Related papers: Two-finger selection theory in the Saffman-Taylor …
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…
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…
The selection of Saffman-Taylor fingers by surface tension has been extensively investigated. In this paper we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele-Shaw cell by small…
A dynamical systems approach to competition of Saffman-Taylor fingers in a channel is developed. This is based on the global study of the phase space structure of the low-dimensional ODE's defined by the classes of exact solutions of the…
We study the singular effects of vanishingly small surface tension on the dynamics of finger competition in the Saffman-Taylor problem, using the asymptotic techniques described in [S. Tanveer, Phil. Trans. R. Soc. Lond. A 343, 155…
We solve numerically the nonlinear differential equation for the Hele-Shaw, Saffman-Taylor problem derived in the preceding work. Stationary solutions with no free phenomenological parameters are found to fit the measured patterns. The…
Nonlinear time-dependent differential equations for the Hele-Shaw, Saffman-Taylor problem are derived. The equations are obtained using a separable ansatz expansion for the stream function of the displaced fluid obeying a Darcian flow.…
We develop a stream function approach for the horizontal Hele-Shaw, Saffman-Taylor finger. The model yields a nonlinear time-dependent differential equation. The finger widths derived from the equation are $1>\lambda>\frac{1}{\sqrt{5}}$, in…
We address pattern selection problems in nonlinear interface dynamics by maximizing the entropy of the most probable (classical) scenario associated with the processes. This variational principle we applied to well-known selection problems…
We present a new class of exact solutions for the so-called {\it Laplacian Growth Equation} describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to common belief, we prove that these solutions…
We present an analytical study for predicting the finger width of the Saffman-Taylor finger in a tapered Hele-Shaw cell. We consider a rectilinear geometry with a constant depth gradient and apply analytical techniques of singular…
We study the dynamics of "finger" formation in Laplacian growth without surface tension in a channel geometry (the Saffman-Taylor problem). Carefully determining the role of boundary geometry, we construct field equations of motion, these…
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,…
We consider Saffman-Taylor channel flow without surface tension on a high-pressure driven interface, but modify the usual infinite-fluid in infinite-channel configuration. Here we include the treatment of efflux by considering a finite…
We analyze the Saffman-Taylor viscous fingering problem in rectangular geometry. We investigate the onset of nonlinear effects and the basic symmetries of the mode coupling equations, highlighting the link between interface asymmetry and…
The three-layer Saffman-Taylor problem introduces two coupled moving interfaces separating the three fluids. A very recent weakly nonlinear analysis of this problem in a radial Hele-Shaw cell setup has shown that the morphologies of the…
We consider the Stefan problem with surface tension, also known as the Stefan-Gibbs-Thomson problem, in an ambient space of arbitrary dimension. Assuming the radial symmetry of the initial data we introduce a novel "probabilistic" notion of…
Being a major limiting factor for the efficiency of various technologies, such as Enhanced Oil Recovery, the viscous fingering (or Saffman--Taylor) instability has been extensively studied, especially for simple Newtonian fluids. Here, we…
Filtration combustion is described by Laplacian growth without surface tension. These equations have elegant analytical solutions that replace the complex integro-differential motion equations by simple differential equations of pole motion…
New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the…