PQ-Calculus of Fibonacci Divisors and Method of Images in Planar Hydrodynamics

TL;DR

This paper introduces a novel golden calculus based on Fibonacci divisors and quantum derivatives, applying it to model flow in annular domains using the method of images in planar hydrodynamics.

Contribution

It develops the golden calculus and hierarchical functions, linking them with hydrodynamic flow modeling in domains bounded by circles with Golden ratio proportions.

Findings

Derived explicit complex potential and velocity fields for point vortices with Golden proportion of images.

Established the connection between Golden periodic functions and flow in annular domains.

Formulated a new mathematical framework for hydrodynamics using Fibonacci-based calculus.

Abstract

By introducing the hierarchy of Fibonacci divisors and corresponding quantum derivatives, we develop the golden calculus, hierarchy of golden binomials and related exponential functions, translation operator and infinite hierarchy of Golden analytic functions. The hierarchy of Golden periodic functions, appearing in this calculus we relate with the method of images in planar hydrodynamics for incompressible and irrotational flow in bounded domain. We show that the even hierarchy of these functions determine the flow in the annular domain, bounded by concentric circles with the ratio of radiuses in powers of the Golden ratio. As an example, complex potential and velocity field for the set of point vortices with Golden proportion of images are calculated explicitly.