Poles and branch cuts in free surface hydrodynamics

TL;DR

This paper investigates the complex dynamics of ideal incompressible fluids with free surfaces using conformal mapping, establishing conditions for the existence of poles and branch points, and proving the nonexistence of certain rational solutions.

Contribution

It provides exact results on pole and branch point solutions in free surface hydrodynamics and proves the nonexistence of specific rational solutions for moving poles.

Findings

Identified conditions for the existence of poles and branch points in fluid dynamics.

Proved the nonexistence of time-dependent rational solutions with moving poles.

Analyzed the complex velocity and conformal mappings in free surface flows.

Abstract

We consider the motion of ideal incompressible fluid with free surface. We analyzed the exact fluid dynamics though the time-dependent conformal mapping $z=x+iy=z(w,t)$ of the lower complex half plane of the conformal variable $w$ into the area occupied by fluid. We established the exact results on the existence vs. nonexistence of the pole and power law branch point solutions for $1/z_w$ and the complex velocity. We also proved the nonexistence of the time-dependent rational solution of that problem for the second and the first order moving pole.