Trace for the Loewner Equation with Singular Forcing

The Loewner equation describes the time development of an analytic map into the upper half of the complex plane in the presence of a "forcing", a defined singularity moving around the real axis. The applications of this equation use the trace, the locus of singularities in the upper half plane. This note discusses the structure of the trace for the case in which the forcing function, xi(t), is proportional to (-t)^beta with beta in the interval (0, 1/2). In this case, the trace is a simple curve, gamma(t), which touches the real axis twice. It is computed by using matched asymptotic analysis to compute the trajectory of the Loewner evolution in the neighborhood of the singularity, and then assuming a smooth mapping of these trajectories away from the singularity. Near the t=0 singularity, the trace has a shape given by [ Re(gamma(t)-gamma(0)) ]^(1-beta) ~ [ beta*Im(gamma(t)) ]^beta ~ O(xi(t))^(1-beta). A numerical calculation of the trace provides support for the asymptotic theory.Comment: 21 pages, 6 figures, submitted to Nonlinearit

Liquid interfaces in viscous straining flows: Numerical studies of the selective withdrawal transition

This paper presents a numerical analysis of the transition from selective withdrawal to viscous entrainment. In our model problem, an interface between two immiscible layers of equal viscosity is deformed by an axisymmetric withdrawal flow, which is driven by a point sink located some distance above the interface in the upper layer. We find that steady-state hump solutions, corresponding to selective withdrawal of liquid from the upper layer, cease to exist above a threshold withdrawal flux, and that this transition corresponds to a saddle-node bifurcation for the hump solutions. Numerical results on the shape evolution of the steady-state interface are compared against previous experimental measurements. We find good agreement where the data overlap. However, the numerical results' larger dynamic range allows us to show that the large increase in the curvature of the hump tip near transition is not consistent with an approach towards a power-law cusp shape, an interpretation previously suggested from inspection of the experimental measurements alone. Instead the large increase in the curvature at the hump tip reflects a logarithmic coupling between the overall height of the hump and the curvature at the tip of the hump.Comment: submitted to JF

Discrete charges on a two dimensional conductor

We investigate the electrostatic equilibria of N discrete charges of size 1/N on a two dimensional conductor (domain). We study the distribution of the charges on symmetric domains including the ellipse, the hypotrochoid and various regular polygons, with an emphasis on understanding the distributions of the charges, as the shape of the underlying conductor becomes singular. We find that there are two regimes of behavior, a symmetric regime for smooth conductors, and a symmetry broken regime for ``singular'' domains. For smooth conductors, the locations of the charges can be determined up to a certain order by an integral equation due to Pommerenke (1969). We present a derivation of a related (but different) integral equation, which has the same solutions. We also solve the equation to obtain (asymptotic) solutions which show universal behavior in the distribution of the charges in conductors with somewhat smooth cusps. Conductors with sharp cusps and singularities show qualitatively different behavior, where the symmetry of the problem is broken, and the distribution of the discrete charges does not respect the symmetry of the underlying domain. We investigate the symmetry breaking both theoretically, and numerically, and find good agreement between our theory and the numerics. We also find that the universality in the distribution of the charges near the cusps persists in the symmetry broken regime, although this distribution is very different from the one given by the integral equation.Comment: 46 pages, 46 figures, submitted to J. Stat. Phy