Vortex Images and q-Elementary Functions

TL;DR

This paper introduces a novel method for analyzing vortex images in an annular domain using q-elementary functions, providing explicit solutions and linking to classical results through conformal mapping.

Contribution

It develops a new analytical approach employing q-logarithmic and Jackson q-exponential functions to solve vortex image problems in annular domains.

Findings

Vortex images are characterized as poles of the q-logarithmic function.

The complex potential is expressed via the Jackson q-exponential function.

A vortex must orbit the cylinders at a specific radius with constant angular velocity.

Abstract

In the present paper problem of vortex images in annular domain between two coaxial cylinders is solved by the q-elementary functions. We show that all images are determined completely as poles of the q-logarithmic function, where dimensionless parameter $q = r^2_2/r^2_1$ is given by square ratio of the cylinder radii. Resulting solution for the complex potential is represented in terms of the Jackson q-exponential function. By composing pairs of q-exponents to the first Jacobi theta function and conformal mapping to a rectangular domain we link our solution with result of Johnson and McDonald. We found that one vortex cannot remain at rest except at the geometric mean distance, but must orbit the cylinders with constant angular velocity related to q-harmonic series. Vortex images in two particular geometries in the $q \to \infty$ limit are studied.