Uniqueness problem for Prandtl spirals

TL;DR

This paper investigates the uniqueness of divergence-free velocity fields with Prandtl spiral vorticity, establishing conditions under which the velocity is uniquely determined and extending the results to unions of spirals.

Contribution

It provides a new uniqueness result for velocity fields with Prandtl spiral vorticity and introduces an innovative conformal mapping approach for the problem.

Findings

Uniqueness of velocity fields with Prandtl spiral vorticity under certain conditions

Extension of uniqueness to unions of concentric logarithmic spirals

A novel conformal map technique simplifying the analysis

Abstract

In this paper, we study the problem of uniqueness of a divergence-free velocity field with vorticity given by the Prandtl spiral. We show that if the class of admissible velocities is restricted to those satisfying the velocity matching condition and an appropriate decay condition at the origin of the spiral, then the velocity field is uniquely determined. We subsequently extend the result to the case of fields with vorticity composed of unions of concentric logarithmic spirals. As a by-product, we derive an alternative way of deriving formula for the velocity corresponding to the Prandtl spirals. The proof relies on an approach that is of independent interest. We construct an explicit conformal map from the exterior of a logarithmic spiral onto a strip. This transformation reduces the problem to establishing the uniqueness of a holomorphic function defined on the strip, under non-standard boundary and decay conditions.