On dynamics of velocity vector potential in incompressible fluids

TL;DR

This paper introduces a quaternionic formulation for the advection of velocity vector potential in incompressible fluids, exploring topological invariants and providing a Nambu-Poisson framework for better understanding fluid dynamics.

Contribution

It presents a novel quaternionic approach and a Nambu-Poisson formalism for the Lagrangian advection equation, linking topology and physics of fluid flow.

Findings

Topological significance of stream-helicity analyzed

Quaternionic formulation of advection equation developed

Nambu-Poisson structure for fluid dynamics established

Abstract

An elegant quaternionic formulation is given for the Lagrangian advection equation for velocity vector potential in fluid dynamics. At first we study the topological significance of a restricted conserved quantity viz., stream-helicity and later more realistic configuration of open streamlines is figured out. Also, using Clebsch parameterisation of the velocity vector potential yet another physical significance for the stream-helicity is provided. Finally we give a Nambu-Poisson formalism of the Lagrangian advection equation for velocity vector potential.