spatial distribution of the disturbances in the central part of the finger front it is possible to predict the size distribution of the lateral branches. In the linear channel the probability of branches larger than the channel width is negligible. For wedges of increasing angle the probability of large secondary branches increases. Finally, for wedges with θ0 larger than approximately 90° infinite fjords separating two long-lived structures are observed. Our experimental results also suggest a generalization of the definition of virtual cells. With this new definition it is possible to show that the increasing complexity of the patterns corresponds to a hierarchy of virtual cells of various sizes.

Laplacian Growth I: Finger Competition and Formation of a Single Saffman-Taylor Finger without Surface Tension: An Exact Result

We study the exact non-singular zero-surface tension solutions of the Saffman-Taylor problem for all times. We show that all moving logarithmic singularities a_k(t) in the complex plane \omega = e^{i\phi}, where \phi is the stream function, are repelled from the origin, attracted to the unit circle and eventually coalesce. This pole evolution describes essentially all the dynamical features of viscous fingering in the Hele-Shaw cell observed by Saffman and Taylor [Proc. R. Soc. A 245, 312 (1958)], namely tip-splitting, multi-finger competition, inverse cascade, and subsequent formation of a single Saffman-Taylor finger.

On the rôle of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension

Small surface tension is known to select a discrete family from the continuum of Saffman–Taylor finger solutions with zero surface tension. Here a method developed recently in Chapman et al . [2] is employed in which a naive perturbation expansion in powers of the surface tension parameter is optimally truncated to show exponentially small terms being switched on across Stokes lines. These terms are responsible for finger selection.

Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. II. Numerical study.

We implement a phase-field simulation of the dynamics of two fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. We demonstrate the use of this technique in different situations including the linear regime, the stationary Saffman-Taylor fingers, and the multifinger competition dynamics, for different viscosity contrasts. The method is quantitatively tested against analytical predictions and other numerical results. A detailed analysis of convergence to the sharp interface limit is performed for the linear dispersion results. We show that the method may be a useful alternative to more traditional methods.

Extremum principles for selection in the Saffman–Taylor finger and Taylor–Saffman bubble problems

We consider the Saffman–Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele–Shaw configuration, and the Taylor–Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of traveling wave solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the channel through which it propagates. To select the “correct” member of the family of solutions we propose two related extremum principles. Employing these principles for a symmetric family with zero surface tension (σ=0) selects the solution (finger or bubble, viscous or inviscid) which happens to be the same as that obtained by taking the limit σ→0 of the nonzero isotropic surface tension problem. For other problems, e.g., perturbation by anisotropic surface tension, the fingers selected by the two approaches are not necessarily the same. We claim that the finger selected by our criteria describes what will be observed on an intermediate time scale in which the effect of perturbations is neglected, whereas the finger selected by taking the limit of vanishingly small perturbations describes what will be observed on the asymptotic time scale O(1/σ) for the problem with the perturbation included. The intermediate time scale is much shorter than the asymptotic time scale. For infinitesimal σ the asymptotic time scale may be well beyond any reasonable observational time scale.

Comment on “Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: An Exact Result”

A Comment on the Letter by Mark Mineev-Weinstein, Phys. Rev. Lett. 80, 2113 (1998). The authors of the Letter offer a Reply.

Comment on “Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: An Exact Result”

A Comment on the Letter by Mark Mineev-Weinstein, Phys. Rev. Lett. 80, 2113 (1998). The authors of the Letter offer a Reply.Received 25 March 1998DOI:https://doi.org/10.1103/PhysRevLett.81.5950©1998 American Physical Society

Comment on “Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: An Exact Result”

A Comment on the Letter by Mark Mineev-Weinstein, Phys. Rev. Lett. 80, 2113 (1998). The authors of the Letter offer a Reply.

Numerical computation of two-dimensional viscous fingering by a conformal mapping method

In this paper we numerically compute the long-time evolution of the nonsingular Saffman-Taylor finger (NSTF) with negligible surface tension using a conformal mapping method. Confirming the one-step theory prediction of the NSTF, we resolve one of the Saffman-Taylor riddles successfully. At the same time, we find a successive averaging operator that not only can reduce residual errors in summing up the Fourier series and speed up calculation greatly but also can be easily applied. Its convergence is proved theoretically.

Selection in the Saffman-Taylor finger problem and the Taylor-Saffman bubble problem without surface tension

We consider the Saffman-Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele-Shaw configuration, and the Taylor-Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the porous channel through which it propagates. To select the correct member of the family of solutions, the conventional approach has been to add surface tension σ and then take the limit σ → 0. We propose a selection criterion that does not rely on surface tension arguments.

Surface tension and dynamics of fingering patterns

We study the minimal class of exact solutions of the Saffman-Taylor problem with zero surface tension, which contains the physical fixed points of the regularized (non-zero surface tension) problem. New fixed points are found and the basin of attraction of the Saffman-Taylor finger is determined within that class. Specific features of the physics of finger competition are identified and quantitatively defined, which are absent in the zero surface tension case. This has dramatic consequences for the long-time asymptotics, revealing a fundamental role of surface tension in the dynamics of the problem. A multifinger extension of microscopic solvability theory is proposed to elucidate the interplay between finger widths, screening and surface tension.

Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: An Exact Result

Using our exact time-depending solutions, we solve the Saffman-Taylor finger selection problem in the absence of surface tension by showing that an arbitrary interface in a Hele-Shaw cell evolves to a single uniformly advancing finger occupying one half of the channel width. This result contradicts the generally accepted belief that surface tension is indispensable for the selection of the one-half-width finger.

Fingering in 2D Parallel Viscous Flow

We study the displacement of miscible fluids of different viscosities (viscosity ratio AI) between two parallel plates, using the BGK lattice gas method. At high Peclet numbers, a symmetric interface develops in the gap between the plates. Above M +J 10, the interface becomes a well-defined finger, the reduced width of which tends to A~n = 0.56 at large AI. Assuming that a miscible displacement at high Peclet numbers is equivalent to an immiscible displacement at high capillary numbers, we extend the calculations of Reinelt & Saffman for immiscible fluids, and find the analytical shape of the finger. The result is compared to the celebrated Saffman-Taylor finger.

Non-singular Saffman-Taylor finger

In this paper the singularity in the Saffman-Taylor solution for viscous finger in Hele-Shaw cell in the absence of surface tension is removed by slight modification. In the present model analytically there exists a unique steady width λ = 0.5 of the viscous finger to which the fingers of other width approach. The conclusion is confirmed by our numerical computation.

An experimental investigation of initial oscillations in a radial Hele-Shaw cell

Hele-Shaw cell is a laboratory device consisting of two parallel plates of glass separated by a thin gap. In this cell, in the flow of two immiscible fluids, when a fluid of higher viscosity is displaced by a fluid of lower viscosity, the less viscous fluid is observed to form “fingers” into the more viscous one due to the unstable interface. The Saffman-Taylor or viscous finger instability has been examined and modeled for over forty years for the rectilinear Hele-Shaw cell and about half as long for the radial Hele-Shaw cell. In this paper, we study, in detail, the early development of viscous instabilities in a radial Hele-Shaw cell. This source flow configuration has been chosen so that the instability can be monitored precisely. The objective of this study is to examine the onset of fingering, i.e. initial number of fingers that form, and the evolution of interface instability. Our experiments suggest that there may be some order in this formation process and one can model this aspect by considering the unsteady velocity components and predicting temporal changes in wavenumber responsible for the initial number of fingers and may be later accounting for the fingertip oscillations and splitting.

Crystalline Saffman–Taylor Fingers

We study the existence and structure of steady-state fingers in two-dimensional solidification, when the surface energy has a crystalline anisotropy so that the energy-minimizing Wulff shape and hence the solid–liquid interface are polygons, and in the one-sided quasi-static limit so that the diffusion field satisfies Laplace’s equation in the liquid. In a channel of finite width, this problem is the crystalline analog of the classic Saffman–Taylor smooth finger in Hele–Shaw flow. By a combination of analysis and numerical Schwarz–Christoffel mapping methods, we show that, as for solutions of the smooth problem, for each choice of Wulff shape there is a critical maximum value of the magnitude of surface tension above which no convex steady-state solutions exist. We then exhibit convergence of convex crystalline solutions to convex smooth solutions as the Wulff shape approaches a circle. We also consider the open dendrite geometry and show that there are no steady-state solutions having a finite number of sides for any crystalline surface energy. This is in striking contrast to the smooth case and an indication that the time-dependent behavior may be more complicated for crystalline surface energies.

Saffman-Taylor fingers with anisotropic surface tension

We have qualitatively explained the experiments of McCloud and Maher for the viscous fingering problem in which an anisotropy in the surface tension parameter was imposed by engraving a grid in one of the plates of the Hele-Shaw cell. We approached the problem in an analytical form by extending solvability theory to incorporate the effect of anisotropy. For the case in which the surface tension has a maximum at the finger tip, our theory provides two possible solutions: one corresponding to the solution of the isotropic case and a new solution which, below a threshold of the surface tension parameter, predicts a wider finger than the isotropic solution. Intuitivelt, we expect the “old” solution, namely the one that does not differ from the isotropic case, to be the selected solution for large values of the surface tension parameter and we expect the new solution to be selected for small values of the surface tension parameter. This was confirmed by dynamical simulations of the interface. The simulations predict that for the case in which the surface tension has a maximum at the finger tip, anisotropy is irrelevant for large values of the surface tension parameter. Furthermore, below a threshold in this surface tension parameter, the selected finger width is systematically wider than the corresponding isotropic case.

Finger competition and viscosity contrast in viscous fingering. A topological approach

Abstract Finger competition in rectangular Hele-Shaw cells is discussed with particular focus on topological features of the velocity field. A reduced description of the interface dynamics is provided in terms of the motion of topological defects which obey simple conservation rules. Based on the existence of a topological invariant, a time-dependent quantity S(t) , directly related to the number of topological defects, is proposed as a global characterization of the finger competition. This quantity is not simply related to the interface shape but may be used as a measure of the ‘complexity’ of evolving patterns and to elucidate the conditions for eventual approach to a single finger stationary solution. Some representative numerical simulations for two extreme limits of the viscosity contrast parameter are presented and analyzed in terms of the dynamics of topological defects. Two specific sequences or histories for the creation/annihilation of defects are then identified, corresponding to radically different behavior of competing fingers. With the help of a local vorticity analysis, we conclude that, in the limit of low viscosity contrast, the Saffman-Taylor finger has a drastically reduced but non-vanishing basin of attraction.

Finger competition and viscosity contrast in viscous fingering. A topological approach

Finger competition in rectangular Hele-Shaw cells is discussed with particular focus on topological features of the velocity field. A reduced description of the interface dynamics is provided in terms of the motion of topological defects which obey simple conservation rules. Based on the existence of a topological invariant, a time-dependent quantity S(t), directly related to the number of topological defects, is proposed as a global characterization of the finger competition. This quantity is not simply related to the interface shape but may be used as a measure of the ‘complexity’ of evolving patterns and to elucidate the conditions for eventual approach to a single finger stationary solution. Some representative numerical simulations for two extreme limits of the viscosity contrast parameter are presented and analyzed in terms of the dynamics of topological defects. Two specific sequences or histories for the creation/annihilation of defects are then identified, corresponding to radically different behavior of competing fingers. With the help of a local vorticity analysis, we conclude that, in the limit of low viscosity contrast, the Saffman-Taylor finger has a drastically reduced but non-vanishing basin of attraction.