Selection of the Taylor-Saffman Bubble does not Require Surface Tension

A new general class of exact solutions is presented for the time evolution of a bubble of arbitrary initial shape in a Hele-Shaw cell when surface tension effects are neglected. These solutions are obtained by conformal mapping the viscous flow domain to an annulus in an auxiliary complex-plane. It is then demonstrated that the only stable fixed point (attractor) of the non-singular bubble dynamics corresponds precisely to the selected pattern. This thus shows that, contrary to the established theory, bubble selection in a Hele-Shaw cell does not require surface tension. The solutions reported here significantly extend previous results for a simply-connected geometry (finger) to a doubly-connected one (bubble). We conjecture that the same selection rule without surface tension holds for Hele-Shaw flows of arbitrary connectivity. We also believe that this mechanism can be found in other, similarly described, selection problems.Comment: 4.5 pages, 3 figure

Whitham-Toda Hierarchy in the Laplacian Growth Problem

The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain. The Laplacian growth equation itself is the quasiclassical version of the string equation that selects the solution to the hierarchy.Comment: 7 pages, no figures, Talk given at the Workshop NEEDS 99 (Crete, Greece, June 1999

Planar elliptic growth

The planar elliptic extension of the Laplacian growth is, after a proper parametrization, given in a form of a solution to the equation for area-preserving diffeomorphisms. The infinite set of conservation laws associated with such elliptic growth is interpreted in terms of potential theory, and the relations between two major forms of the elliptic growth are analyzed. The constants of integration for closed form solutions are identified as the singularities of the Schwarz function, which are located both inside and outside the moving contour. Well-posedness of the recovery of the elliptic operator governing the process from the continuum of interfaces parametrized by time is addressed and two examples of exact solutions of elliptic growth are presented.Comment: 27 page

Velocity selection (without surface tension) in multi-connected Laplacian growth

We predict a novel selection phenomenon in nonlinear interface dynamics out of equilibrium. Using a recently developed formalism based on the Schottky-Klein prime functions, we extended the existing integrable theory from a single interface to multiple moving interfaces. After applying this extended theory to the two-dimensional Laplacian growth, we derive a new rich class of exact (non-singular) solutions for the unsteady dynamics of an arbitrary assembly of air bubbles within a layer of a viscous fluid in a Hele-Shaw cell. These solutions demonstrate that all bubbles reach an asymptotic velocity, $U$, which is {\it precisely twice} greater than the velocity, $V$, of the uniform background flow, i.e., $U=2V$. The result does not depend on the number of bubbles. It is worth to mention that contrary to common belief, the predicted velocity selection does not require surface tension.Comment: 5 pages, 1 figure. Updated versio