Steady free surface potential flow of an ideal fluid due to a singular sink on the flat bottom

TL;DR

This paper analyzes a steady free-surface flow of an ideal fluid caused by a singular sink on a flat bottom, proving existence and uniqueness of solutions and describing the free boundary's shape and behavior near the sink.

Contribution

It introduces a novel mathematical approach using Levi-Civita technique to prove solution existence and characterizes the free boundary's geometry in this flow problem.

Findings

Unique solution exists for Froude number above a threshold.

The free boundary has a cusp over the sink and is analytic elsewhere.

Inclination angle of the free boundary is less than π/2 except at the cusp.

Abstract

A two-dimensional steady problem of a potential free-surface flow of an ideal incompressible fluid caused by a singular sink is considered. The sink is placed at the horizontal bottom of the fluid layer. With the help of the Levi-Civita technique, the problem is rewritten as an operator equation in a Hilbert space. It is proven that there exists a unique solution of the problem provided that the Froude number is greater than some particular value. The free boundary corresponding to this solution is investigated. It has a cusp over the sink and decreases monotonically when going from infinity to the sink point. The free boundary is an analytic curve everywhere except at the cusp point. It is established that the inclination angle of the free boundary is less than $\pi/2$ everywhere except at the cusp point, where this angle is equal to $\pi/2$. The asymptotics of the free boundary near the cusp point is investigated.