Vortex motion around a circular cylinder above a plane

The study of vortex flows around solid obstacles is of considerable interest from both a theoretical and practical perspective. One geometry that has attracted renewed attention recently is that of vortex flows past a circular cylinder placed above a plane wall, where a stationary recirculating eddy can form in front of the cylinder, in contradistinction to the usual case (without the plane boundary) for which a vortex pair appears behind the cylinder. Here we analyze the problem of vortex flows past a cylinder near a wall through the lenses of the point-vortex model. By conformally mapping the fluid domain onto an annular region in an auxiliary complex plane, we compute the vortex Hamiltonian analytically in terms of certain special functions related to elliptic theta functions. A detailed analysis of the equilibria of the model is then presented. The location of the equilibrium in front of the cylinder is shown to be in qualitative agreement with the experimental findings. We also show that a topological transition occurs in phase space as the parameters of the systems are variedComment: 17 pages, 8 figure

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